Class 10th Mathematics Gujarat Board Solution
Exercise 6.1- According to the definition of similarity of triangles, which are the conditions…
- For ΔPQR and ΔXYZ, the correspondence PQR ↔ YZX is a similarity. m∠P = 2m∠Q and…
- The correspondence ABC ↔ PQR between Δ ABC and Δ PQR is a similarity. AB : PQ =…
- Δ PQR ~ ΔDEF for the correspondence PQR ↔ EDF. If PQ + QR = 15, DE + DF = 10 and…
- In ΔABC and ΔPQR, ABC ↔ QPR is a similarity. The perimeter of AABC is 15 and…
- In ΔXYZ ~ ΔDEF consider the correspondence XYZ ↔ EDF. If deltaxyz/deltadef = 3/4…
- Using the definition of similarity prove that all the isosceles right angled…
- For ΔABC and ΔXYZ, ABC ↔ XYZ is a similarity. If ab/4 = bc/6 = ac/3 , AC = 3 and…
- State whether the following statements are true or false. Give reasons for your…
- ΔABC ~ ΔPQR for the correspondence ABC ↔ QRP. If m∠A = 50, m∠C = 30, then m∠R…
- ΔLMN ~ ΔXYZ for the correspondence LMN ↔ ZYX. If m∠Z = 50, m∠X = 40, then m∠L…
- If the correspondence ABC ↔ EFD is a similarity in AABC and ADEF, then of the…
- ABC ~ PQR is a similarity in ΔABC and ΔPQR. If the perimeter of ΔABC is 12 and…
Exercise 6.2- In ΔABC, a line parallel to bar bc intersects bar ab and bar ac in D and E…
- In ΔABC, the bisector of ∠C intersects bar ab in F. If 2AF = 3FB and AC = 7.2…
- In ΔXYZ, the bisector of ∠Y intersects bar zx in P. (1) If XP : PZ = 4 : 5 and…
- In Δ ABC, the bisector of ∠A intersects bar bc in D. Prove that bd = bc x…
- □ ABCD is a trapezium such that vector ab || bar cd . m in bar ad and n in bar…
- In ΔABC, D and E are the mid-points of bar bc and bar ac respectively. bar ad…
- In ΔPQR, x in bar qr , such that Q-X-R. A line parallel to bar pr and passing…
- In ΔABC, x in bar bc and B-X-C. A line passing through X and parallel to vector…
- In ΔABC, the bisector of ∠A intersects bar bc in D and the bisector of ∠ADC…
- In ΔABC, D is the mid-point of bar bc and P is the mid-point of bar ad . vector…
Exercise 6.3- In figure 6.27, Prove that BR = CS.
- □ ABCD is a parallelogram. Prove that ABD and BDC are similar.
- In □ABCD, M and N are the mid-points of bar ad and bar bc . If vector ab ∥ bar…
- In □ ABCD, A—P—D, B—Q—C. If AB ∥ PQ and PQ || DC, prove that AP X QC = PD X BQ.…
- In Δ ABC, the bisector of ∠ A intersects BC in D. The bisector of ∠ADB…
- In Δ ABC, Δ PQR and ΔXYZ correspondences ABC ↔ PQR, PQR ↔ XYZ are similarity.…
- State giving reasons, whether the following statements are true or false: In all…
Exercise 6.4- ∠B is a right angle in ΔABC and bar bd is an altitude to hypotenuse. AB = 8, BC…
- In □m ABCD, T ∈ bar bc and vector dt intersects bar bd in M and vector dc in O.…
- In □m ABCD, M is the mid-point of bar bc . vector dm and vector ab intersect in…
- P and Q are the mid-points of vector ab and bar ac in ΔABC. If the area of ΔAPQ…
- □ABCD is a rhombus. vector ac integrate erdection bar bd = {0}. Prove that the…
- In □PQRS, bar pr integrate erdection bar qs = {T}, PS = QR, bar pq | bar rs .…
- In ΔABC, a line parallel to bar bc , passes through the mid-point of vector ab .…
- P, Q, R are the mid-points of the sides of ΔABC. X, Y, Z are the mid-points of…
- In Δ ABC, x in bar bc , y in bar c_a , z in bar ab such that bar xy | vector ab…
- Two triangles are similar. Prove that if sides in one pair of corresponding…
Exercise 6- In ΔABC, p in bar ab such that ap/pb = m/n , m , n are positive real numbers. A…
- D, E and F are the mid-points of bar bc , bar c_sa and vector ab respectively in…
- Can two similar triangles have same area? If yes, in which case they have the…
- The correspondence ABC ↔ DEF is similarity in ΔABC and ΔDEF. bar am is an…
- Explain with reasons, whether the following statements are true or false: (1)…
- bar ad and bar be are the altitudes of ΔABC. If AB = 12, AC = 9.9, AD = 8.1 and…
- D, E, F are respectively the mid-points of bar pq , bar qr , bar pr of ΔPQR.…
- In ΔABC, bar am and bar cn are altitudes. If AB = 12, BC = 15 and AM = 9.6,…
- Areas of two similar triangles are 25 and 16. The ratio of the perimeters of…
- Area of ΔABC = 36 and area of ΔPQR = 64. The correspondence ABC ↔ PQR is a…
- In ΔABC, A—M—B, A—N—C and bar mn || bar bc . If AM = 8.4, AN = 6.4, CN = 19.2,…
- The correspondence ABC ↔ PQR is a similarity in Δ ABC and ΔPQR. AB = 16, AC =…
- The correspondence ABC ↔ XYZ is a similarity in ΔABC and ΔXYZ. ABC = 72, BC =…
- □ ABCD is trapezium in which bar ad || bar bc . The diagonals intersect in P.…
- In ΔABC, m∠B = 90 and bar bd is an altitude. The correspondence BDA ↔ _____…
- The correspondence ABC ↔ ZXY is a similarity in ΔABC and ΔXYZ. If AB = 12, BC =…
- ABC ↔ DEF is a similarity in ΔABC and ΔADEF, m∠A = 40, m∠E + m∠F =A. 40 B. 80…
- In ΔABC, M ∈ bar ab , N ∈ bar ac such that bar mn || bar bc ,……….is not true.A.…
- In ΔABC, B—M—C and A—N—C, bar mn || bar ab . If NC : NA = 1 : 3 and CM = 4,…
- In ΔXYZ and ΔPQR, XYZ ↔ PQR is similarity. XY = 12, YZ = 8, ZX = 16, PR = 8.…
- The bisector of ∠P in ΔPQR intersects bar qr in D. If QD: RD = 4 : 7 and PR =…
- The lengths of the sides bar bc , bar ca , bar ab of ΔABC are in the ratio 3 :…
- The correspondence ABC ↔ YZX in ΔABC and ΔXYZ is similarity. m∠B + m∠C = 120.…
- Correspondence ABC ↔ DEF of bar ac ABC and ΔDEF is similarity. If AB + BC = 10…
- The lengths of the sides of ΔDEF are 4, 6, 8. ΔDEF ~ ΔPQR for correspondence…
- The bisector of ∠B intersects bar ac in D. If BA = 12 and BC = 16, AD = 9, then…
- In Δ ABC, bar pq || bar bc , P ∈ bar ab , Q ∈ bar ac . If 4AP = AB and AQ = 4,…
- In ΔABC, the correspondence ABC ↔ BAC is similarity …….. of the following is…
- In ΔABC, A—P—B, A—Q—C and bar pq || bar bc . If PQ = 5, AP = 4, AB = 12, then…
- In Δ PQR, P—M—Q and P—N—R. If PQ = 18, PM = 12, PR = 9 and NR = …, then bar mn…
- According to the definition of similarity of triangles, which are the conditions…
- For ΔPQR and ΔXYZ, the correspondence PQR ↔ YZX is a similarity. m∠P = 2m∠Q and…
- The correspondence ABC ↔ PQR between Δ ABC and Δ PQR is a similarity. AB : PQ =…
- Δ PQR ~ ΔDEF for the correspondence PQR ↔ EDF. If PQ + QR = 15, DE + DF = 10 and…
- In ΔABC and ΔPQR, ABC ↔ QPR is a similarity. The perimeter of AABC is 15 and…
- In ΔXYZ ~ ΔDEF consider the correspondence XYZ ↔ EDF. If deltaxyz/deltadef = 3/4…
- Using the definition of similarity prove that all the isosceles right angled…
- For ΔABC and ΔXYZ, ABC ↔ XYZ is a similarity. If ab/4 = bc/6 = ac/3 , AC = 3 and…
- State whether the following statements are true or false. Give reasons for your…
- ΔABC ~ ΔPQR for the correspondence ABC ↔ QRP. If m∠A = 50, m∠C = 30, then m∠R…
- ΔLMN ~ ΔXYZ for the correspondence LMN ↔ ZYX. If m∠Z = 50, m∠X = 40, then m∠L…
- If the correspondence ABC ↔ EFD is a similarity in AABC and ADEF, then of the…
- ABC ~ PQR is a similarity in ΔABC and ΔPQR. If the perimeter of ΔABC is 12 and…
- In ΔABC, a line parallel to bar bc intersects bar ab and bar ac in D and E…
- In ΔABC, the bisector of ∠C intersects bar ab in F. If 2AF = 3FB and AC = 7.2…
- In ΔXYZ, the bisector of ∠Y intersects bar zx in P. (1) If XP : PZ = 4 : 5 and…
- In Δ ABC, the bisector of ∠A intersects bar bc in D. Prove that bd = bc x…
- □ ABCD is a trapezium such that vector ab || bar cd . m in bar ad and n in bar…
- In ΔABC, D and E are the mid-points of bar bc and bar ac respectively. bar ad…
- In ΔPQR, x in bar qr , such that Q-X-R. A line parallel to bar pr and passing…
- In ΔABC, x in bar bc and B-X-C. A line passing through X and parallel to vector…
- In ΔABC, the bisector of ∠A intersects bar bc in D and the bisector of ∠ADC…
- In ΔABC, D is the mid-point of bar bc and P is the mid-point of bar ad . vector…
- In figure 6.27, Prove that BR = CS.
- □ ABCD is a parallelogram. Prove that ABD and BDC are similar.
- In □ABCD, M and N are the mid-points of bar ad and bar bc . If vector ab ∥ bar…
- In □ ABCD, A—P—D, B—Q—C. If AB ∥ PQ and PQ || DC, prove that AP X QC = PD X BQ.…
- In Δ ABC, the bisector of ∠ A intersects BC in D. The bisector of ∠ADB…
- In Δ ABC, Δ PQR and ΔXYZ correspondences ABC ↔ PQR, PQR ↔ XYZ are similarity.…
- State giving reasons, whether the following statements are true or false: In all…
- ∠B is a right angle in ΔABC and bar bd is an altitude to hypotenuse. AB = 8, BC…
- In □m ABCD, T ∈ bar bc and vector dt intersects bar bd in M and vector dc in O.…
- In □m ABCD, M is the mid-point of bar bc . vector dm and vector ab intersect in…
- P and Q are the mid-points of vector ab and bar ac in ΔABC. If the area of ΔAPQ…
- □ABCD is a rhombus. vector ac integrate erdection bar bd = {0}. Prove that the…
- In □PQRS, bar pr integrate erdection bar qs = {T}, PS = QR, bar pq | bar rs .…
- In ΔABC, a line parallel to bar bc , passes through the mid-point of vector ab .…
- P, Q, R are the mid-points of the sides of ΔABC. X, Y, Z are the mid-points of…
- In Δ ABC, x in bar bc , y in bar c_a , z in bar ab such that bar xy | vector ab…
- Two triangles are similar. Prove that if sides in one pair of corresponding…
- In ΔABC, p in bar ab such that ap/pb = m/n , m , n are positive real numbers. A…
- D, E and F are the mid-points of bar bc , bar c_sa and vector ab respectively in…
- Can two similar triangles have same area? If yes, in which case they have the…
- The correspondence ABC ↔ DEF is similarity in ΔABC and ΔDEF. bar am is an…
- Explain with reasons, whether the following statements are true or false: (1)…
- bar ad and bar be are the altitudes of ΔABC. If AB = 12, AC = 9.9, AD = 8.1 and…
- D, E, F are respectively the mid-points of bar pq , bar qr , bar pr of ΔPQR.…
- In ΔABC, bar am and bar cn are altitudes. If AB = 12, BC = 15 and AM = 9.6,…
- Areas of two similar triangles are 25 and 16. The ratio of the perimeters of…
- Area of ΔABC = 36 and area of ΔPQR = 64. The correspondence ABC ↔ PQR is a…
- In ΔABC, A—M—B, A—N—C and bar mn || bar bc . If AM = 8.4, AN = 6.4, CN = 19.2,…
- The correspondence ABC ↔ PQR is a similarity in Δ ABC and ΔPQR. AB = 16, AC =…
- The correspondence ABC ↔ XYZ is a similarity in ΔABC and ΔXYZ. ABC = 72, BC =…
- □ ABCD is trapezium in which bar ad || bar bc . The diagonals intersect in P.…
- In ΔABC, m∠B = 90 and bar bd is an altitude. The correspondence BDA ↔ _____…
- The correspondence ABC ↔ ZXY is a similarity in ΔABC and ΔXYZ. If AB = 12, BC =…
- ABC ↔ DEF is a similarity in ΔABC and ΔADEF, m∠A = 40, m∠E + m∠F =A. 40 B. 80…
- In ΔABC, M ∈ bar ab , N ∈ bar ac such that bar mn || bar bc ,……….is not true.A.…
- In ΔABC, B—M—C and A—N—C, bar mn || bar ab . If NC : NA = 1 : 3 and CM = 4,…
- In ΔXYZ and ΔPQR, XYZ ↔ PQR is similarity. XY = 12, YZ = 8, ZX = 16, PR = 8.…
- The bisector of ∠P in ΔPQR intersects bar qr in D. If QD: RD = 4 : 7 and PR =…
- The lengths of the sides bar bc , bar ca , bar ab of ΔABC are in the ratio 3 :…
- The correspondence ABC ↔ YZX in ΔABC and ΔXYZ is similarity. m∠B + m∠C = 120.…
- Correspondence ABC ↔ DEF of bar ac ABC and ΔDEF is similarity. If AB + BC = 10…
- The lengths of the sides of ΔDEF are 4, 6, 8. ΔDEF ~ ΔPQR for correspondence…
- The bisector of ∠B intersects bar ac in D. If BA = 12 and BC = 16, AD = 9, then…
- In Δ ABC, bar pq || bar bc , P ∈ bar ab , Q ∈ bar ac . If 4AP = AB and AQ = 4,…
- In ΔABC, the correspondence ABC ↔ BAC is similarity …….. of the following is…
- In ΔABC, A—P—B, A—Q—C and bar pq || bar bc . If PQ = 5, AP = 4, AB = 12, then…
- In Δ PQR, P—M—Q and P—N—R. If PQ = 18, PM = 12, PR = 9 and NR = …, then bar mn…
Exercise 6.1
Question 1.According to the definition of similarity of triangles, which are the conditions for correspondence DEF ↔ ZXY between ΔDEF and ΔXYZ to be a similarity?
Answer:Given, ∆DEF ↔ ∆ZXY,
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAF0AAABrCAIAAACaMBkkAAAAAXNSR0IArs4c6QAABr9JREFUeF7tnH9MVWUYx0NjShajLC0rFl1SwwaxiCgBg0guITq0qY3NplkJOdplCBQsJxvJRcZNpvgjS2Jz0E8GRVyKLgWYRjgmC8riQqOy1ExGGjoC+t5OIyb33nPPPe97zvuenfPHHbu8932f53Oe933O97znPD4TExPX6Mc0AjN0Jk4J6FycB4bOReciZcnQ40WPFz1epBDQ40UKLX190eOFbLxkZGT4TDukDMFlWx+p+shoNMbHx+fk5HDprsdGS1tfSkpK0LPmocBHCfHS3t4eExMjNb48PkNsNZQQL4DS1tbGlvnUrPGUS3BwcHV1dXR0NDVL2OrYo3mElASrKyoq2LKdqjVYL9wfCBODwSDWSmv/F59HlZWVdrv9qiuY/v5+qmdL9c49mkeurPz02A+PP3yP6j7QMEA8XtyMWrC7qfv7X2mYpXqfsrhcGf07y/zR70OXVHeDuAGyuMCa5GWLM1+tByDilqnboVwumWlLb/T3K6xoVtcN4qPL5TJz5oxiU1Kv/eybtV8TN07FDuVygek3zJlVlrvicG3n0a4fVfSE7NAEuMCgoNtvQtTkWRoHfvmDrH1q9UaGC6xfGn7XxtQIpKc/L11RyxmC4xLjAps2pT4YYpiHqBkbGydooipdkeQCB17JSLgwPLK3+pgqzhAclDCXWb7XWnJT6lt6G1q/I2il8l0R5gIH5s+9Hump6MBnvfYzyvtDakTyXGBZ6MLb8jbH4TqYX4lAhQvQrIwLSYxeiPTEqUSgxQVosjbE+M32LTpgIxXbSvZDkQskQml2cte3p9+qO6GkS0TGoshFkAjlL6889H4HdxKBLhdBIhRlGvPLm376bYjImVSmE+pc4EZsRFBacjjS01+XR5XxSv4oSnCBlc8+Gbko6JY8y8e8SASFuDgkQnrCmfMX979zXP7JVKAH5bhcN9u3/KVVtc09XEgE5bgIEmFXdnLxoRb2JYKiXIAm/N4Fpg0xJvOHjEsEpbkAzeqE++IiDZAILK/BKnABmm0bl0Ei7NjH7i6COlwgEcymJEiEmsaTCiQXL4ZQhwsMDfD3wx2svdVfftU96IXdtH+iGhc4Fhw4t3Dr8lxLI4MSQU0uQIMFeL0xDOmJNYmgMheg2bIuCtoSEoH21JDUv/pcYO6OrcshEbDWSDKdamMmuEAiWHJSIBGs7aeoeut550xwgbkL5vkXZyUVHbSdGjjnufX0WrLCBR5GLLkDEiFzZx0LEoEhLoJEiI24O69M/Z1ctrgATd4zj+ITE4reHPGkZ+a4OB40yko63j34blO3Jw5QasMcF/h5c8AcpKfyI0c7e36m5LZotyxygdG4GYz7ntmlDafPDov6QKMBo1zgKp6YhkRAelJFIrDLRZAId94akL/bSiMi3PfJNBdHYnrRCLW9/22ldxFY5yLsItRYT+JdBCWjhnUugkTA/n/hvua+wfOKoeGAiyARXnjqEazBQ8MjyqDhgwtYrE8KiwoNROZWZheBGy5Ak/9c/Nj4ePEbnysQMjxxgUTAdmVrZ/8Hzd/QRsMTF0EiID1ZqtpoSwTOuAgSARMK9yKoSgT+uACNMXpRasISUwnFXQQuuQAN0jaejti+5xNKCw2vXICj2PQEXut5/b0OGmg45uLYRchNqao/0do5QBwNx1zAAmobEiG/3EpcIvDNBWgeCg3csjYKO7lkJQL3XIAmbUU4HsPC/j9BiaAFLkCzPT1h5PLorsNfkFpoNMIFEgEvPbV02ElJBI1wESQC0hMkAh7Dkh812uECFiGG+XgfbFtpA56OkIlGU1zAIjl2sUMiyH7QSGtcHLsIa6NQMgIvWMoJGQ1yEW7ToGSEHImgQS4IE0gEpKcjDV1eSwRtcgGaf98HS4RE8K5khGa5AA1KRmxeE4n3wbwoGaFlLkDz9KoHIBG82EVwxwW12YRCocJRU1ODb7ir0Jb/fDwkQlnVfxUs4QIcmXQK7uAblP68Onm5KciHsnVojU+hzdS/hW/CVlu4qOd37sLFxzYdrLP1wFrBqUmzExMTzWbzdC9E6tchXmw2m9VqdVp29/41r8m5RlD4t6ipUbVzHW6bo6wlwgROIXAKCgr6+vqcWCJ6wlEEUjhEW3LUACBQ3xKfqBDr1Oz/I8qVV0Lgufo9RyymmipASU9Pd2W/eB1IVJjFgWBzHm8KTwZyw2G5xcl2VTlXJE9jfQEUTEV8Ts1N5MxjtSfRfDTZQGOzyb07LuMFKzbWWmEeCgf+Rilr7q5fvAtI8fXFu355/5XGdYDXp0fn4hydzkXnImVS6fGix4seL1IIOG/7D67qNnDSoruAAAAAAElFTkSuQmCC)
There are mainly two conditions for correspondence DEF
ZXY between
DEF and
XYZ to be a similarity.
⟹The corresponding angles are congruent.
i.e. m∠D = m∠X ∠D ≅ ∠X
m∠E = m∠Y ∠E ≅ ∠Y
m∠F = m∠Z ∠F ≅ ∠Z
⟹The lengths of the corresponding sides are in proportion.
i.e. ![](data:image/png;base64,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)
Question 2.For ΔPQR and ΔXYZ, the correspondence PQR ↔ YZX is a similarity. m∠P = 2m∠Q and m∠X = 120. Find m∠Y.
Answer:Given, ΔPQR and ΔXYZ
P = 2
Q and
X = 120
If the correspondence PQR
YZX is a similarity then
m∠P = m∠X ... (i)
m∠Q = m∠Y ... (ii)
m∠R = m∠Z ... (iii)
From eq. (i) and (ii)
∠P = 120
Because m
P = 2m
Q
Hence, 2
Q = 120
Q = 60
And ∠Q = ∠Y
Hence, ∠Y = 60.
Question 3.The correspondence ABC ↔ PQR between Δ ABC and Δ PQR is a similarity. AB : PQ = 4 : 5. If AC = 6, then find PR. If QR = 15, then find BC.
Answer:Given, ΔABC and ΔPQR
And AB : PQ = 4 : 5
AC = 6 and QR = 15
The correspondence ABC
PQR between
ABC and
PQR is a similarity.
Hence, ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEIAAAAjCAMAAAAXIiwlAAAAAXNSR0IArs4c6QAAAHVQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjo6OmaQOma2OpC2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDoAkDpmkNv/tmYAtmY6ttu2ttv/tv//25A625Bm27Zm27aQ29uQ2////7Zm/9uQ/9u2//+2///bJ4Te6gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABdklEQVRIS+2V0VqDMAyFW6aIm6BOhDkZEwq8/yN6krQwL2zw88JdmAvbSf/kNLQHY64lxuIOUlxmEbfHFaoEMGZ4sjbZ06y/37zTgL+DfxjP4wGXPXRTnWPt9PpWvPgUpk47VUYAlrX9ztQkjFS4R0qmhAfGggRwHCqmTU+9yHURAVhSSBtRnHtRSqNiEYAlRQM5E5EspdF7EQCGDH4N3MpDIrtZoWIBGrsHnpvSEm1t+sEb2mq9mGagMycgN3wu/kPvQE395UDHIzEvCxMt9Y8BLaHyfO1GflnmenC+uskzBDk44Va3zgDIuKM7RRe0hV+4DPeuZBeNRwBoPGfkXDSbylzMz2W68QUgjHMK8aALP/xWymWKls8t/eeUVAKLE+kbAcBeK58dnlW+/ioVAlBt/9lhy6T65Jpre8G1ATbk25jJNezhhJJH3YgHsJmx2BzpU+pfZIsXza85HgHAiHqNTb8sx9a6QT9c0RI4oKqFayr/6Pkna60d2taFOukAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
PR = ![](data:image/png;base64,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)
PR = 7.5
And ![](data:image/png;base64,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)
![](data:image/png;base64,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)
BC = ![](data:image/png;base64,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)
PR = 12
Question 4.Δ PQR ~ ΔDEF for the correspondence PQR ↔ EDF. If PQ + QR = 15, DE + DF = 10 and PR = 6, find EF.
Answer:Given,
PQR and
DEF are similar for the correspondence PQR
EDF.
PQ + QR = 15 and DE + DF = 10
PR = 6
If the correspondence PQR
DEF is a similarity.
Hence, ![](data:image/png;base64,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)
And ![](data:image/png;base64,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)
From these above two equations
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAH8AAAArCAMAAABmfKfAAAAAAXNSR0IArs4c6QAAAG9QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjpmOmaQOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDoAkNv/tmYAtpA6tpBmttvbttv/tv//25A627Zm27aQ2////7Zm/7aQ/9uQ/9u2//+2///bW1UfjQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACV0lEQVRYR+2YW1vDIAyGS3XaqVO3Wk87uY3//xsl4RygZNOb6bjiYXn5QggstGn+eZO9gDb9gDisn4S4njMi4gwjfBz0phF0mN0q4e7qs2mW7byRr+KBzCP7BRkJDA3evtS99kohJHul3yzFotl1oCx7OleiHxp6vOqANw0h3d+qVQ9aGLpRS/RDQ48z9WH6EHK+yB72oFFhAIeCRvUjQ4Nz4p+sHyEcXXc3m8PsZgOquDlj+pGhwWnO5ILhlKwmQpiM7f1G6Wb0Bzwd0PwKiX7029gmOCWjqVNWRwU7p8Z/P5vgzlWaU0JNC/lRl1bkuJXzTxkivhVky7KexPoW8qMm8QcdBt8S/dAQ8fTMFvffx9xAQboNAu8fetuk909gqPGdyt9a+IPEDiBICsvKt06ISXKTpfreEHBwYAjys+CIVwogastaCEBsw2pYYgOVSpv9MwNiGzLmCk1WnVCXAaOxDRlzXUwuEfi9COT+tEqzu/+3H3dq/mcFatBZ/n5M/M9ygX/Iaf0Waafvak3mXSKSCoCsN8fUC+AsZcrdfY9Fv64LvrpaAYZ2EbOs6+vZCWXrT10MmVpsqOlru5A5PNYfYDnKaeKrw9eCUcSTAsjYjTGZFM1TZhSLeuyvklq2pO8Z1oFIlJAyo7JHfUhGtr5hsj6X10+oWB+K+Ug/e5cxfK7q25Xmo8La/7E9K+sTymWFOoBH51+ZGcu/iLJn7vjzpz8YFHwu6xPK3AotnHn7Lqmd/5ThHIAsZW7FO/j+ZO/f2gcYyjDeXm721inxKM7CLjYnR+AbI4RDqV/8PJYAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
EF ![](data:image/png;base64,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)
EF = 4
Question 5.In ΔABC and ΔPQR, ABC ↔ QPR is a similarity. The perimeter of AABC is 15 and perimeter of ΔPQR is 27. If BC = 7 and QR = 9, find PR and AC.
Answer:Given, In
ABC and
PQR, ABC
QPR is a similarity
The perimeter of
ABC is 15 and perimeter of
PQR is 27
And BC = 7 and QR = 9
AB + BC + CA = 15
AB + 7 + CA = 15
AB + CA = 8 ...... (i)
PQ + QR + RP = 27
PQ + 9 + RP = 27
PQ + RP = 18 ...... (ii)
Due to similarity ![](data:image/png;base64,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)
![](data:image/png;base64,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)
Hence, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
PR = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADUAAAAgCAMAAACFI8BqAAAAAXNSR0IArs4c6QAAAH5QTFRFAAAAAAAAAAA6AABmADpmADqQAGaQAGa2OgAAOgA6OjoAOmZmOma2OpC2OpDbZgAAZgA6ZjoAZjpmZmaQZpDbZrbbZrb/kDoAkDo6kDqQkLbbkNv/tmYAtmY6tv//25A625Bm27Zm27aQ29u22////7Zm/9uQ/9u2//+2///bQm2URwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABMUlEQVQ4T82UfVPCMAzG04EyfJsTsArDOvdC+/2/IElapFPMteedZ/7ZbcvTJ/mlLcBfRlcrdQfgtMJYJDrbxy20xQ7caw/uZZeoorRx6bMH9EwP88C5eVZtsBhSuyILE0RZVtTM8ExdZVjZioijKsWqvVGqWKcD40xTbABa5dldDPv0xt8P4UmMNffQXPU/qpyek8xW5xmeVFL/TuOatopTBkUV4kYSAmVdvZhU81EiDRIRwSjiVXBDT1sYSwRhCLhoNlVxzchEoMG/uzrOCE02M0/3YpxonBsLLjJ5tplANOq+l6d8WHmTMTz5hXbUfCvCkH+6PU/SIH95gPEy77e1V8lj+OZsfqPCCnNuG+9FvMqMKj9VTgsH7ktjrrmm7YU3aIYXEcdD7fYIHo/Pv4sj640VJEXMG4EAAAAASUVORK5CYII=)
PR = 14.4
Hence, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
AC = ![](data:image/png;base64,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)
AC = 5
Question 6.In ΔXYZ ~ ΔDEF consider the correspondence XYZ ↔ EDF.
If
find
and
.
Answer:Given,
XYZ and
DEF are similar for the correspondence XYZ
EDF
And ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMoAAAA3CAYAAABNadVvAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAfESURBVHja7V29bhs5EN4HSHuAA5iuzg9g0y6TKljvGamNyAtVMVIYAYEgdbB2Z+DgBzh36S5PcMWlvcZv4CL3BHmHOw1XXFHUkByuVuIKnuIrbC2l4c/MfDOr/VTc3t4WDAYjDF4EBoMdhcFgR2Ew2FEYDHYUxtpoVHV0LIrHoij+K/ar7x/u7l6419zdfXhRluqC14sd5VnippZvi0L8PK3VO3AYUYinSjVH+HXyx7RpXnodrpm+lEXxQzucgbz8ahyt2i++L73mQJQfP2PXyfrmbWtb8TM0HuaB2b7VQDOzQZbqamcdpdvE+cZtCzoSy/ILFqXHkEng8MEBjV17KYuv5tDG5uu71uyBkJf3St3sLa5ddcBLKe7hOrNuH0vx2f7b/jzjWDHbNrWGp1X9CewCW+pKNDFbOEIj0BtMOIg54DukPoeyMwTpeutafaCF+OZGfHOtfbBUKa/sqKzHVvX7kAPnWGOwa1pOL1DHDawTO4azYG10yUMHKA5MPWAQ3YGatZE7Pp/usMzrHf23lL9j47osM7dDO46ovlEysJnDtlkCZe6hdU2MZAteGism20N3+mnBUdvNcvksRCX435k8eDDv4/JmE7lg89Fx1jWNmhy2dq0eDp9Netykeu3yaXvhsLFvlHoVmwup5lBqz55bIY4fJ6o5DNYJgYMG63dcHP8NWSfFudpr5/NCMsnKeZjNUd2ovbMD+RDLcIuayd98yEpnZxRxsBrFFJFlKb9AtDL/czmzOawH8uwBNhxLbabQnDTqELivXnhn89uNW6YZ9jhI7aqZHJo6Bj4XujwYNaDY5EZK6ljKXEJrqtdvzuXdyN6HdtnUsQs6hMPZHWRCBjJOdXJy8gelzlhQwbj928wkqj59p5shtTofzFEwbukW3FgBjh1Kc5jquv4tpZA3kayuqvc6+5jitqqb6VS9sh26yxZEm9xx1LGxuaTUBSl2eGmEkw2oRX1K86RzKkr90zGE8VBarNs3CPXyRSa3C4NFPTczmPeScna4PdHFbIS9ueaAQmQ340xkM10M25mW/o7YhI2jjKXMJVgTIBHWzMk+VNh6eNcNdbzwoTYOJkTxRM1aSbVPpg5XGu31zzs5+rmTtTfQjkhggE5rsPAW57bHGPpG7e64TrnYhNXssUI9IjZhtIsyljKX1DYvNncK7fJ1qbB18o0zNU3oUIecfEwdrnS25Hf8xI7L8sK4fHrBcaEgFU8tp1dHlPeiUABfZgrRJ6pNGN2hjI3NJUZdVu5bIHSMSolC3SdfUe86F+U+jY8y7kKHK1Z/r+Uovj6zm35D9MDUFJRWXIh2xWoKlz5RbPLRrthY6PjE5pKyMb7IT6VdMAffNd3htuYITQq3BRxqJrgOELIn1uGC18dExbS9gTn3ogm++w12HWOoid4M4L52zx2hcGjrETpaM+7vG+ce7o4+QWE/v6lEssmiXfYGxsZS5kIpnE0A8aV/Cu2yW8IxGgQ0sev2IBHUvm71LMy7jIHoG+pwXV9f/9rWBHm6X8Y208GE/6m6Ot/f3/8nVG8l3eiyYX+Qa4j9HRr3uhhVsQ+MXaD7aNdyW3qxCG7DIWST2XwhTv5y24ShsX1pl++9zddEMIeKZS1sj2hAqCzyXSzq/Rz8PRBkomOr88DXPdlRKLf3GZvmzuO59/BcQaYIY2ztPQdAdh17x4gdZQB6wUiDXu9Z9oZGgfk2Lq/LiB0Fe5aAs8rm0RbZsO7LNRpj5BmFwWCwozB2rF6jdPI2UdPxBjAY7CgMxgYcpd8NKwZjeGzksK9hA0cLBtcoTL0YDHYUBoMdhcFgR2Ew2FEYmwRrDrOjMCIYUnM4qCssxJPvGQ2KHrH36dSEZ2O2HXxiD+CNM2qy5rB3M4fUHG6vbRVI7PdVanLkExHsrinlFdaK1RpohXjExoWe29dPzGZ45omqEsOR2tnI56Y57Hv8ePGkIv6ZIbkiVZYXqWPA7mrSvM6Rpc16saMQospz1ByOCUn4MhNFgCL1s3IxlxNx8ud02mbHtRwlt+bwunrDIZv02IDm8DpzIUWzzJrDMTrnkxvCxkFkDjklNkbv91w8JE+WFvewdxSVm6QiMpfmcF+9YapNuFDFMHOJpfycmsMxtUefwqQrVwRrpSNzwD73s4yISK6HAfX6z58eHcxRcmsO99EbTrGpr1bxLmsO0x1pVdU/NdP5u16Zfm3LUK6lQLymo+TWHO6rN0y1CRs71FzChyav5rB5LUTNMEfDKBR0wFIpXspvqmyi5rPnNIyjZNYc7qM37EbiFM3hIeeSWhdsU3M4prXr63phzgx1XigzoGNm7+/7Ra6Nt9kdwY5BHCW35nAfveGlaJqoOTzUXML1Xl7NYQrtwrJJijj3WLtdcaFAPHjQJplJc7iv3jDVJmzsUHOJNUZyag7HaJevY0YR566lbJYyB1HQOyfWzii5NYf76g2TbUI0h4eaS7Tmy6A5bNrYqFPOaCYEAtNmdyll6JeD21+uqs71WKcBkvItgZ11lNyaw+voDVNs8mkOrzsXCk/Oojksfvk3+GTgfB3cdVyio8TveKF2jey3GwdxFNYcHsPmsULnWBClCKwOmeuuMWsO74SjsOZwhm4Maw7vjqOw5nAesObwjmYUBoPBjsJgsKMwGOwoDEYG/A/yHBI95Z8WxAAAAABJRU5ErkJggg==)
By similarity law,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 7.Using the definition of similarity prove that all the isosceles right angled triangles are similar.
Answer:Prove that all the isosceles right angled triangles are similar
![](data:image/jpeg;base64,/9j/4AAQSkZJRgABAQEAYABgAAD/2wBDAAoHBwgHBgoICAgLCgoLDhgQDg0NDh0VFhEYIx8lJCIfIiEmKzcvJik0KSEiMEExNDk7Pj4+JS5ESUM8SDc9Pjv/2wBDAQoLCw4NDhwQEBw7KCIoOzs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozv/wAARCAB0AH4DASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD0vxD4jXQXtQbbz1kbdOQ+3yYQyq0nQ5wXXjjjPPGDtEgdTiuc1HwxdaxquoTXOqz2trcWy2iQ2wibfFgl9/mRtgksR8p6AZ9A+8tLtfCNnZanMstyJbSGaWFiN586MFgeCCevsTTSu7CeiOhorNtLua1uV07UH3SNn7PcYwJwOx9HA6jv1HcDSoasCYUUUUhhRRRQAUUUUAFFFFABRRRQAUUUUAFZuvf8g+L/AK/bX/0fHWlWbr3/ACD4v+v21/8AR8dVH4kJ7Fu7tIb63aCddyHkEHBUjoQexHY1UtLua2uF0/UG3Sn/AFFxjAnA7H0cdx36juBpVBd2kN9btBOuUPIIOCpHQg9iOxoT6MGieis2zvJre4XTtQYGYj9zPjAuAP5OO4/EcdNKk1YE7hRRRSGFFFFABRRRQAUUUUAFFFFABWbr3/IPi/6/bX/0fHWlWbr3/IPi/wCv21/9Hx1UfiQnsaVFFFSMgvLOG+t2gnUlSQQQcMpHRgexHrVSzvJoLkadqDAz4zDNjAuFH8mHcfiOOmlVe8s4b63MMwOMhlZThkYdGU9iPWqT6MTRYorOs7yaG4Gn6gR9owTDMBhbhR3How7j8Rx00aTVgTuFFFFIYUUUUAFFFFABRRRQA13WNGd2CqoyzE4AHrWHrWsaXLYxrHqVo7C7tmIWdScCdCT16AAn8K3qzde/5B8X/X7a/wDo+OrhbmVyZXsWra+s7zd9lu4J9vXypA2PyqxRRUu3QoKKKKQFe9s4b+3MEwOM5VlOGRh0ZT2IqtZXs0dwNP1AgXOCYpQMLcKO49GHdfxHFaNV72yhv7cwzAjkMjqcMjDoynsRVJ9GJosUVnWV7NHcf2fqBH2kDMcoGFuFHcejDuPx6Vo0mrAmFFFFIYUUUUAFFFFABWbr3/IPi/6/bX/0fHWlWbr3/IPi/wCv21/9Hx1UfiQnsaVFFFSMKKKKACiiigCve2UN/B5Uu4EHcjocNGw6Mp7EVWsr2ZJ/7P1DaLoDMcgGFuFH8Q9CO69vpWjVa+sor+38qXcpB3JIhw0bDoynsRVJ9GJos0Vn2N9KJ/sF/tW7UZR1GFnUfxL7+q9vpg1oUmrAncKKKKQwooooAKzde/5B8X/X7a/+j460qzde/wCQfF/1+2v/AKPjqo/EhPY0qKKKkYUUUUAFFMlljgheaVgkcalmY9AByTT6ACiiigCtfWMV/B5cm5WU7o5EOGjYdGU+v/6qgsb2UTmwv9q3aDKsows6/wB5ff1Hb6EGtCq19YxX8IjkLI6HdHKhw0bdmB/znoeKpPoxNdSzRWfY30vnGwvwqXaDKsows6/3l/qO30wToUmrAncKKKKQzhvEus6lD4k1CxttS1WBYNMjntodP09bnfKzSD5v3T4B2qOSo681tXuoR3XhSy1GWe32PLZyySxvmIfvoyxDf3evNS3vhw3OsS6rbaxf2E80CQSC3EJVlUsR/rI2IOXPQ1o6dYW+l6fBYWilYLdAiAsScD1J6mmnZ3E9UVv+Ej0L/oNaf/4FJ/jR/wAJHoX/AEGtP/8AApP8a0qKfuhqZv8Awkehf9BrT/8AwKT/ABo/4SPQv+g1p/8A4FJ/jWlRR7oamDrGv6NNol9FFq9jJI9tIqotyhLEqcADPJq5/wAJHoX/AEGtP/8AApP8afrv/IA1H/r1l/8AQDV+qfLyr+uxOtzN/wCEj0L/AKDWn/8AgUn+NH/CR6F/0GtP/wDApP8AGtKip90rUzf+Ej0L/oNaf/4FJ/jR/wAJHoX/AEGtP/8AApP8a0qKPdDUxL7VfD1/AI31uxR0bfFKl1HujYdGHP8A+scGn6Nr9tqMslkbu2mu4RljbyBklX+8uCce46j6YJ2KKd1a1hWdzmdCOr67YW2uNrU1stywlSyjgiMKxbuFJK7yxUckOOTwMDFdNWJb+Gls5/8ARNV1C3svO877DG8flA53EAlPMClsnaHxyR04rbqCgooooAKKKKACiiigBOtLRRQAUUUUAFFFFABRRRQAUUUUAf/Z)
![](data:image/jpeg;base64,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)
According to these figures,
In ∆ABC, AB = BC and m∠B = 90
In ∆XYZ, XY = YZ and m∠Y = 90 i.e. ∆ABC and ∆XYZ are isosceles right angled triangles.
To prove: ∆ABC and ∆XYZ are similar triangles.
Proof: ∆ABC and ∆XYZ are both isosceles right angled triangles in which m∠B = m∠Y = 90
Also AB = BC and XY = YZ, then
Question 8.For ΔABC and ΔXYZ, ABC ↔ XYZ is a similarity. If
, AC = 3 and XY = 5, find YZ and XZ.
Answer:Given, For ΔABC and ΔXYZ, ABC ↔ XYZ is a similarity
![](data:image/png;base64,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)
And AC = 3 and XY = 5
Hence, ![](data:image/png;base64,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)
AB = 4 and BC = 6
Due to similarity of both the triangles,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⟹ YZ =
= 7.5
And,
⟹ XZ =
= 3.75
Question 9.State whether the following statements are true or false. Give reasons for your answer:
(1) If ΔPQR and ΔABC are similar and none of them is equilateral, then all the six correspondences between ΔPQR and ΔABC are similarities.
(2) All congruent triangles are similar.
(3) All similar triangles are congruent.
(4) If the correspondences ABC ↔ BAC is similarity, then AABC is an isosceles triangle.
(5) The correspondence PQR ↔ YZX between ΔPQR and ΔYZX is a similarity. If m∠P = 60, m∠R = 40, then m∠Z = 80.
Answer:(1) False
For equilateral triangles, all the six correspondences are similarity. But in triangles other than equilateral, the measures of all the angles are not same.
(2) True
Congruent triangles are equal with respect to size and shape, while for triangles to be similar, it is sufficient that their shapes are same.
(3) False
Similar triangles are equal in shape but not in size. For triangles to be congruent, they must be equal in both, shape as well as size. Hence, all similar triangles are not congruent.
(4) True
For ∆ABC, the correspondence ABC↔BAC is a similarity.
∴ ∠A≅∠B
∴BC≅AC
Hence, two sides of ABC are congruent and therefore ∆ABC is an isosceles triangle.
(5) Given statement is true.
Question 10.ΔABC ~ ΔPQR for the correspondence ABC ↔ QRP. If m∠A = 50, m∠C = 30, then m∠R =
A. 80
B. 50
C. 30
D. 100
Answer:The triangle ABC and triangle PQR are similar in correspondence ABC↔QRP
∴ m
Q = m
A = 50
And m
P = m
C = 30
In ∆PQR,
m
P + m
Q + m
R = 180
30 + 50 + m
R = 180
m
R = 100
Question 11.ΔLMN ~ ΔXYZ for the correspondence LMN ↔ ZYX. If m∠Z = 50, m∠X = 40, then m∠L + m∠N =
A. 10
B. 90
C. 110
D. 80
Answer:In ∆LMN and ∆XYZ, the correspondence LMN
ZYX is a similarity.
∴ m
L = m
Z = 50
And m
N = m
X = 40
Hence, m
L + m
N = 50 + 40 = 90
Question 12.If the correspondence ABC ↔ EFD is a similarity in AABC and ADEF, then of the following is not true.
A. ![](data:image/png;base64,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)
B. ![](data:image/png;base64,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)
C. ![](data:image/png;base64,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)
D. ![](data:image/png;base64,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)
Answer:the correspondence ABC
EFD is a similarity in ∆ABC and ∆DEF
∴ ![](data:image/png;base64,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)
Question 13.ABC ~ PQR is a similarity in ΔABC and ΔPQR. If the perimeter of ΔABC is 12 and perimeter of a ΔPQR is 20, then AB : PQ =
A. 3 : 5
B. 5 : 3
C. 4 : 3
D. 3 : 4
Answer:Given, ABC
PQR is a similarity in
ABC and
PQR and the perimeter of
ABC is 12 and perimeter of a
PQR is 20
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Hence, AB : PQ = 3 : 5
According to the definition of similarity of triangles, which are the conditions for correspondence DEF ↔ ZXY between ΔDEF and ΔXYZ to be a similarity?
Answer:
Given, ∆DEF ↔ ∆ZXY,
There are mainly two conditions for correspondence DEF ZXY between
DEF and
XYZ to be a similarity.
⟹The corresponding angles are congruent.
i.e. m∠D = m∠X ∠D ≅ ∠X
m∠E = m∠Y ∠E ≅ ∠Y
m∠F = m∠Z ∠F ≅ ∠Z
⟹The lengths of the corresponding sides are in proportion.
i.e.
Question 2.
For ΔPQR and ΔXYZ, the correspondence PQR ↔ YZX is a similarity. m∠P = 2m∠Q and m∠X = 120. Find m∠Y.
Answer:
Given, ΔPQR and ΔXYZ
P = 2
Q and
X = 120
If the correspondence PQRYZX is a similarity then
m∠P = m∠X ... (i)
m∠Q = m∠Y ... (ii)
m∠R = m∠Z ... (iii)
From eq. (i) and (ii)
∠P = 120
Because mP = 2m
Q
Hence, 2Q = 120
Q = 60
And ∠Q = ∠Y
Hence, ∠Y = 60.
Question 3.
The correspondence ABC ↔ PQR between Δ ABC and Δ PQR is a similarity. AB : PQ = 4 : 5. If AC = 6, then find PR. If QR = 15, then find BC.
Answer:
Given, ΔABC and ΔPQR
And AB : PQ = 4 : 5
AC = 6 and QR = 15
The correspondence ABC PQR between
ABC and
PQR is a similarity.
Hence,
PR =
PR = 7.5
And
BC =
PR = 12
Question 4.
Δ PQR ~ ΔDEF for the correspondence PQR ↔ EDF. If PQ + QR = 15, DE + DF = 10 and PR = 6, find EF.
Answer:
Given, PQR and
DEF are similar for the correspondence PQR
EDF.
PQ + QR = 15 and DE + DF = 10
PR = 6
If the correspondence PQRDEF is a similarity.
Hence,
And
From these above two equations
EF
EF = 4
Question 5.
In ΔABC and ΔPQR, ABC ↔ QPR is a similarity. The perimeter of AABC is 15 and perimeter of ΔPQR is 27. If BC = 7 and QR = 9, find PR and AC.
Answer:
Given, In ABC and
PQR, ABC
QPR is a similarity
The perimeter of ABC is 15 and perimeter of
PQR is 27
And BC = 7 and QR = 9
AB + BC + CA = 15
AB + 7 + CA = 15
AB + CA = 8 ...... (i)
PQ + QR + RP = 27
PQ + 9 + RP = 27
PQ + RP = 18 ...... (ii)
Due to similarity
Hence,
PR =
PR = 14.4
Hence,
AC =
AC = 5
Question 6.
In ΔXYZ ~ ΔDEF consider the correspondence XYZ ↔ EDF.
If find
and
.
Answer:
Given, XYZ and
DEF are similar for the correspondence XYZ
EDF
And
By similarity law,
Question 7.
Using the definition of similarity prove that all the isosceles right angled triangles are similar.
Answer:
Prove that all the isosceles right angled triangles are similar
According to these figures,
In ∆ABC, AB = BC and m∠B = 90
In ∆XYZ, XY = YZ and m∠Y = 90 i.e. ∆ABC and ∆XYZ are isosceles right angled triangles.
To prove: ∆ABC and ∆XYZ are similar triangles.
Proof: ∆ABC and ∆XYZ are both isosceles right angled triangles in which m∠B = m∠Y = 90
Also AB = BC and XY = YZ, then
Question 8.
For ΔABC and ΔXYZ, ABC ↔ XYZ is a similarity. If , AC = 3 and XY = 5, find YZ and XZ.
Answer:
Given, For ΔABC and ΔXYZ, ABC ↔ XYZ is a similarity
And AC = 3 and XY = 5
Hence,
AB = 4 and BC = 6
Due to similarity of both the triangles,
⟹ YZ =
= 7.5
And,
⟹ XZ =
= 3.75
Question 9.
State whether the following statements are true or false. Give reasons for your answer:
(1) If ΔPQR and ΔABC are similar and none of them is equilateral, then all the six correspondences between ΔPQR and ΔABC are similarities.
(2) All congruent triangles are similar.
(3) All similar triangles are congruent.
(4) If the correspondences ABC ↔ BAC is similarity, then AABC is an isosceles triangle.
(5) The correspondence PQR ↔ YZX between ΔPQR and ΔYZX is a similarity. If m∠P = 60, m∠R = 40, then m∠Z = 80.
Answer:
(1) False
For equilateral triangles, all the six correspondences are similarity. But in triangles other than equilateral, the measures of all the angles are not same.
(2) True
Congruent triangles are equal with respect to size and shape, while for triangles to be similar, it is sufficient that their shapes are same.
(3) False
Similar triangles are equal in shape but not in size. For triangles to be congruent, they must be equal in both, shape as well as size. Hence, all similar triangles are not congruent.
(4) True
For ∆ABC, the correspondence ABC↔BAC is a similarity.
∴ ∠A≅∠B
∴BC≅AC
Hence, two sides of ABC are congruent and therefore ∆ABC is an isosceles triangle.
(5) Given statement is true.
Question 10.
ΔABC ~ ΔPQR for the correspondence ABC ↔ QRP. If m∠A = 50, m∠C = 30, then m∠R =
A. 80
B. 50
C. 30
D. 100
Answer:
The triangle ABC and triangle PQR are similar in correspondence ABC↔QRP
∴ mQ = m
A = 50
And mP = m
C = 30
In ∆PQR,
mP + m
Q + m
R = 180
30 + 50 + mR = 180
mR = 100
Question 11.
ΔLMN ~ ΔXYZ for the correspondence LMN ↔ ZYX. If m∠Z = 50, m∠X = 40, then m∠L + m∠N =
A. 10
B. 90
C. 110
D. 80
Answer:
In ∆LMN and ∆XYZ, the correspondence LMN ZYX is a similarity.
∴ mL = m
Z = 50
And mN = m
X = 40
Hence, mL + m
N = 50 + 40 = 90
Question 12.
If the correspondence ABC ↔ EFD is a similarity in AABC and ADEF, then of the following is not true.
A.
B.
C.
D.
Answer:
the correspondence ABC EFD is a similarity in ∆ABC and ∆DEF
∴
Question 13.
ABC ~ PQR is a similarity in ΔABC and ΔPQR. If the perimeter of ΔABC is 12 and perimeter of a ΔPQR is 20, then AB : PQ =
A. 3 : 5
B. 5 : 3
C. 4 : 3
D. 3 : 4
Answer:
Given, ABC PQR is a similarity in
ABC and
PQR and the perimeter of
ABC is 12 and perimeter of a
PQR is 20
Hence, AB : PQ = 3 : 5
Exercise 6.2
Question 1.In ΔABC, a line parallel to
intersects
and
in D and E respectively. Fill in the blanks shown in the table :
![](data:image/png;base64,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)
Answer:Given, In ΔABC, a line parallel to
intersects
and
in D and E respectively as shown in fig.
![](data:image/png;base64,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)
In
ABC, a line parallel to BC intersects AB and AC in D and E respectively.
∴
...... (i)
...... (ii)
…… (iii)
Also, AB = AD + DB and AC = AE + EC
1) AB = AD + DB
AB = 3.6 + 2.4
∴AB = 6
Now from eq (i)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
AE = ![](data:image/png;base64,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)
AE = 2.7
But, AC = AE + EC
AC = 2.7 + 1.8 = 4.5
AC = 4.5
2) AC = AE + EC
4.2 = AE + 3.15
AE = 1.05
From eq (ii)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
AD = ![](data:image/png;base64,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)
AD = 1.55
Then AB = AD + DB
∴ 6.2 = 1.55 + DB
DB = 4.65
3) AC = AE + EC
∴ AC = 6.4 + 8.0
AC = 14.4
∴
From eq (i)
![](data:image/png;base64,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)
AD = ![](data:image/png;base64,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)
AD = 9.6
AB = AD + DB
AB = 9.6 + 12
AB = 21.6
4) From eq (ii)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
AC = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADgAAAAgCAMAAABw3UvaAAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOgA6OjoAOmZmOmaQOma2OpDbZgAAZgA6ZjoAZjo6ZjpmZjqQZra2ZrbbZrb/kDoAkDo6kDqQkGaQkLbbkNv/tmYAtmY6tmZmtv//25A625Bm27Zm29uQ29u22////7Zm/9uQ/9u2//+2///bEI0WHQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABPklEQVRIS+1Ua1ODMBBMtIq2gPigqUppI8oj+f//z727gMyUGVBn+qmZYQLcbvb2bnJKnX9ZrfVVEXTbaHuagS9v8bPLtN7UEmWYHUH97l6+StrcM+M+1hmI3iSqSxOOCmxMtLkRonvc4unzsCC6FJE9SYPCMEo1lgya2AciWC/ghkVEb+K6I/IYFpy5p6M3+Zv4aCNJixV6jzHFBCY5I3+KQ1vrFRNPFF2WA0cnCAxGCy5Sc33kI3qPBfuUc/c3NTBCZBzBfIluQFiInxFv6kB1CVVlhURVEbeDcYANPi4vExXgpvxhXYp5rgp4Q+2Ry/6FSROu+by8f635ttEFeXhX1TDK5qk0HgZUe9fPngXEIMhI+zND5pmNOKRVLbYI8EjQ/oanRJCmBHltJqb7dNJBEESXUmcWE+dr8A/EN+xsGa2SfPyaAAAAAElFTkSuQmCC)
AC = 13.8
AB = AD + DB
18.4 = 7.2 + DB
DB = 18.4 – 7.2
DB = 11.2
Then, AC = AE + EC
∴ 13.8 = 5.4 + EC
EC = 8.4
5) From eq (iii)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
AB = ![](data:image/png;base64,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)
AB = 6.8
AB = AD + DB
6.8 = AD + 3.4
AD = 3.4
AC = AE + EC
5.1 = AE + 2.55
AE = 2.55
![](data:image/png;base64,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)
Question 2.In ΔABC, the bisector of ∠C intersects
in F. If 2AF = 3FB and AC = 7.2 find BC.
Answer:Given,
![](data:image/jpeg;base64,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)
Given, 2AF = 3FB
∴![](data:image/png;base64,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)
The bisector of ∠C intersects AB in F.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
BC = ![](data:image/png;base64,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)
BC = 4.8
Question 3.In ΔXYZ, the bisector of ∠Y intersects
in P.
(1) If XP : PZ = 4 : 5 and YZ = 6.5, find XY.
(2) if XY : YZ = 2 : 3 and XP = 3.8, find PZ and ZX.
Answer:Given, XP : PZ = 4 : 5
XY : YZ = 2 : 3
![](data:image/jpeg;base64,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)
In ∆XYZ, the bisector of angle Y intersects ZX in P.
(1) ∴ ![](data:image/png;base64,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)
![](data:image/png;base64,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)
XY = ![](data:image/png;base64,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)
XY = 5.2
(2) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEAAAAAgCAMAAACVQ462AAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OgBmOjoAOjo6OjpmOma2OpDbZgAAZgA6ZjoAZjpmZpCQZpDbZrbbZrb/kDoAkDo6kJDbkLaQkNv/tmYAtmY6tmZmttuQtv//25A625Bm27Zm2/+22////7Zm/9uQ/9u2//+2///bnjTsCgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABSUlEQVRIS+2VyVbDMAxFpbQljCmUIUADTRji1v//gTzbssOGSBwWsECb5DhXT/L0QvTrMTA31PFiVzNzdaP2I3x1ynxHIzP5dtGTO97SiOcrBpXI/H59REhuiA7rxt8jLwjEASUKX6HoRYBdfRbSrAKZp271tuljtSfUTgIvUNUj8WhdaHd+tXqHABZxackXnmgIaRDa9JhW6sASmc8Ccd0GvoWApXxa58ATdctQEkcAHXD1WOPF0EHmt1DhNIf/+HoFOhyMFLPHq1Dyoi3pd3lNb+67cQo/KfFHcp14YXzOb1jo2LfihdeFFy90Jz0sAj6nRPbCiRcr2z/gZlru45i8cOInL3S16smhveKFwhcv9K0+gc9emPnihYNlAkFBvDDz2QtjQ88WW0sChRcv9C2GD5cWgeiFhcf+p9+CfnFlg5MX2nnlXHwA+1ceXSCe/EUAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
PZ = ![](data:image/png;base64,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)
PZ = 5.7
ZX = PZ + XP
ZX = 5.7 + 3.8
ZX = 9.5
Question 4.In Δ ABC, the bisector of ∠A intersects
in D. Prove that
and ![](data:image/png;base64,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)
Answer:Given, In ΔABC, the bisector of ∠A intersects
in D
TO PROVE :
1) ![](data:image/png;base64,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)
2) ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAI4AAAA0CAYAAABcrAAbAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAUuSURBVHja7Zy/TttAHMevOzwAKJcNdnJh7YTCQemIhLEyARlQ5IVKGRgMG1LhBZhgoy9ApcLYDvAEVIInKM8Aze8cO/bFZ5+ds+OQ3/BVJRKn9+dzvz/38x05PT0lKFRW4SCgEBwUgoNCcFAIzkfrPCGfVMrz/EcaCwRHobOzzhyn5K4/SG99vQ/+FaK0+XP78HhZ9Wyv15l37NUdSunv4DnKHuCZ48O1Bms5+1Xo44nNtph9spX0nbx9mXmT67TYPoDjD7DrWEu8Ru4Joa/ccVfk73etVYsS8gqDazlO8Dk8xwh5Dv/WJNXrtRcbhD7G9cFEX2YenG6LHhHCXtquuxBeqcIKsd1r6bvf4O+U2eeds7O5uBUu/9aktMvItQp+E32ZaWiEuwLrIgHig0Nb3aNgFVrsaxxMYblue4HV1y/jJiLJ4qlWNViEPG4PFkOtVvujmngTfZlpcFyHr4CpDk8cBIaD1frPX61iEAl5SVrBwfeY7WaFd4PSG/l3vcllz1mtl+gT3bixOXXjwDHVF3RTMLi93iIAc3y4vbzO6pcAzart7KS5LpOWLwzPwE08W467lPV31uvsEmAR8Nf4vWwxTPUF3VQomwJgKLfdTq83PxovFBv0wgpv0uZNu832IavLCo3s9lTgmOoLuqnBAAJIIi2FLCM04EPAkk27EQsoshz6N8+kQupM2e4FWE5IseNiN5N9QTclxQD+ivQD47LAgYmvk/oTt/kepfxHlgBbgDLck/IVCe4RHJNuKsaUe0BFB71oVxVA4wfjfWvIGP8uu0x16k0vwiD4AbAMDroqQ25KHtjAxEsDOwgo3wRoismE3+SW+zlPWyB1licS/k9wPWnPi91hKWWPyxZN92XmoAE4/M2xtcPjhl+b6VqtLw1KHkXJgdnn8a4NtuMbj7Ad7z/ndjoLNmNu3jKDzei5avUnlQy8NvdjInBrIQDE31vsQMDf6h7E1Z1M9CX5w4QiYJaiXlWKgcFqi1U/o6LN2/DW+0jWYvPNJqW3Ul3rNk8GNK5sRq6GtSV+By5Xqr9FPjPdl+Q4wGvEu1wEJPX6E5jRTid5c2paioGoAmIcCLzkYNFxrBVGyYNv7tWpZfWLgajCwIkvlg23rkdT2mkqBqIKACcpbVWldkUVA1FTBI4qbZX3PHxIjBXQUoLyj/DG3YcGxwNDDYFcMDNVQItkDAlSAY2aIDiB9VC4qbDF8SewjGIgquLgxL3MlBTjlFkMzLpvhBpf2uCkvXoI77QOsirxspHRAtqYMY6Om0NlkxY4Om5qYG3ewm7JlKvCGGdKXVWSm/JqIV6tQ/68yGIgquLgDCvE0SKgKIBxvucVAqOvVsYEzMaLgagKgxNMuso10OYtt53NtI27KhUDURPYx0GhEBwUglOpAZyiMojJ8g1OviKrLOqw/qTaPc4lCwiOhoo+rB9nBcpqt0hcMl6ygOBoqujD+mHBRmuDNO5MvJeU1u5o5qx3yQKCo6kyDusXAU5auyPuSvOSBQRHU2Ud1jcNTlq7R76rcckCgqOpMg/rmwRHp92xbirlkgUER1NlHdaXU2AvoO2DM5jIrOmxTrtH3JTGJQsIjobKPKyvqPq/j/yN2Vcm2p3kplSXLCA4Wils+Yf1Tbgq3XanZVNhS5r1FZUZT73LP6xvApws7Y6Ar3nJAoKToEkd1h8XnKztDn+ue8kCgqMIUCd5WD8vOHnbnfeSBQQnKUCdwGH90b0h9ksHnDztHveSBQQHZd5y4yCgEBwUgoOqtv4DSfla528XH38AAAAASUVORK5CYII=)
![](data:image/png;base64,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)
Proof: In
ABC, the bisector of
A intersects
in D
∴ ![](data:image/png;base64,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)
By applying componendo
∴ ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAH8AAAAqCAMAAACtIHRlAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZpCQZpC2ZrbbZrb/kDoAkDo6kGYAkNv/tmYAtmY6tpA6ttvbttv/tv/btv//25A627aQ2////7Zm/9uQ/9u2/9vb//+2///bJnUr7QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACEElEQVRYR+1Y7U7DMAxsymCFAeNrFCiDUdpB8/4PSGInWbKlcWskNKTmx9RJPl98cb3csmxamSzPGpTh806IfGG+pJQxEFkKvfJLGhKGBkStyF+ArBZXjVybL5Zelo+HO7GQbjnPMrkW6pNafmhIVN0X1xq9LXQa+YSbSfJbiCw1pFtaAROb8EJDou52gwdQ71WOyWL1O4hNOqB+LzQkqudZffKhifTnwYrxW4jCKObv59MYcC/TLjQk0vm3hTrjHhUj/A6i+HX/nb1Thw862tCQaHvRYBkR/graG1rcbwoHQaCsogcXqx9CQ6Ia8ivph+vvIMg/tv8Cou5GV9YKdQDxMg719yDI37PzUAAv1CdqofXh5WyFfg3p98+DsOr3iTCB4ocDyFdqmFDzx4fgUKlg48TyQj2ipereRtFjB7wWQswewmG6r7+OtZANzt/ZiiK37W9Co0R0jiliUmBS4NgV6PnRim/b/cD99oFQpSf9sWvJ2d8o/TkEE+aPFBjvcno35tkffMzP38gqGC6nN6dnf+Dxq6QvLQyXkxDA3VTtTZC6NXNcDsmv72t450MXllgcl5PmR6fkLpBp18hyOUQDglMy/DvbHwexXE66fnRKA/lZLoc8f6X5MP15LofgB6fk+i/pGnguZ0T9adfKdDnE/AGnZOZPHvlfx8GZLoeav9opmfm7GPKnATmi/3nADziPQ6uSaoWVAAAAAElFTkSuQmCC)
By applying invertendo
![](data:image/png;base64,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)
BD
...... (i)
Similarly,
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAE0AAAAqCAMAAADBJiZgAAAAAXNSR0IArs4c6QAAAIFQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZpCQZpC2ZrbbZrb/kDoAkDo6kGYAkNv/tmYAtmY6tpA6ttvbttv/tv/btv//25A627aQ2////7Zm/9uQ/9u2/9vb//+2///bJnUr7QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABrElEQVRIS+1WXU/DMAxMCmyFAeVrC1AGo7SD5v//QBzbiRweWo8VaRLLQ5WtvosdX3ox5sCHd7MWUvTOhlFchh/DQ4TStDh/jYjOFs9h3ldz4FxbeI4NEYrTT2evGVPflzj1LvD0FWY6llwKJZR3lJHpbzexVApR5BYXDsUgm9mWBGvmpjl5j7l9PZ2FuSY3CmU2Lsm7FRCvkC1s6OxtjCoPZTZu5faiZX7839e8AYOcIjRna1AXodR9usDN629CLzoLpcbu4CZqeuodhKYuBF10qAfSzx4KMbhBqSOwClHWSYcD6YlQVm8R+lhBE1vggZ3b0Mk6XY6VmQ5hCOWTtdAoYZz4GPHPd6BGDeLHmr6GgyMFZ5PfoewYbOL3u1U68eIHTscGAVl+3IESFho7zQw+g0WvN429av1aI6zM4HNY9HoyWP+okKk0+ByWvL7RZEVdkwafw6LXowMphzD4HJa8XnWd4dWEJeew5PWTsCWvn6JS4fWqG8iPLoDBS5jw+g59eQeFEJGASa+H++ESbqoanUiDFzDh9XBBeSnB6x/UJ4sNXg1Tyu9Pwr4BMCsy2y5p5i0AAAAASUVORK5CYII=)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
DC =
PROVED.
Question 5.□ ABCD is a trapezium such that
||
.
and
such that
||
. Prove that
.
Answer:Given, ABCD is a trapezium such that
||
.
and
such that
|| ![](data:image/png;base64,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)
![](data:image/jpeg;base64,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⟹ First draw DB to intersect MN at P.
Proof : AB || CD and MN || AB
∴ MN || CD
Now, MN || AB and M-P-N
Hence, MP || AP
Similarly, as MN || CD and M-P-N
PN || CD
In ∆ABD, MP || AB
... (i)
In ∆BCD, PN || CD
... (ii)
From eq (i) and (ii)
PROVED.
Question 6.In ΔABC, D and E are the mid-points of
and
respectively.
and
intersect in G. A line m passing through D and parallel to
intersects
in K. Prove that AC = 4CK.
Answer:Given, In
ABC, D and E are the mid-points of
and
respectively.
and
intersect in G. A line m passing through D and parallel to
intersects
in K
![](data:image/jpeg;base64,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)
To prove : AC = 4CK
Proof : In ∆ABC, E is the midpoint of AC.
∴ CE =
AC
∴ AC = 2CE ...... (i)
In ∆ABC, D is the midpoint of BC.
∴CD =
CB
∴
-...... (ii)
In ∆CBE, C-D-B, C-K-E
∴![](data:image/png;base64,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)
From eq (ii)
∴![](data:image/png;base64,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)
Hence, CE = 2CK
Substituting this into eq (i)
AC = 2 (2CK)
AC = 4CK
Question 7.In ΔPQR,
, such that Q-X-R. A line parallel to
and passing through X intersects
in Y. A line parallel to px and passing through Y intersects
in Z. Prove that ![](data:image/png;base64,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)
Answer:Given, In
PQR,
, such that Q-X-R. A line parallel to
and passing through X intersects
in Y. A line parallel to px and passing through Y intersects
in Z
![](data:image/jpeg;base64,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)
To prove : ![](data:image/png;base64,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)
Proof : In
PQR, XY || PR
∴
...... (i)
In
PQX, YZ || PX
∴
...... (ii)
From eq (i) and (ii),
![](data:image/png;base64,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)
Question 8.In ΔABC,
and B-X-C. A line passing through X and parallel to
intersects
in Y. A line passing through X and parallel to
intersects
in Z. Prove that CY2 = AC • CZ.
Answer:Given, In
ABC,
and B-X-C. A line passing through X and parallel to
intersects
in Y. A line passing through X and parallel to
intersects
in Z as shown in figure.
![](data:image/png;base64,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)
To prove : CY2 = AC × CZ
Proof : In ∆ABC, XY || AB
∴
...... (i)
In ∆YBC, XZ || BY
∴
...... (ii)
From eq (i) and (ii)
![](data:image/png;base64,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)
∴ CY2 = AC × CZ
Question 9.In ΔABC, the bisector of ∠A intersects
in D and the bisector of ∠ADC intersects
in E. Prove that AB × AD × EC = AC × BD × AE.
Answer:Given, In
ABC, the bisector of ∠A intersects
in D and the bisector of ∠ADC intersects
in E as shown in figure.
![](data:image/png;base64,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)
To prove: AB × AD × EC = AC × BD × AE
Proof: In ∆ABC, the bisector of ∠A intersects BC in D.
∴
...... (i)
In ∆ADC, the bisector of ∠ADC intersects AC in E.
∴
...... (i)
By multiplying eq (i) and (ii)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALMAAAAqCAMAAADPo5pKAAAAAXNSR0IArs4c6QAAAJBQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjpmZmaQZpCQZpC2ZrbbZrb/kDoAkDo6kGYAkNv/tmYAtmY6tpA6ttvbttv/tv/btv//25A627Zm27aQ29u22////7Zm/7aQ/9uQ/9u2/9vb//+2///bbt74sAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAADMUlEQVRYR+1ZXWPTMAxMCqyFDQrlYw0QNkZgDS3J//93yJKcSrYSexvlpfHD0i053UWW5fpWFPP4bxnoq4sdkPVV6cbiyv2SGAqyeHmbel6HJ6ZcmBRXlluiasvFV3ft1ktQflPCz9RQkENVvkkBdHhkyoMppuL3ijXXH1fI2VdObbfGF6PRvf+B1wNf/d81pK/opaeHCE8fFSyPqahJc/fuzhcHaRZ57qsXTnS3vlJ6AkixX2XMjdfsJhM1K1gmE8tolkXzzCnDSH++oEo/sHDVa7g7GhLMzVi2RXjWrKY0h+knp6avtvDCLuW0NC6+K1IIdb9Z6mUpIRiFl2SqNobwrFnDUkwOzZr3r3Y8Vxipr4PahDcRBY6yAki25iG8qdnlLMHUsuYGO5wrDmMNUvbDSCFEr9vJ2qBHrdrIY6IV+NYt+db1Pb+asbhFQd9vtOgIArWV0exE+GENShjWRooJZbX4FLXmuNdRpcJdWdARpAjryUx11Os0LJNpyC2oguSS8FpuELxKgv6HVXWEHKoFd/rJRSjC854iYVlM1CzW0ChcIqGi72jvfn4tmA8fKMF7vqLWELK41K3GVs5fDVx43rsVLIsJNvyMzWu6fc135wzMGZgzcL4ZqLHP4/HpJN10CP/kD6k5ejKBD5AiOpv7p66Ns0nk/KKPy8DxLPlrAw3qctpIClygHAjKejiOv76SeRTweK+maMrXu/4m1VWVC5QHQdEPx9FZA82jkMe7QmSt9J+PO4Hp7kgXKIaMT/UkboLJmUchz+AKNVGGTXdHukAxJKWZ6CPcFBOEDJ/3rlBfqRM3F2HsI4mTvgUZFT2Ns3wkRDjzKOQZXCFlRXlmw90RjooJGROdwJlMbB6FPIMrZAuIPZkTaTbcH2Ry5lEobXCF7Ike1ezi/MPasHwkzk5UG8IVsqwVw92RLlCWG8PFksCNMwFe8whXqEU7RvY600dibgxjQFJrcARn+Uieaat5hsUMPQOM92v454ToeJa74/cGdIFiyHivk+5RhBtnQvNIPi9dIRD9bQVG0qfj3m24O4ELFEEm0oxHF3aPQtw4Ex138nke9/1lRoUZ+AvAqoYA81jYiAAAAABJRU5ErkJggg==)
∴ ![](data:image/png;base64,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)
∴ AB × AD × EC = AC × BD × AE
Question 10.In ΔABC, D is the mid-point of
and P is the mid-point of
.
intersects
in Q. Prove that (i) CQ = 2AQ (ii) BP = 3PQ
Answer:Given: In ΔABC,
BD = DC [∵ D is mid-point of BC]
AP = PD [∵ P is mid-point of AD]
Construction: Draw a line l from D parallel to AC and let it intersect BQ at S and AB at R.
Draw line m from D parallel to AC intersecting AC at T.
Draw a line from C parallel to BQ.
Extend AD and let it intersect the parallel drawn from C at point E.
![](data:image/jpeg;base64,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)
Proof:
In ΔADT,
PQ||DT [∵ by construction]
AP = PD [∵ Given]
Hence, by mid-point theorem,
AQ = QT …(1)
In ΔBQC,
BD = DC [∵ given: d is mid-point of BC]
DS|| QC [∵ by construction]
Hence, by mid-point theorem,
QT = TC …(2)
By eq. (1), (2) and (3) we get that,
AQ = QT =TC …(4)
⇒ AQ = 2QC [As QC = QT + TC]
In ΔPDB and ΔCDE,
∠CDE = ∠BDP [vertically opposite angles]
BD = DC [Given]
∠PBD = ∠ECD [alternate interior angles between BP||CE]
Therefore, ΔCDE ≅ ΔPDB.
CE = BP [By CPCT] …(5)
In ΔAPQ and ΔAEC,
∠A = ∠A [common angle]
∠APQ = ∠AEC [Corresponding angles when PQ||CE]
∠AQP = ∠ACE [Corresponding angles when PQ||CE]
By AAA similarity,
ΔAPQ ∼ ΔAEC
And,
![](data:image/png;base64,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)
[from (4)]
[From (5)]
PQ = 3BP
Hence proved.
In ΔABC, a line parallel to intersects
and
in D and E respectively. Fill in the blanks shown in the table :
Answer:
Given, In ΔABC, a line parallel to intersects
and
in D and E respectively as shown in fig.
In ABC, a line parallel to BC intersects AB and AC in D and E respectively.
∴ ...... (i)
...... (ii)
…… (iii)
Also, AB = AD + DB and AC = AE + EC
1) AB = AD + DB
AB = 3.6 + 2.4
∴AB = 6
Now from eq (i)
AE =
AE = 2.7
But, AC = AE + EC
AC = 2.7 + 1.8 = 4.5
AC = 4.5
2) AC = AE + EC
4.2 = AE + 3.15
AE = 1.05
From eq (ii)
AD =
AD = 1.55
Then AB = AD + DB
∴ 6.2 = 1.55 + DB
DB = 4.65
3) AC = AE + EC
∴ AC = 6.4 + 8.0
AC = 14.4
∴ From eq (i)
AD =
AD = 9.6
AB = AD + DB
AB = 9.6 + 12
AB = 21.6
4) From eq (ii)
AC =
AC = 13.8
AB = AD + DB
18.4 = 7.2 + DB
DB = 18.4 – 7.2
DB = 11.2
Then, AC = AE + EC
∴ 13.8 = 5.4 + EC
EC = 8.4
5) From eq (iii)
AB =
AB = 6.8
AB = AD + DB
6.8 = AD + 3.4
AD = 3.4
AC = AE + EC
5.1 = AE + 2.55
AE = 2.55
Question 2.
In ΔABC, the bisector of ∠C intersects in F. If 2AF = 3FB and AC = 7.2 find BC.
Answer:
Given,
Given, 2AF = 3FB
∴
The bisector of ∠C intersects AB in F.
BC =
BC = 4.8
Question 3.
In ΔXYZ, the bisector of ∠Y intersects in P.
(1) If XP : PZ = 4 : 5 and YZ = 6.5, find XY.
(2) if XY : YZ = 2 : 3 and XP = 3.8, find PZ and ZX.
Answer:
Given, XP : PZ = 4 : 5
XY : YZ = 2 : 3
In ∆XYZ, the bisector of angle Y intersects ZX in P.
(1) ∴
XY =
XY = 5.2
(2)
PZ =
PZ = 5.7
ZX = PZ + XP
ZX = 5.7 + 3.8
ZX = 9.5
Question 4.
In Δ ABC, the bisector of ∠A intersects in D. Prove that
and
Answer:
Given, In ΔABC, the bisector of ∠A intersects in D
TO PROVE :
1)
2)
Proof: In ABC, the bisector of
A intersects
in D
∴
By applying componendo
∴
By applying invertendo
BD ...... (i)
Similarly,
DC = PROVED.
Question 5.
□ ABCD is a trapezium such that ||
.
and
such that
||
. Prove that
.
Answer:
Given, ABCD is a trapezium such that ||
.
and
such that
||
⟹ First draw DB to intersect MN at P.
Proof : AB || CD and MN || AB
∴ MN || CD
Now, MN || AB and M-P-N
Hence, MP || AP
Similarly, as MN || CD and M-P-N
PN || CD
In ∆ABD, MP || AB
... (i)
In ∆BCD, PN || CD
... (ii)
From eq (i) and (ii)
PROVED.
Question 6.
In ΔABC, D and E are the mid-points of and
respectively.
and
intersect in G. A line m passing through D and parallel to
intersects
in K. Prove that AC = 4CK.
Answer:
Given, In ABC, D and E are the mid-points of
and
respectively.
and
intersect in G. A line m passing through D and parallel to
intersects
in K
To prove : AC = 4CK
Proof : In ∆ABC, E is the midpoint of AC.
∴ CE = AC
∴ AC = 2CE ...... (i)
In ∆ABC, D is the midpoint of BC.
∴CD = CB
∴ -...... (ii)
In ∆CBE, C-D-B, C-K-E
∴
From eq (ii)
∴
Hence, CE = 2CK
Substituting this into eq (i)
AC = 2 (2CK)
AC = 4CK
Question 7.
In ΔPQR, , such that Q-X-R. A line parallel to
and passing through X intersects
in Y. A line parallel to px and passing through Y intersects
in Z. Prove that
Answer:
Given, In PQR,
, such that Q-X-R. A line parallel to
and passing through X intersects
in Y. A line parallel to px and passing through Y intersects
in Z
To prove :
Proof : In PQR, XY || PR
∴ ...... (i)
In PQX, YZ || PX
∴ ...... (ii)
From eq (i) and (ii),
Question 8.
In ΔABC, and B-X-C. A line passing through X and parallel to
intersects
in Y. A line passing through X and parallel to
intersects
in Z. Prove that CY2 = AC • CZ.
Answer:
Given, In ABC,
and B-X-C. A line passing through X and parallel to
intersects
in Y. A line passing through X and parallel to
intersects
in Z as shown in figure.
To prove : CY2 = AC × CZ
Proof : In ∆ABC, XY || AB
∴ ...... (i)
In ∆YBC, XZ || BY
∴ ...... (ii)
From eq (i) and (ii)
∴ CY2 = AC × CZ
Question 9.
In ΔABC, the bisector of ∠A intersects in D and the bisector of ∠ADC intersects
in E. Prove that AB × AD × EC = AC × BD × AE.
Answer:
Given, In ABC, the bisector of ∠A intersects
in D and the bisector of ∠ADC intersects
in E as shown in figure.
To prove: AB × AD × EC = AC × BD × AE
Proof: In ∆ABC, the bisector of ∠A intersects BC in D.
∴ ...... (i)
In ∆ADC, the bisector of ∠ADC intersects AC in E.
∴ ...... (i)
By multiplying eq (i) and (ii)
∴
∴ AB × AD × EC = AC × BD × AE
Question 10.
In ΔABC, D is the mid-point of and P is the mid-point of
.
intersects
in Q. Prove that (i) CQ = 2AQ (ii) BP = 3PQ
Answer:
Given: In ΔABC,
BD = DC [∵ D is mid-point of BC]
AP = PD [∵ P is mid-point of AD]
Construction: Draw a line l from D parallel to AC and let it intersect BQ at S and AB at R.
Draw line m from D parallel to AC intersecting AC at T.
Draw a line from C parallel to BQ.
Extend AD and let it intersect the parallel drawn from C at point E.
Proof:
In ΔADT,
PQ||DT [∵ by construction]
AP = PD [∵ Given]
Hence, by mid-point theorem,
AQ = QT …(1)
In ΔBQC,
BD = DC [∵ given: d is mid-point of BC]
DS|| QC [∵ by construction]
Hence, by mid-point theorem,
QT = TC …(2)
By eq. (1), (2) and (3) we get that,
AQ = QT =TC …(4)
⇒ AQ = 2QC [As QC = QT + TC]
In ΔPDB and ΔCDE,
∠CDE = ∠BDP [vertically opposite angles]
BD = DC [Given]
∠PBD = ∠ECD [alternate interior angles between BP||CE]
Therefore, ΔCDE ≅ ΔPDB.
CE = BP [By CPCT] …(5)
In ΔAPQ and ΔAEC,
∠A = ∠A [common angle]
∠APQ = ∠AEC [Corresponding angles when PQ||CE]
∠AQP = ∠ACE [Corresponding angles when PQ||CE]
By AAA similarity,
ΔAPQ ∼ ΔAEC
And,
[from (4)]
[From (5)]
PQ = 3BP
Hence proved.
Exercise 6.3
Question 1.In figure 6.27, ![](data:image/png;base64,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)
Prove that BR = CS.
![](data:image/jpeg;base64,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)
Answer:In Δ ABC
Given, A-P- B, B- R-S-C and A-Q-C
and![](data:image/png;base64,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)
To prove – BR = CS
Proof : In Δ ABC given ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEUAAAAXCAMAAABqFXLnAAAAAXNSR0IArs4c6QAAAHtQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjpmOmaQOma2OpC2OpDbZgAAZjoAZjo6ZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkNv/tmYAtmY6tpA6tpBmttv/tv//25A627Zm27aQ2////7Zm/9uQ/9u2/9vb//+2///bgxMTnwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABPElEQVQ4T91UaVODMBDdxZbGC7UtKlaLpQf5/7/QvUKTAh0+qjudTghv3769APh3VuCI5c1lru0YNv9VVfGlZHT7xap2T4izVaKvLR7pp1eGze4lWQZnd5Z3Wyzoxt1sAepsBX6D5qOeMQs9EJYQ9A81PjR+k72HAHq3hoNjf1+GN30WXzK2LajyB8dH/5qw7ElBpe58PFuiJbAQQZ3EotimxZecFViUwNNnOb3Nt+Sl4GDCsnN5I0JFMNOOsHAncupEAJ9Z6AWXfRILV5cz77NY6KkZCcNgRiKsq+76enWFwcBJXeTBmlMldRvsERVQwV2no2JWKFMXSxmauooJaKwYbP3mqQ7r6z8d4tzmyLQObIDuiP8g8Oylt/k6LBdfhCSjeEKunve4aI7LaF6el902TiYB+HaoO/tH7Qcz2h9nAF4xIgAAAABJRU5ErkJggg==)
∴
………..eq(1)
(∵ If a line
to one side of a Δ intersects the other two sides of the Δ in distinct points, the segments of the other sides of the Δ in the same half plane of the line are proportional to the corresponding sides of the Δ)
Similarly, ![](data:image/png;base64,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)
∴
………………..2
From 1 and 2, we get
……..eq(3)
Similarly, ![](data:image/png;base64,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)
∴
……………4
From 3 and 4
![](data:image/png;base64,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)
⇒ BR = CS hence proved
Question 2.□ ABCD is a parallelogram. Prove that ABD and BDC are similar.
Answer:![](data:image/jpeg;base64,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)
Given ABCD is a ∥gram
To prove : - Δ ABD ∼ Δ BDC
Proof :- For the correspondence ABD ↔ CDB between ΔABD and Δ BDC
∠ BAD ≅ ∠ DCB (opposite ∠s of a ∥gram![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAXCAMAAADwWqV6AAAAAXNSR0IArs4c6QAAADZQTFRFAAAAADqQAGa2OgA6Ojo6OmaQOpDbZgAAZrb/kDoAkNv/tmYAttv/25A62////9uQ//+2///bTNiKjgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAP0lEQVQYV2NgIAB4WXkgKvhZOKAMZi4Ig5eRDyIDFRDkZINIcDNBaagCbnQaIi/ADlEP0w81D8qF2ge3H9WdAJMhAcAJLkqyAAAAAElFTkSuQmCC)
∠ABD ≅ ∠CDB (alternate ∠s)
∠ADB ≅ ∠CDB (alternate ∠s)
AB = CD (opposite sides of a ∥ gram ABCD )
∴ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAADMAAAAhCAMAAABDYWOIAAAAAXNSR0IArs4c6QAAAHtQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjo6OmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZpC2ZpDbZrbbZrb/kDoAkDpmkGY6kNv/tmYAtmY6ttu2ttv/tv//25A625Bm27aQ29uQ2////7Zm/9uQ/9u2//+2///bxHgZwQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABC0lEQVQ4T71UyxKCMAxMERVREBVR66OigP3/LzQJyKgHGtRxDwwM2Ww3aQLwP9ziMYqVgUIMjzLdYjI4YWSBz4rpbtjNIV43HNCjq5uB+aegKTvplHNiu7HfcTgU5CcSydTeMT37yUR+TARgKZTFjMRPxf73Xn1AmU6mKFyp0YXPGHb6yRc7d31eIqpEcSN7oAyPpi8H0/+eo6kgDCxRi0an/fd46XL457P1KPaHdXsuiETNbul2+SuKxSuhvJRHwpuKxgd3x+xqdcTXOg8kowDtBuDxKQPJbN9inDkCc2zWfHUafOdIZvsTHV4FgEuhhx8wKsX1WXOqRFY3OGNn/Lo//lLWH8kV+D7mDjgZE521qygXAAAAAElFTkSuQmCC)
Also AD = CB (opposite sides of a ∥ gram ABCD)
∴
= 1
And BD = DB (common)
∴
= 1
Hence, for the correspondence, ABD ↔ CDB between ΔABD and ΔBDC
The corresponding angles are congruent and the lengths of its corresponding sides are in proportion.
∴for the correspondence ABD ↔ CDB, ΔABD and ΔBDC are similar.
Question 3.In □ABCD, M and N are the mid-points of
and
. If
∥
prove that
∥![](data:image/png;base64,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)
Answer:![](data:image/jpeg;base64,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)
Given: In □ ABCD, M and N are the mid points of
and
and
∥ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABYAAAAXCAMAAAA4Nk+sAAAAAXNSR0IArs4c6QAAAGZQTFRFAAAAAAAAAAA6AABmADo6ADqQAGa2OgAAOjpmOmaQOpC2OpDbZgAAZjoAZpC2ZrbbZrb/kDoAkDo6kNv/tmYAtmY6tpA6ttvbttv/tv//25A627aQ2////7Zm/9uQ/9u2//+2///bDnvD6AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAiUlEQVQoU61Q2Q7CMAxzOnZBGcvGTXfk/3+StAXBBHuYND9EjeU6ToA1YOkP0uXOjz2RKZ2wtzPFJTrcaevkbFqMNgMGpp1n+1wbSNNCOLxYFSoO1SPSUSec3Kb0aFOnhlomauF5esYEx9+RPmEXcn4CvnSa84DvdUwd455yok31Xr68Lj/fGj+ex5EJywMw7jsAAAAASUVORK5CYII=)
To Prove:
∥ ![](data:image/png;base64,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)
Construction:- extend
towards D and
towards C to intersect each other at P ( ∵ AB > CD)
Proof:- M is the mid-point of ![](data:image/png;base64,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)
∴ DA = 2 DM …………1
Similarly, N is the mid-point of ![](data:image/png;base64,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)
∴ CB = 2 CN…………..eq(2)
In Δ PAB ,
∥ ![](data:image/png;base64,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)
∴ ![](data:image/png;base64,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)
( ∵ If a line ∥ to one side of a Δ intersects the other two sides of the Δ in distinct points, the segments of the other sides of the Δ in the same half plane of the line are proportional to the corresponding sides of the Δ)
∴
(from 1 and 2)
∴![](data:image/png;base64,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)
In Δ PMN, P-D-M , P-C-N and
![](data:image/png;base64,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)
And ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEMAAAAXCAMAAABnCwKgAAAAAXNSR0IArs4c6QAAAIdQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZpCQZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkNv/tmYAtmY6tpA6ttvbttv/tv/btv//25A627aQ2////7Zm/9uQ/9u2/9vb//+2///bdgIFXAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABNUlEQVQ4T91T7VLCMBBMgkIRNSIiRKkoxLaYvP/zeblrvrCBGX+JN51Opt1sdm8vjP2jknygxs2gQzMM/lPdsArFW4W2xG3qxMh7eKLcz0dAzBrCium7/9NysXFrIyfAtOXwDnXEofldY7cAR+xBcU9fP1W4tMrtNjJtac7RVQ5hXzY91io6nZn5zpshjrIO3W/x5zEiZUxPmB7tvY6v12u39pXpsApxUbMXbdUa6NahqeOPNPCMI7FJvsESGu9umuDO9bQOeqnPSS5FDo2JOpFne1ryYh5cZ1sOZogjAR7rYFFj76XDRFs0RKNxNtsW5yFmS6Q9n5FghojqMDc/+gFosYIpjDMmXBZwFUGIu5GjHc361aqYC5zwVgFi6Wd9lmWY7jvBUYKd/G4Wz9md+xXJ5Wz6Bk70It/n3ARSAAAAAElFTkSuQmCC)
∴
∥ ![](data:image/png;base64,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)
Question 4.In □ ABCD, A—P—D, B—Q—C. If AB ∥ PQ and PQ || DC, prove that
AP X QC = PD X BQ.
![](data:image/jpeg;base64,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)
Answer:Given: In □ ABCD, A—P—D, B—Q—C,
∥
and ![](data:image/png;base64,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)
To prove: AP×QC = PD×BQ
Proof:
∥
and ![](data:image/png;base64,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)
∴
∥
∥ ![](data:image/png;base64,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)
So,
,
,
are three ∥ lines
and
and
are their transversal
∴ ![](data:image/png;base64,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)
(∵ if three or more than three ∥ lines are intercepted by two transversal, the segments cut off on the transversal between the same parallel lines are proportional)
∴ AP × QC = PD × BQ hence proved
Question 5.In Δ ABC, the bisector of ∠ A intersects BC in D. The bisector of ∠ADB intersects AB in F and the bisector of ∠ ADC intersects in E. Prove that AF X AB X CE = AE × AC × BF.
Answer:![](data:image/jpeg;base64,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)
Given : In Δ ABC, the bisector of ∠ A intersects
in D. The bisector of ∠ADB intersects
in F and the bisector of ∠ ADC intersects
in E
To prove : AF × AB × CE = AE × AC × BF
Proof : in Δ ABC, the bisector of ∠A intersects
at D
∴
1
(∵ in a Δ the bisector of an angle divides the side opposite to the angle in the segments whose lengths are in the ratio of their corresponding sides)
Similarly, In Δ ADB, the bisector of ∠D intersects
at F
∴
………………2
And in Δ ADC , the bisector of ∠D intersects
at E
∴
…………..eq(3)
Multiplying 1, 2 and 3, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
And
(∵
)
∴ ![](data:image/png;base64,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)
∴ AF × BD × CE =AE × CD × BF
Question 6.In Δ ABC, Δ PQR and ΔXYZ correspondences ABC ↔ PQR, PQR ↔ XYZ are similarity. Prove that ABC ↔ XYZ is similarity.
Answer:Given:- In Δ ABC, Δ PQR and ΔXYZ correspondences ABC ↔ PQR, PQR ↔ XYZ are similarity
To Prove:- ABC ↔ XYZ is similarity
Proof:- In ΔABC and ΔPQR , the correspondence ABC ↔ PQR is a similarity
∴ ∠A ≅ ∠P, ∠ B ≅ ∠Q , ∠C ≅ ∠R ………..1
…………………..eq(2)
In Δ PQR and ΔXYZ , the correspondence PQR ↔ XYZ is a similarity
∴ ∠ P ≅∠X , ∠ Q ≅ ∠Y , ∠R≅ ∠Z………………3
……………………………………..4
From 1 and 3 , we get
∠A ≅ ∠x, ∠ B ≅ ∠Y , ∠C ≅ ∠Z
Multiplying 2 and 4, we get
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Thus for the correspondence ABC ↔ XYZ between ΔABC and ΔXYZ, the corresponding angles are congruent and the lengths of corresponding sides are in proportion.
∴ the correspondence ABC ↔ XYZ between ΔABC and ΔXYZ is a similarity.
Question 7.State giving reasons, whether the following statements are true or false: In all the following questions the line does not contain a side of the triangle.
(1) A line can be drawn in the plane of a triangle not intersecting any of the sides of a triangle.
(2) A line can be drawn in the plane of a triangle which is not passing through any of the three vertices and intersecting all the three sides of the triangle.
(3) If a line drawn in the plane of a triangle intersects the triangle at only one point, the line passes through a vertex of the triangle.
(4) If a line intersects two of the three sides of a triangle in two distinct points and does not intersect the third side, then the line is parallel to the third side.
(5) In the plane of Δ ABC, a line / can be drawn such that
= {P},
= {Q} and
= ϕ.
Answer:(1) The given statement is true because for a line in the plane of a Δ, there are 3 possibilities
i) The line does not intersect the Δ
ii) The line intersects the Δ at one point
iii) The line intersects the Δ at two points
The first possibility is satisfied; hence the given statement is true.
![](data:image/jpeg;base64,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(2) The given statement is false because according to the theorem, if a line lying in the plane of a Δ and not passing through any vertex intersects one side, then it does intersect one more side but does not intersect the third side. Thus, a line not passing through a vertex of a Δ cannot intersect all the three sides.
(3) The given statement is true because a line intersecting a Δ and not passing through any vertex will intersect the Δ at two points.
![](data:image/jpeg;base64,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)
(4) The given statement is false because if a line intersects two sides of a Δ and does not intersect the third side, it can intersect the line containing the third side.
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(5) The given statement is true because in the plane of ΔABC, if line l intersects
and
at distinct points P and Q respectively, then according to the theorem, if a line lying in the plane of a Δ and not passing through any vertex intersects one side, then it does intersect one more side but does not intersect the third side, it cannot intersect
.
In figure 6.27,
Prove that BR = CS.
Answer:
In Δ ABC
Given, A-P- B, B- R-S-C and A-Q-C
and
To prove – BR = CS
Proof : In Δ ABC given
∴ ………..eq(1)
(∵ If a line to one side of a Δ intersects the other two sides of the Δ in distinct points, the segments of the other sides of the Δ in the same half plane of the line are proportional to the corresponding sides of the Δ)
Similarly,
∴ ………………..2
From 1 and 2, we get
……..eq(3)
Similarly,
∴ ……………4
From 3 and 4
⇒ BR = CS hence proved
Question 2.
□ ABCD is a parallelogram. Prove that ABD and BDC are similar.
Answer:
Given ABCD is a ∥gram
To prove : - Δ ABD ∼ Δ BDC
Proof :- For the correspondence ABD ↔ CDB between ΔABD and Δ BDC
∠ BAD ≅ ∠ DCB (opposite ∠s of a ∥gram
∠ABD ≅ ∠CDB (alternate ∠s)
∠ADB ≅ ∠CDB (alternate ∠s)
AB = CD (opposite sides of a ∥ gram ABCD )
∴
Also AD = CB (opposite sides of a ∥ gram ABCD)
∴ = 1
And BD = DB (common)
∴ = 1
Hence, for the correspondence, ABD ↔ CDB between ΔABD and ΔBDC
The corresponding angles are congruent and the lengths of its corresponding sides are in proportion.
∴for the correspondence ABD ↔ CDB, ΔABD and ΔBDC are similar.
Question 3.
In □ABCD, M and N are the mid-points of and
. If
∥
prove that
∥
Answer:
Given: In □ ABCD, M and N are the mid points ofand
and
∥
To Prove: ∥
Construction:- extend towards D and
towards C to intersect each other at P ( ∵ AB > CD)
Proof:- M is the mid-point of
∴ DA = 2 DM …………1
Similarly, N is the mid-point of
∴ CB = 2 CN…………..eq(2)
In Δ PAB , ∥
∴
( ∵ If a line ∥ to one side of a Δ intersects the other two sides of the Δ in distinct points, the segments of the other sides of the Δ in the same half plane of the line are proportional to the corresponding sides of the Δ)
∴ (from 1 and 2)
∴
In Δ PMN, P-D-M , P-C-N and
And
∴ ∥
Question 4.
In □ ABCD, A—P—D, B—Q—C. If AB ∥ PQ and PQ || DC, prove that
AP X QC = PD X BQ.
Answer:
Given: In □ ABCD, A—P—D, B—Q—C, ∥
and
To prove: AP×QC = PD×BQ
Proof: ∥
and
∴ ∥
∥
So, ,
,
are three ∥ lines
and
and
are their transversal
∴
(∵ if three or more than three ∥ lines are intercepted by two transversal, the segments cut off on the transversal between the same parallel lines are proportional)
∴ AP × QC = PD × BQ hence proved
Question 5.
In Δ ABC, the bisector of ∠ A intersects BC in D. The bisector of ∠ADB intersects AB in F and the bisector of ∠ ADC intersects in E. Prove that AF X AB X CE = AE × AC × BF.
Answer:
Given : In Δ ABC, the bisector of ∠ A intersects in D. The bisector of ∠ADB intersects
in F and the bisector of ∠ ADC intersects
in E
To prove : AF × AB × CE = AE × AC × BF
Proof : in Δ ABC, the bisector of ∠A intersects at D
∴ 1
(∵ in a Δ the bisector of an angle divides the side opposite to the angle in the segments whose lengths are in the ratio of their corresponding sides)
Similarly, In Δ ADB, the bisector of ∠D intersects at F
∴ ………………2
And in Δ ADC , the bisector of ∠D intersects at E
∴ …………..eq(3)
Multiplying 1, 2 and 3, we get
And(∵
)
∴
∴ AF × BD × CE =AE × CD × BF
Question 6.
In Δ ABC, Δ PQR and ΔXYZ correspondences ABC ↔ PQR, PQR ↔ XYZ are similarity. Prove that ABC ↔ XYZ is similarity.
Answer:
Given:- In Δ ABC, Δ PQR and ΔXYZ correspondences ABC ↔ PQR, PQR ↔ XYZ are similarity
To Prove:- ABC ↔ XYZ is similarity
Proof:- In ΔABC and ΔPQR , the correspondence ABC ↔ PQR is a similarity
∴ ∠A ≅ ∠P, ∠ B ≅ ∠Q , ∠C ≅ ∠R ………..1
…………………..eq(2)
In Δ PQR and ΔXYZ , the correspondence PQR ↔ XYZ is a similarity
∴ ∠ P ≅∠X , ∠ Q ≅ ∠Y , ∠R≅ ∠Z………………3
……………………………………..4
From 1 and 3 , we get
∠A ≅ ∠x, ∠ B ≅ ∠Y , ∠C ≅ ∠Z
Multiplying 2 and 4, we get
Thus for the correspondence ABC ↔ XYZ between ΔABC and ΔXYZ, the corresponding angles are congruent and the lengths of corresponding sides are in proportion.
∴ the correspondence ABC ↔ XYZ between ΔABC and ΔXYZ is a similarity.
Question 7.
State giving reasons, whether the following statements are true or false: In all the following questions the line does not contain a side of the triangle.
(1) A line can be drawn in the plane of a triangle not intersecting any of the sides of a triangle.
(2) A line can be drawn in the plane of a triangle which is not passing through any of the three vertices and intersecting all the three sides of the triangle.
(3) If a line drawn in the plane of a triangle intersects the triangle at only one point, the line passes through a vertex of the triangle.
(4) If a line intersects two of the three sides of a triangle in two distinct points and does not intersect the third side, then the line is parallel to the third side.
(5) In the plane of Δ ABC, a line / can be drawn such that = {P},
= {Q} and
= ϕ.
Answer:
(1) The given statement is true because for a line in the plane of a Δ, there are 3 possibilities
i) The line does not intersect the Δ
ii) The line intersects the Δ at one point
iii) The line intersects the Δ at two points
The first possibility is satisfied; hence the given statement is true.
(2) The given statement is false because according to the theorem, if a line lying in the plane of a Δ and not passing through any vertex intersects one side, then it does intersect one more side but does not intersect the third side. Thus, a line not passing through a vertex of a Δ cannot intersect all the three sides.
(3) The given statement is true because a line intersecting a Δ and not passing through any vertex will intersect the Δ at two points.
(4) The given statement is false because if a line intersects two sides of a Δ and does not intersect the third side, it can intersect the line containing the third side.
(5) The given statement is true because in the plane of ΔABC, if line l intersects and
at distinct points P and Q respectively, then according to the theorem, if a line lying in the plane of a Δ and not passing through any vertex intersects one side, then it does intersect one more side but does not intersect the third side, it cannot intersect
.
Exercise 6.4
Question 1.∠B is a right angle in ΔABC and
is an altitude to hypotenuse. AB = 8, BC = 6. Find the area of ΔBDC.
Answer:![](data:image/png;base64,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)
In ∆ABC, ∠B is a right angle, AB = 8 and BC = 6
![](data:image/png;base64,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)
![](data:image/png;base64,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)
= 24 …(1)
In ∆ABC, ∠B is a right angle and BD is an altitude,
In ∆ABC,
∠A + ∠C = 90⁰ and,
In ∆BDC,
∠DBC + ∠C = 90⁰
So, ∠A = ∠DBC …(2)
In ∆ABC and ∆BDC,
∠DAB ≅ ∠DBC [from (2)]
∠ADB ≅ ∠BDC [by right angles]
The correspondence ADB ↔ BDC is a similarity by AA corollary.
Areas of similar triangles are proportional to the squares of their corresponding sides.
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFUAAAAqCAMAAAD1X+d8AAAAAXNSR0IArs4c6QAAAJZQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjpmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZpCQZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kLaQkLbbkNv/tmYAtmY6tpA6ttvbttv/tv/btv//25A627Zm27aQ29v/2////7Zm/9uQ/9u2/9vb//+2///bPF7u/QAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACLklEQVRIS+1Xa1PCMBBMyisIiOIDqiAFwYptkfz/P+fd5RKDKAnjyMgM+UIo181m75JbhDixodNmAZR1KmEk7Wc3l0kXfwiP1c2Ug/RMEZgQpUzo4abfEmKdyks713MJT4JjPZC1FxNVqu4bx2e3CoGALGLoFNcw802fF96HXHUWOaOWamgjN9dLKwExqxR8WNQYrkIwKm2WR97ipwbJEKT5+2ODdxZQgVFLObKBOh0BPfzKqJQ8k7rmIigqBTBqVpsMpKQMVxfM7AsqypCZNAaHQQUmvUKsKEk5kqIk7igQly3HlSs0a4KKV0iHFGHUChez9WBLZj9dy5UqJq8BHs0oe4xK2z6gsnxdUQjg6nYNpPgUJJg5s0xGJyI8cqM/1GQhXmHbfUh0ARig7NKc2A7m3ZSArLui3oOsxwpD7xEWSqDxFGZxjjgr8PcKZFTDVNSBS8cF/jgJsf3mxdArR/g9XoEjkPnnS/CVhybF9y7AegW3WtKJ8y4c3nDN01zPZFJ87wLtrFfoeagwnWZ58gSXvOtHXivxvQtZDaEf4nosEDI2yrYOz6S4bgiA3DNi82cNhTVQnknxOrdO45qrXdWw5NbtOhSZFM+7RNk2fyNZbYrZ2eJqTMpvUPUY+uDEubedbJFhOVQBQ5v8CQ3PpPjeJdZhbaXzM8W7XAmwJH8RXVkE7tlXz6T43gUc9xDkjz4FeLqU+xfhmRQ+seRdgOYMHEn9LvrEVqp9kqblAyHCPTz+gx2ZAAAAAElFTkSuQmCC)
…(3)
Now, ADB + BDC = ABC
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
BDC = 8.64
The area of ∆BDC is 8.64.
Question 2.In □m ABCD, T ∈
and
intersects
in M and
in O. Prove that AM2 = MT • MO.
Answer:In
ABCD, T ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAARCAYAAAA7bUf6AAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAACrSURBVDjLY2hsbGSgFDMMfkOAgAmImbFgRoKGgBSVZ4Wqm8jJbQWy/6Bj49gGb4KG1Eca+8rJmWyNLC42IMs75eWxUiayJstj6+slyQ6ThmhjH1m3jFIcYUFcmOS4yRYD1fzFFhZEhwnIEFm3nBKKorgh1tibUcZ9T1p5OS/ZhtTXx0oaMzDcNoys9cPmd6ITG8g1DAyyr0wjcyLJTmxgF+VFqlKU2IZ2LgYADTuC+Of3PLcAAAAASUVORK5CYII=)
and
intersects
in M and
in O.
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)
To prove:
AM2 = MT •MO
In ∆AMB and ∆OMD
∠AMD ≅ ∠OMD (Vertically opposite angles)
∠MBA ≅ ∠MDO (Alternate angles)
The correspondence AMD↔OMB is a similarity
…(1)
In ∆AMD and ∆TMB
∠AMD ≅ ∠TMB (Vertically opposite angles)
∠MAD ≅ ∠MTB (Alternate angles)
The correspondence AMD↔TMB is a similarity
…(2)
Multiplying (1) and (2),
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Hence, AM2 = MT •MO
Question 3.In □m ABCD, M is the mid-point of
.
and
intersect in N. Prove that DN = 2MN.
Answer:In
ABCD, M is the mid-point of
.
and
intersect in N.
![](data:image/png;base64,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)
To prove : DN = 2MN
Proof: M is the mid-point of BC
MB = MC
…(1)
In ∆MBN and ∆MCD
∠BMN ≅ ∠CMD (Vertically opposite angles)
∠MNB ≅ ∠MDC (Alternate angles)
The correspondence MBN↔MCD is a similarity
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFQAAAAqCAMAAAAanYxCAAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGZmAGa2OgAAOgBmOjpmOmaQOma2OpC2OpDbZgAAZgA6ZjoAZmY6ZpC2Zra2ZrbbZrb/kDoAkDo6kGYAkNu2kNvbkNv/tmYAtmY6tpA6ttvbttv/tv//25A627aQ2/+22////7Zm/9uQ/9u2/9vb//+2///bYn8x8gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB/ElEQVRIS+1W21LDIBAFerHRam9a8ZJYbbSJlv//PXdZ0gJJu/ShznTGfUlIOIcDLOwR4mJiO5Xy0aotpZwIaKpcCKObr4cnEiERAqFuNwCBf0P3nFBzZB800rFoIQFoVgSfLVCbqK7mlnS2QOVGJ5CGSKNJDUrczl6mwGb0Az6g+arhcxppgGxISWle9NaiGn450ryCtU0jDZCW9Od5sCbSCiZcTLYNqShUnkjqI2mnhh+4DUAKY9TX6z1pnY0SSX2kVWpAEJGKsrfAbKI1hY+lXCZtVIAMNyoXdQb8HqnRg3nK7gdIIjUaNoikFdD2SEXF534LGSht0tyRUrNwx+x48rvZNEi7ppA6eIIIjy/2mNpmnbHTj5DumPaX3EH8//+/AmdagcJWClst8LpgY9e9/cJgDyPZQc/X4dTpn0/JRTB3+gx1886Lj5AA+JxDzo07HQpctd8ar1ouIocCle1uY1au8AUOpak0CVkeeRuowVijnqiadvgMuPmxx/GIkKWnI3IopNRZIo7UdyhURl1EDsWRGjRUrFLf25Az25O2fAaW7yRSDxmT+g7ltOl7yHj6vkNxpHXG5xTaEM/bkIuiiByKI/V7HFraCIkOdJdSsUOxya9YL+FZeHIoRqsl2HPU2+kz1Ni6TCajQm8DrG+ZlP17Nms44j/8/wubj0fWfKf58QAAAABJRU5ErkJggg==)
(from 1)
MN = MD
Now, D-M-N
DN = DM + MN
DN = DM + MN
DN = MN + MN (MD = MN)
DN = 2MN
Hence proved.
Question 4.P and Q are the mid-points of
and
in ΔABC. If the area of ΔAPQ = 12√3, find the area of ΔABC.
Answer:![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJsAAACVCAIAAADACR+rAAAAAXNSR0IArs4c6QAADRxJREFUeNrtnXtQU2caxo/WbrFbtFtLOp2d7Uy7f3Qv43btCnjp2rUsKB210HqBWgiXcpVqW7tFpUkQEtDWS7UVEdtycdqKEGCBqlw1UCAJJKACFkMAyYUk3BJIwjXJDk0XKYEQycnJ+U6+5y84Muicx+f83vf9vpMPMUARSwi8BdBRKOgozlTE632Fxq27p4SOEkE6vYF8vvmvH9VSstuho0RQq1jtncy/yu/1pPPkqjHoKPBKYAqpV6bSGZTSnFUlhY6CreGRyc0JDUEpzRks6c5Tt3advgUdBVuVzf2b4uuZbHkuW55SKlr3CbdFrIaOAiz/s7eP5gqnv33r1C3KlXboKKgandAFfHmHK1RNX8njyENTW9Sjk9BRQkkPMwoFHbWbeiT3Pz1x8t3oD5KOfSrqEkJHwZZQcO/V7QFLNsYhXqlLXqH+w9O/rbUZOgouHvUhEe8t8Ti1KppHiqx9OpqHeJ3bExCm101AR4GUXNrt6vX28rd/cImodgm76RJe9XhA2V+2RIrvC6GjQGqgT+bm5f+YX6FLxI9TjkZUL997/aUtQXJpN3QUVB2Ooy7bRH0qikuK5q6Kql+2mXHgw48J1604kqPDKuWWHTuXrA5A1sctXU3e4LGtr1cBKyOQH7wDA/SEeOonR76+eCGZkXjkcKxEKoWOAqz8/Hwqldrf1zc1AhwdodPpWVlZej186oIpmUxGo9FycnKmLSwpKTl06JBQCGtdMFVQUEClUsVi8fSVwcHBhISES5cuQUeBDGh8fHxOTs6s68aYdnR0QEeBJKhIJDKtlRISErKysnQ6HXQUPILO+aeEjylC1IDOJOhMDQ4OJiYmZmVlQUfBUE9PD41Gy83NNfMz169fJ3DRixAvoBQKZb6AGtXf32+kKSF7U0I5aiSo+YASnqYIwQJqhqAOQlPiOCqVSi0M6MyYEo+mxHG0oKBgQYLO6k0JGVOCOGo5QQlPU4I4akmJa4amRCp6ieDodA+6CGOIR1OEGAG1sMSdM6Z0Oj0zMxM6iruALvo3lJSUxMbGEoamwDual5e36IDOKnqJsSADtqOLK3GJXfSC7ag1BDWlKTF6U4AdtZ6gpjQlQNGLgB5Q040KVtIU9N4UVEfRIijxaIoAHVCJRILi71QqlQToTYF0dHqZBfUnJAFGSAigAV3EFPehaAodBbXENVVpaSnQNEVADCjqBJ2TpoAWvQhwAaVSqbYgKGFoioAYUFsQdKaAHiGB5KgtelDi0RQBK6A2KnHnjGliYmJmZiZwCzLAOGrNRgVraApcTBGAAmrTEpcwNEUACiiTycT47zXSFKyiFwEooNgQdKaMIySwelMEBpRgNAXAUeNOIiwJakpTgBZk8O7ow77NYjuatre3Q0cBJii4NMW1o7ZeZiEkTXHtqH0JOidN8R9T7BxVqMZE/SODaks/sFgqlWKwzPJQNI2NjcU/TbFwdGJSf/Zad+iFlg+y2oLPN39f0wMKQWfFFIjtDVg4erL4ftRXdwfU4waDoVOh/Xdiw/WmPjz3oOaLXjMjpLEJXXffSLNouFmkbhWrJyb1BHRUqZnYmsxv6Hhwbk4+V77r9K2xCZ15glIoFDwQ1DSmZmh6ranP9Qgn7nI7Jbs94mLr/vSfeofGieYop13lncyXKR+cCymUaV7/9RXTgOKKoJbHlMmW+5xomv7Wk87LZcuI5uit+8Nbk/iSgdHpK/d6NB6JDWYcxU+JO8cjx+ye3jyuYuepB+clvnmiKZMlJZqjYxM6n8+aChumPk5cKNcKejTnSkV+Z26PjuvmK3FxSFDTmM5Z9BY29P6TVl/E6/2B33v66v1txxvbpBoCVkaFDQrvZH5ly8DXlRL3OM6zkSwmR26eoPgpcc2MkEy3NxTzetce4Zy51v3F9e4z17qxtxO7frReqHo/s+3Qd/eCzzf7nmwKTW2RDY6ZISjOOwRjb2o6QsrjKOx+Sq19Zkb8ziFjM2MaUBqNhk+CWkLT7FrZa4kNOr3jOWpmSIRngi44QmrsHDpXItJDR/Ff4pqhqSPOjBaWflwuFTHo9Ly8PAM4Kisro1AoXcJ7Bv04dPQXNTfxQvcdXPc62d3DN3Cvfy9QZyZptZqwELLbv7a5e5PJEe/zuLWO7ujtJt7qzf5LPc84+Rc/+salFWv8cq5cBsjR4qLCZ1z9Ht2R4eT/wxKvL1/YsLuBU+u4jup1k2FR+xGPz11iGl3Cq0j7uMv9i//26lttrXckEokY35JIJB2Cu+u8djntziPtq5/698fwEa/zfkGR+slJB3VUNdD72o6A5btyXSJqpk6TDLv5VHjdU+4hke8GJycnH8W3kpKS9kWEuqx+/Xdh1S7hrJ/PN61e/vbV9b4xAwqJgzo6Pqr18QtZui2DFM11CWORImueCL7xBzff/Jzvq6urb+JbVVVVxQW5f3Tf/nhAKSmy1iXsJima84jP5a07Q0a1asflaPb33650D3MOLFkVxV0RzELcDh6KowHE0UTGsSVrDzgH3VgVxXUmlzu7R6anf+Pote7+/TErXvR47OWQZ9e9czA2Tj2kAsjRUa2WSjv67Lq9j60JWflnr0Ay2dFrXZlMdiw56WLqOR67SnD3jgFMtbe18NhV311Kj6fRurq6HNpR45BIoZAbwJdaPUyn0zMyMuz+vqndHDUus4AyxbVEOHmRDbFvQEGZ4loipVLJYDAyMjIc0VGwllkeKqZ239OLwICiKDy8IWMHR4lHUFOa2vENGcReAZVKpYR0VKVSGYteR3GUqATFD00R7AMKxE4ia2Tf18IR7AMK1kYF4HpTBOOAErLEnbM3NdIU+6IXwT6ghDxrGT80xc5RJpNJeILOiimDwcCepgjGATU4ksrKyrCnKUaOWn+6GbhFL8YLMggMKAY0xTKmWDhqJChRh0SW0BTLERICA4oBTbEsehEMAuogPShOaGpbR4m9zIJPmiK2DqjDEnSmVCoVZjS1oaMSicTBCWpKU4FAALCjjjYkspCmth6C2spRWOLOGVMMRkiI7QJK4I0KVvamNo2pTRydJqiDLLPgiqaIjQIKCTpf0WvrXUjoOwoJaklMbTdCQmwRUAcfEi1Y9BppaqMREgIDSrCYIqgHFA6JLC968e6oRCKh0WgwoJb3praIKYJuQCFBH4qm6enpqNMUNUeNyywwoA9LU9RHSAiKAYUExQNN0XEULrNYE1N0R0gIWgGFU1xrelMUx6UoOGr8ZHkY0MWpvLwc3d4UgQG1O03RXTe11lGJREKhUOAyi/W9KVo0RawPKFxmQavoRSUVVjkKp7g4pKlVjubm5sIeFF2aWj9CQmBAcdWbWh/TxTsKh0ToyrinNz09HWVHh0Ymsutk2XWyVrEa9qB4o2nP4NjVxt5cjjyHLau407+wo5XNA+FprccLO1PLRb4nm67UyeYrvmCJawst+IZMWrl47WH2V5XitAqxz2dNFyvEOhOHHjjaNzy+7XjjlTrZ/93tJ6c0z3nsNiSovWh6oVwcntZq/LqI1+tBb1CPTM7rKLddtf1448xDJBWqsTlP8oUEtRdNv66U7Pi0sbFzqLFz6MOsthPFXeaeujda+rcweNrxBQ63gMss2NB0zhFSJku69jA7gSlMYAo9Gbz96T9pRnXzOlonUHrSedoZGRX0aNSjkzCg2NN0vhFSWoU4NLXF+LWof2QDhXune3heR/uHx31PNJXd/qWCYt9Tuh5mC+UaU4Lm5+fD+26Xojet/IGjmtFJTwavxaQl+VWtW8RTeDF4F8rF2XUyDzrvfKlo1lGaMKAY03RWTM+XitzjOPn1ciZXHvPN3WMFnROT+gX60aauoZRSUUqpqKZNaUpQuMyCcdE7i6Ydcm36TUlauTi1TPRtdY+1MyMYUCy16F1ICzuqHVaKOu7e4rHpiUeLiorgvcZMLBaLSvmEW8sSCe+qVQPoOFpRcnWzT7CLW8AzL/ts2uzRKWyHNxoz9SrkXlu3kv6+/Wm3gI3byMWF+dY6WlPNem693zKfb58Mq10RVPnI+o9jgTrXDHTFJyYv33jQmVzxZFjtMt/LpLV7blSULN5RvV733vsfIa+dIsXwps4ejPhx5bs1JNfdhXk5jXx+A5QtxefzrxXlP+fq4xzM+vnsRBYphod4ngsM36/XTS7S0fEx7Zv+oU4+WaSoup/PB73xdBRn5YaYd/x3M+h0KpQtxWDQyXv3rHxp56pItvFwVpeIGqddTO+9B0bVykU/dfUf/ucQsukYaeoM36n/Kc7BVb93fbOqsrRdIGiDsqUEAkFd9c3n3X1+G1hBimJP3f+YRmTz6Yh9BwwLbXIwx9Hbjfw/bfJbuiXlCXKl057/PrImPOn4CYg3zPT52S9/sybUaXfBE0GVS73Tnt+wp55TZ22t28Tj7CRHP7/Rz3Vr4NmzX4yPjcIbjZn0uomvLqa5eQe+8IrfG3sj62qq0OlHDboJjVIxplHBW2wXjWmHNEqFYXLcwp//H6NHEZKaxuq8AAAAAElFTkSuQmCC)
In ∆ABC, P and Q are mid-points of AB and AC respectively.
![](data:image/png;base64,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)
The correspondence ∆APQ↔∆ABC is a similarity by SSS theorem.
Now,
Areas of similar triangles are proportional to the squares of their corresponding sides.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
ABC = 12√3 × 4
ABC = 48√3
The area of ∆ABC is 48√3.
Question 5.□ABCD is a rhombus.
= {0}. Prove that the area of ΔOAB =
(area of □ ABCD).
Answer:Given:
ABCD is a rhombus.
= {0}.
![](data:image/png;base64,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)
To prove: Area of
OAB =
(area of □ ABCD)
Proof:
O is the mid-point of AC as well as BD.
Further, in rhombus ABCD,
AB ≅ BC ≅ CD ≅ DA
In ∆OAB and ∆OBC,
OA ≅ OC
OB ≅ OB
AB ≅ CB
∆OAB and ∆OBC are congruent by SSS theorem for congruence.
Thus, their areas are equal.
Similarly, ∆OAB, ∆OBC, ∆OCD and ∆ODA are all congruent triangles having equal areas.
∆OAB = ∆OBC = ∆OCD = ∆ODA
Now, ABCD = ∆OAB + ∆OBC + ∆OCD + ∆ODA
ABCD = ∆OAB + ∆OAB + ∆OAB + ∆OAB
ABCD = 4∆OAB
![](data:image/png;base64,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)
Thus,
![](data:image/png;base64,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)
Question 6.In □PQRS,
= {T}, PS = QR,
. Prove that ΔTPS is similar to ΔQTR.
Answer:![](data:image/png;base64,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)
Given: In
PQRS, PR ∩ QS = {T}, PS = QR and PQ||RS.
To prove: ∆TPS and ∆QTR are similar.
Construction: Through R, draw a line parallel to PS to intersect PQ at M.
Proof:
In PMRS, PM||RS (P-M-Q) and PS||MR (construction).
Thus, PMRS is a parallelogram.
PS = MR
Further, PS = QR (given)
In ∆RMQ, ∠RMQ ≅ ∠RQM …(i)
Further, the corresponding angles formed by transversal PQ to PS||MR are congruent.
∠RMQ ≅ ∠SPM (corresponding angles)
∠RQM ≅ ∠SPM (by (i))
∠RQP ≅ ∠SPQ (P-M-Q)
In ∆SPQ and ∆RQP,
∠SPQ ≅ ∠RQP
PQ ≅ QP (same line segment)
SP ≅ RQ (given)
The correspondence ∆SPQ↔∆RQP is a congruence by SAS theorem of congruence.
∠PSQ ≅ ∠QRP
∠PST ≅ ∠QRT (S-T-Q and R-T-P)
In ∆TPS and ∆QTR
∠STP ≅ ∠RTQ (Vertically opposite angles)
∠PST ≅ ∠QRT
∆TPS and ∆QTR are similar by AA corollary.
Thus, by AA corollary the correspondence ∆TSP↔∆TRQ is a similarity.
Question 7.In ΔABC, a line parallel to
, passes through the mid-point of
. Prove that the line bisects
.
Answer:![](data:image/png;base64,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)
Given: In ∆ABC, a line parallel to BC, passes through the mid-point of AB.
To prove: Line m bisects AC.
Proof: In the plane of ∆ABC, line m is parallel to BC and intersects AB at a point other than a vertex of the triangle.
Thus, m intersects AC.
Let m ∩ AC = {E}
In ∆ABC, D is the mid-point of AB.
AD = DB
![](data:image/png;base64,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)
In ∆ABC, A-D-B, A-E-C and DE||BC.
![](data:image/png;base64,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)
![](data:image/png;base64,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)
AE = EC and A-E-C.
E is the mid-point of AC.
Thus, line m bisects AC.
Question 8.P, Q, R are the mid-points of the sides of ΔABC. X, Y, Z are the mid-points of the sides of ΔPQR. If the area of ΔXYZ is 10, find the area of APQR and the area of ΔABC.
Answer:![](data:image/png;base64,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)
In ∆ABC, P, Q, R are the mid-points of the sides AB, BC and CA respectively.
![](data:image/png;base64,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)
The correspondence ∆ABC↔∆QRP is a similarity.
Areas of similar triangles are proportional to the squares of their corresponding sides.
![](data:image/png;base64,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)
∆ABC = ∆POR
Similarly, X,Y and Z are the mid-points of the sides of ∆PQR, we get
∆PQR = 4∆XYZ
∆PQR = 4×10
∆PQR = 40
Thus, the area of ∆PQR is 40 sq. units.
∆ABC = 4∆PQR
∆ABC = 4×40
∆ABC = 160
Hence, the area of ∆ABC is 160 sq. units.
Question 9.In Δ ABC,
such that
. Prove that X, Y, Z are the mid-points of
,
,
respectively.
Answer:![](data:image/png;base64,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)
Given: In ∆ABC, X∊BC, Y∊CA, Z∊AB.
Also, XY||AB, YZ||BC and ZX||AC.
To prove: X,Y and Z are the mid-points of BC, CA and AB respectively.
Proof: In ∆ABC, YZ||BC.
…(1)
In ∆ABC, XY||AB.
…(2)
From (1) and (2),
…(3)
In ∆ABC, ZX||AC.
…(4)
From (3) and (4),
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEkAAAAqCAMAAADIzYYaAAAAAXNSR0IArs4c6QAAAGZQTFRFAAAA///b//+22///tv///9vbtv/b/9uQttv/kNv/ttuQ/7ZmZrb/Zrbb25A6tpA6ZpCQOpDbtmYAkGYAOma2AGa2kDoAOjqQADqQADpmZgBmADo6ZgA6ZgAAOgAAAABmAAA6AAAAIf+aGwAAAAF0Uk5TAEDm2GYAAAFSSURBVHja7ZXrTsQgEIUHWq3rruhWwbradnn/lzTcOoBYpiZmjdnzj4RDhzNDP4A/KqaGBgD4qK3mtrrf7ju/NLlnr887e9LRLg7VT/NRArBHLXNP/zSZNT+1ZpOtr1aUBLc19fDTrVrswpZHPElmHiFBhGy6SQLxpJvntzb1MHWAbgoXrcaNkQ93mad7bXy5JrojUGtivbsUegS2kSlC3EnisYd/7JY+CsIE4ElMmc+jZ++nUoZm2MaSa0KPT4iPcwvu4qIeupvM3gSEnlHrwQ39fD9tey3vD2YCiJ6rrjLvXAdV/1u6pF/wXOZ2/1sI0YisP+J2BFEkK0FfuR1BNP7P15ta5raDaEbW9VDK3PYQTcm6rm+47SGakrWSd5HbAaIJWddV5vYC0Q2Jl7m9QDQmayXvIrdziFJqKnMbIRqRlQLznNsI0Yis9Qe2mdsX0SfZqzKV1USEbQAAAABJRU5ErkJggg==)
AZ2 = BZ2
AZ = BZ
Hence, A-Z-B and AZ = BZ.
Thus, Z is the mid-point of AB.
Similarly, Y is the mid-point of AC and X is the midpoint of BC.
Therefore, X, Y and Z are the mid-points of BC, CA and AB respectively.
Question 10.Two triangles are similar. Prove that if sides in one pair of corresponding sides are congruent, then the triangles are congruent.
Answer:Given: The correspondence ∆ABC↔∆PQR is a similarity and AB ≅ PQ.
To prove: ∆ABC and ∆PQR are congruent.
Proof:
The correspondence ∆ABC↔∆PQR is a similarity.
![](data:image/png;base64,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)
Further, AB ≅ PQ
AB = PQ
![](data:image/png;base64,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)
![](data:image/png;base64,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)
AB = PQ, BC = QR, AC = PR
AB ≅ PQ, BC ≅ QR, AC ≅ PR
The correspondence ∆ABC↔∆PQR is a congruence by SSS theorem of congruence.
Hence, ∆ABC and ∆PQR are congruent.
∠B is a right angle in ΔABC and is an altitude to hypotenuse. AB = 8, BC = 6. Find the area of ΔBDC.
Answer:
In ∆ABC, ∠B is a right angle, AB = 8 and BC = 6
= 24 …(1)
In ∆ABC, ∠B is a right angle and BD is an altitude,
In ∆ABC,
∠A + ∠C = 90⁰ and,
In ∆BDC,
∠DBC + ∠C = 90⁰
So, ∠A = ∠DBC …(2)
In ∆ABC and ∆BDC,
∠DAB ≅ ∠DBC [from (2)]
∠ADB ≅ ∠BDC [by right angles]
The correspondence ADB ↔ BDC is a similarity by AA corollary.
Areas of similar triangles are proportional to the squares of their corresponding sides.
…(3)
Now, ADB + BDC = ABC
BDC = 8.64
The area of ∆BDC is 8.64.
Question 2.
In □m ABCD, T ∈ and
intersects
in M and
in O. Prove that AM2 = MT • MO.
Answer:
In ABCD, T
and
intersects
in M and
in O.
To prove:
AM2 = MT •MO
In ∆AMB and ∆OMD
∠AMD ≅ ∠OMD (Vertically opposite angles)
∠MBA ≅ ∠MDO (Alternate angles)
The correspondence AMD↔OMB is a similarity
…(1)
In ∆AMD and ∆TMB
∠AMD ≅ ∠TMB (Vertically opposite angles)
∠MAD ≅ ∠MTB (Alternate angles)
The correspondence AMD↔TMB is a similarity
…(2)
Multiplying (1) and (2),
Hence, AM2 = MT •MO
Question 3.
In □m ABCD, M is the mid-point of .
and
intersect in N. Prove that DN = 2MN.
Answer:
In ABCD, M is the mid-point of
.
and
intersect in N.
To prove : DN = 2MN
Proof: M is the mid-point of BC
MB = MC
…(1)
In ∆MBN and ∆MCD
∠BMN ≅ ∠CMD (Vertically opposite angles)
∠MNB ≅ ∠MDC (Alternate angles)
The correspondence MBN↔MCD is a similarity
(from 1)
MN = MD
Now, D-M-N
DN = DM + MN
DN = DM + MN
DN = MN + MN (MD = MN)
DN = 2MN
Hence proved.
Question 4.
P and Q are the mid-points of and
in ΔABC. If the area of ΔAPQ = 12√3, find the area of ΔABC.
Answer:
In ∆ABC, P and Q are mid-points of AB and AC respectively.
The correspondence ∆APQ↔∆ABC is a similarity by SSS theorem.
Now,
Areas of similar triangles are proportional to the squares of their corresponding sides.
ABC = 12√3 × 4
ABC = 48√3
The area of ∆ABC is 48√3.
Question 5.
□ABCD is a rhombus. = {0}. Prove that the area of ΔOAB =
(area of □ ABCD).
Answer:
Given: ABCD is a rhombus.
= {0}.
To prove: Area of OAB =
(area of □ ABCD)
Proof:
O is the mid-point of AC as well as BD.
Further, in rhombus ABCD,
AB ≅ BC ≅ CD ≅ DA
In ∆OAB and ∆OBC,
OA ≅ OC
OB ≅ OB
AB ≅ CB
∆OAB and ∆OBC are congruent by SSS theorem for congruence.
Thus, their areas are equal.
Similarly, ∆OAB, ∆OBC, ∆OCD and ∆ODA are all congruent triangles having equal areas.
∆OAB = ∆OBC = ∆OCD = ∆ODA
Now, ABCD = ∆OAB + ∆OBC + ∆OCD + ∆ODA
ABCD = ∆OAB + ∆OAB + ∆OAB + ∆OAB
ABCD = 4∆OAB
Thus,
Question 6.
In □PQRS, = {T}, PS = QR,
. Prove that ΔTPS is similar to ΔQTR.
Answer:
Given: In PQRS, PR ∩ QS = {T}, PS = QR and PQ||RS.
To prove: ∆TPS and ∆QTR are similar.
Construction: Through R, draw a line parallel to PS to intersect PQ at M.
Proof:
In PMRS, PM||RS (P-M-Q) and PS||MR (construction).
Thus, PMRS is a parallelogram.
PS = MR
Further, PS = QR (given)
In ∆RMQ, ∠RMQ ≅ ∠RQM …(i)
Further, the corresponding angles formed by transversal PQ to PS||MR are congruent.
∠RMQ ≅ ∠SPM (corresponding angles)
∠RQM ≅ ∠SPM (by (i))
∠RQP ≅ ∠SPQ (P-M-Q)
In ∆SPQ and ∆RQP,
∠SPQ ≅ ∠RQP
PQ ≅ QP (same line segment)
SP ≅ RQ (given)
The correspondence ∆SPQ↔∆RQP is a congruence by SAS theorem of congruence.
∠PSQ ≅ ∠QRP
∠PST ≅ ∠QRT (S-T-Q and R-T-P)
In ∆TPS and ∆QTR
∠STP ≅ ∠RTQ (Vertically opposite angles)
∠PST ≅ ∠QRT
∆TPS and ∆QTR are similar by AA corollary.
Thus, by AA corollary the correspondence ∆TSP↔∆TRQ is a similarity.
Question 7.
In ΔABC, a line parallel to , passes through the mid-point of
. Prove that the line bisects
.
Answer:
Given: In ∆ABC, a line parallel to BC, passes through the mid-point of AB.
To prove: Line m bisects AC.
Proof: In the plane of ∆ABC, line m is parallel to BC and intersects AB at a point other than a vertex of the triangle.
Thus, m intersects AC.
Let m ∩ AC = {E}
In ∆ABC, D is the mid-point of AB.
AD = DB
In ∆ABC, A-D-B, A-E-C and DE||BC.
AE = EC and A-E-C.
E is the mid-point of AC.
Thus, line m bisects AC.
Question 8.
P, Q, R are the mid-points of the sides of ΔABC. X, Y, Z are the mid-points of the sides of ΔPQR. If the area of ΔXYZ is 10, find the area of APQR and the area of ΔABC.
Answer:
In ∆ABC, P, Q, R are the mid-points of the sides AB, BC and CA respectively.
The correspondence ∆ABC↔∆QRP is a similarity.
Areas of similar triangles are proportional to the squares of their corresponding sides.
∆ABC = ∆POR
Similarly, X,Y and Z are the mid-points of the sides of ∆PQR, we get
∆PQR = 4∆XYZ
∆PQR = 4×10
∆PQR = 40
Thus, the area of ∆PQR is 40 sq. units.
∆ABC = 4∆PQR
∆ABC = 4×40
∆ABC = 160
Hence, the area of ∆ABC is 160 sq. units.
Question 9.
In Δ ABC, such that
. Prove that X, Y, Z are the mid-points of
,
,
respectively.
Answer:
Given: In ∆ABC, X∊BC, Y∊CA, Z∊AB.
Also, XY||AB, YZ||BC and ZX||AC.
To prove: X,Y and Z are the mid-points of BC, CA and AB respectively.
Proof: In ∆ABC, YZ||BC.
…(1)
In ∆ABC, XY||AB.
…(2)
From (1) and (2),
…(3)
In ∆ABC, ZX||AC.
…(4)
From (3) and (4),
AZ2 = BZ2
AZ = BZ
Hence, A-Z-B and AZ = BZ.
Thus, Z is the mid-point of AB.
Similarly, Y is the mid-point of AC and X is the midpoint of BC.
Therefore, X, Y and Z are the mid-points of BC, CA and AB respectively.
Question 10.
Two triangles are similar. Prove that if sides in one pair of corresponding sides are congruent, then the triangles are congruent.
Answer:
Given: The correspondence ∆ABC↔∆PQR is a similarity and AB ≅ PQ.
To prove: ∆ABC and ∆PQR are congruent.
Proof:
The correspondence ∆ABC↔∆PQR is a similarity.
Further, AB ≅ PQ
AB = PQ
AB = PQ, BC = QR, AC = PR
AB ≅ PQ, BC ≅ QR, AC ≅ PR
The correspondence ∆ABC↔∆PQR is a congruence by SSS theorem of congruence.
Hence, ∆ABC and ∆PQR are congruent.
Exercise 6
Question 1.In ΔABC,
such that
are positive real numbers. A line passing through P and parallel to
intersects
in Q. Prove that (m + n)2 (area of ΔAPB) = m2(area of ΔABC)
Answer:We have
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Given: In ∆ABC,
such that
, where m and n are positive real numbers.
QP ∥ BC
Prove that: (m + n)2 (area of ∆APB) = m2 (area of ∆ABC)
Proof: Since ![](data:image/png;base64,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)
[by invertendo componendo invertendo rule]
⇒ ![](data:image/png;base64,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)
In ∆APQ and ∆ABC,
∠APQ ≅ ∠ABC [∵, corresponding angles]
∠AQP ≅ ∠ACB [∵, corresponding angles]
⇒ The correspondence APQ ↔ ABC is a similarity.
Remember the property, areas of similar triangles are proportional to the squares of the corresponding sides.
∴ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
By cross-multiplication, we get
(m + n)2 (Area of ∆APB) = m2 (Area of ∆ABC)
Thus, proved.
Question 2.D, E and F are the mid-points of
,
and
respectively in ΔABC. Prove that the area of □ BDEF = 1/2 area of ΔABC.
Answer:We have
![](data:image/jpeg;base64,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Given: D is the midpoint of
, E is the midpoint of
& F is the midpoint of
.
⇒ AF = FB, AE = EC & BD = DC
To Prove: Area of □BDEF = 1/2 Area of ∆ABC
Proof: Since, given is that D, E and F are midpoints of
respectively.
We know, AE = EC.
⇒ AE = FD [∵, EC = FD]
⇒
[∵,
] …(i)
Also, we know that, AF = FB.
⇒ AF = ED [∵, FB = ED]
⇒
[∵,
] …(ii)
Now, in ∆AFE and ∆DEF, we have
AF ≅ ED, AE ≅ FD & FE ≅ EF
⇒ By SSS theorem of congruence, we can say that the congruency AFE ↔ DEF is a similarity.
Similarly, in ∆AFE and ∆FBD, we have
AF ≅ FB, AE ≅ FD & FE ≅ BD
⇒ By SSS theorem of congruence, we can say that the congruency AFE ↔ FBD is a similarity.
Similarly, the correspondence AEF ↔ EDC is a similarity.
So, ∆AFE, ∆DEF, ∆FBD and ∆EDC are all congruent triangles.
⇒ ∆AFE = ∆DEF = ∆FBD = ∆EDC …(iii)
Now,
BDEF = ∆DEF + ∆FBD
⇒ BDEF = ∆AFE + ∆AFE [from equation (iii)]
⇒ BDEF = 2 ∆AFE …(iv)
And
∆ABC = ∆AFE + ∆DEF + ∆FBD + ∆EDC
⇒ ∆ABC = ∆AFE + ∆AFE + ∆AFE + ∆AFE [from equation (iii)]
⇒ ∆ABC = 4 ∆AFE
⇒ ∆ABC = 2 (2 ∆AFE) …(v)
Comparing equations (iv) and (v), we get
∆ABC = 2 (BDEF)
⇒ BDEF = 1/2 ∆ABC
⇒ Area of □BDEF = 1/2 (Area of ∆ABC)
Thus, proved.
Question 3.Can two similar triangles have same area? If yes, in which case they have the same area?
Answer:Yes, two similar triangles can have same area.
Note, when two triangles are congruent then all corresponding sides as well as corresponding angles of one triangle are equal to those of the other triangle.
⇒ If two triangles are congruent, they are similar too.
And congruent triangles have same area too.
Hence, the similar triangles need to be congruent to have similar areas.
Question 4.The correspondence ABC ↔ DEF is similarity in ΔABC and ΔDEF.
is an altitude of ΔABC and
is an altitude of ΔDEF. Prove that AB X DN = AM X DE.
Answer:We have
![](data:image/jpeg;base64,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)
Given: The correspondence ABC ↔ DEF is similarity in ∆ABC and ∆DEF.
is an altitude of ∆ABC and
is an altitude of ∆DEF.
To Prove: AB × DN = AM × DE
Proof: Since, the correspondence ABC ↔ DEF is a similarity.
By definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
So, we can write ∠B ≅ ∠E. …(i)
In ∆ABM and ∆DEN,
From result (i),
∠B ≅ ∠E
Also, ∠M ≅ ∠N [since, they are right angles of ∆ABM and ∆DEN respectively]
⇒ By AA-corollary, the correspondence ABM ↔ DEN is a similarity in ∆ABC and ∆DEF]
Then, again by definition of similarity of correspondences in triangles we can say that,
![](data:image/png;base64,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)
By cross-multiplication, we get
⇒ AB × DN = AM × DE
Hence, proved.
Question 5.Explain with reasons, whether the following statements are true or false:
(1) AAA criterion of similarity of triangles cannot be the criterion for congruence of triangles.
(2) SAS criterion for congruence of triangles cannot be a criterion for similarity of triangles.
(3) Two congruent triangles have the same area.
(4) Two similar triangles always have the same area.
(5) Area of similar triangles are proportional to the squares of measures of their corresponding angles.
Answer:(1). Similarity of triangles and congruence of triangles are closely related, but has a fine line of difference.
Similar triangles are the same shape, but not necessarily the same size. The triangles are congruent if, in addition to this, their corresponding sides are of equal length.
By AAA criterion of similarity of triangles, we get the shapes of triangles equal but, for congruence of triangles their sizes must be equal too.
∴ AAA criterion of similarity of triangles cannot be the criterion for congruence of triangles.
Hence, the statement is true.
(2). For congruence of triangles, the corresponding sides are of equal length while for similarity of triangles, measure of the corresponding sides of two triangles must be proportional.
SAS criterion for congruence of triangles not only makes sure that the triangles have equal length of corresponding sides but also makes sure that the triangles have same shape.
∴ SAS criterion for congruence of triangles can be a criterion for similarity of triangles.
Hence, the statement is false.
(3). Congruency of triangles ensure that the triangles have same shape as well as equal lengths of corresponding sides.
⇒ The shape and size of two congruent triangles are equal.
∴ Two congruent triangles have the same area.
Hence, the statement is true.
(4). Similarity of triangles indicate that they have the same shape but their sizes are not same.
Area of triangles are equal when their size are equal, shape of triangle cannot determine its area.
∴ Two similar triangles doesn’t always have the same area.
Hence, the statement is false.
(5). Area of similar triangles are proportional to the squares of their corresponding sides.
Since, the measure of corresponding angles of similar triangles are always equal.
∴ Area of similar triangles are not proportional to the squares of their corresponding angles.
Hence, the statement is false.
Question 6.Fill in the blanks so that the following statements are true:
and
are the altitudes of ΔABC. If AB = 12, AC = 9.9, AD = 8.1 and BE = 7.2, perimeter of ΔABC = _____.
Answer:We have
![](data:image/jpeg;base64,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)
Given that: AD and BE are altitudes of ∆ABC. ⇒ ∠ADC = ∠
In ∆ADC and ∆BEC,
∠ADC = ∠BEC [∵, ∠ADC = ∠BEC = 90° as AD and BE ]
∠ACD = ∠BCE [∵, ∠ACD and ∠BCE are same angles of the same triangles, so they obviously are equal]
⇒ By AA-corollary, we can say that ∆ADC ∼ ∆BEC for the correspondence of ADC ↔ BEC.
By definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAF8AAAAgCAMAAABD5lTTAAAAAXNSR0IArs4c6QAAAIpQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjo6OmaQOma2OpC2OpDbZgAAZgA6ZjoAZjo6ZjqQZpC2Zra2ZrbbZrb/kDoAkDo6kDpmkGYAkNv/tmYAtmY6tmZmtpA6ttu2ttv/tv//25A625Bm27aQ29uQ2////7Zm/9uQ/9u2/9vb//+2///bJz1rfAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABvUlEQVRIS+VW7VKDMBBMqq1oLbYqRW2wH7QWKnn/13PvLoA6gzJp8I8305lAyWZvb5OLUv80qviKMz/NtR4ljQhlpBHjrZKB9lanuL7YYXIZ3R6tmbUwBV6faO09/Z/74tvnTbzEZDM5MkRGD9X9URE+v+WBdxRTZUCyih31arFU1WKlGLa8w2o0MLSqV2QrBmjwAf6IJQBLqs+QFA988aV6oFzzp0pwKqx/itTO4p8DywKFfqginhr+XNZa/8JLHBiEEs9GK5XrBE8z1p5q8JW/NWDAPh1N2QcHcvONWKI7MAfQEBg091DqMlGvKC35R4QDQuN/3ih2rTlRDTevMTdgiIRVDC5lREP7NAg+oPOwzEUE5v/+Mt5hdNaW69BUCjzZOo2+fWW4+uwAT9WYvzWYzjXoFc2ivw6cPow9lD5sGtKekhhGH9Gm0HSEBPan7C9D0DZFJwq8v9z5gC1O1DPa7Q89q9zLCiE+kl7MJ447x0KgthiuF9OJf4j6erw/hboXc8ctI9+e1bmg68XS0W366TLRn+NPX7peXOPLvShc1L14KP51Lxb84Pq3vZivDvPA/ml78Rv5/6835wf3vSn21yeoLQAAAABJRU5ErkJggg==)
⇒ ![](data:image/png;base64,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)
⇒ BC = 8.8
So, perimeter of ∆ABC is given by
Perimeter of ∆ABC = AB + BC + AC
⇒ Perimeter of ∆ABC = 12 + 8.8 + 9.9
⇒ Perimeter of ∆ABC = 30.7
Thus, perimeter of ∆ABC = 30.7.
Question 7.Fill in the blanks so that the following statements are true:
D, E, F are respectively the mid-points of
,
,
of ΔPQR. The correspondence DEF ↔ _____ is similarity.
Answer:We have
![](data:image/jpeg;base64,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)
Given: D is the midpoint of PQ ⇒ PD = DQ = 1/2 PQ
But PD = DQ = FE [∵, F and E are also midpoints of PR and QR respectively]
So, EF = 1/2 PQ
Or
…(i)
E is the midpoint of QR ⇒ QE = ER = 1/2 RQ
But QE = ER = DF [∵, D and F are also midpoints of PQ and PR respectively]
So, DF = 1/2 RQ
Or
…(ii)
F is the midpoint of PR ⇒ PF = FR = 1/2 RP
But PF = FR = DE [∵, D and E are midpoints of PQ and QR respectively]
So, DE = 1/2 RP
Or
…(iii)
Comparing equations (i), (ii) & (iii), we get
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKsAAAAtCAMAAADm32vuAAAAAXNSR0IArs4c6QAAAHtQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjpmOmaQOma2OpDbZgAAZgA6ZjoAZjo6ZpDbZrbbZrb/kDoAkGY6kNvbkNv/tmYAtpA6tpBmttvbttv/tv//25A627Zm27aQ2////7Zm/7aQ/9uQ/9u2//+2///bbxeZHAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAC0ElEQVRYR+2YXXebMAyGIVtb1m5Z24RtbOsoySD+/79w+rINJaeSkuUi56Art9GDXoQsg4pisctkINQl2zauVj+0SOy5uv8NjnZKXKdYudWCwe/dk0g6rO/gz7/VtuBVq2tlz6Euv4C/nTqCYVjNhsfywx92CjUqLJqtrA5f1byKZ6jxtpi3UEcwDKtY//DSvtGaVWtw9uwrkCn3aqCiqxcrplpfMTf2qOJ5WN/uHFR0zRiF1W2kFTdK0mrCRWuoo1YTFbUKFsO6tILOPWu14lOtVuqN1hjWqzXvMlOGZjVgouY1oMskj/+zt6Bp2as87y0fNtOaepblXmOfyz3LQqX2OML0njXOK/fyPp0FlqhyFqywOTJvsjmGYXWLx1M+I3EF+9pgcsY+vMiGtFHxjE0YHdTqwRO+VeD38dmga3FZMrBkYMnAFWWgkS8cSyfMt3UZKkmZLJRkHodK7RGcGUu7/FX9fpmneTwFp8W6qnQuYs/MgLwS3uPrXdE9wsvXRr/iBDJTcUSDsaZXeDdiQFU39OpI779dhdOCdrUpwk8apigmEL182qkYC7FRWCVWU26KYU3TDP4caWGu1FeokocpimXIQ41ijZea1qgvat1DNhsWiUvNWCt5OigRSNjoClowFoXfOilFoeapEU1wbHmFm/NQMVb+NMTHabIm10BX3e5ochML2KK1w5rxUKyVsLg0fqnJJyTtyNXnnSsqQ1gyTq3pazCFtaQ11FyWaRTheZoIDesbqBkPRbEY80xAxhJjeaZdotcQ3eCeBnYOioMy5pjWFM2dTAIyJA2Ay/h9I4i7m4NigeMJsxaHfm+xqPd5hE7/xK4LZ4GeVpm19LAjPdQYs09r+k84PG+oz+VhS/gFEw4+zbS08sC2wf1lpiiWYN/NMx4IQjZv+pwqr51GeaPM/KHyd4N7dnQadbbY16qEZuu10yhvlMX/Ehn4B3nxStyftnmYAAAAAElFTkSuQmCC)
We can also arrange it as,
![](data:image/png;base64,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)
∴ The correspondence DEF ↔ RPQ is a similarity by SSS theorem.
Thus, answer is RPQ.
Question 8.Fill in the blanks so that the following statements are true:
In ΔABC,
and
are altitudes. If AB = 12, BC = 15 and AM = 9.6, then CN = _____.
Answer:We have
![](data:image/jpeg;base64,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)
Given: AM ⊥ BC and CN ⊥ AB.
AB = 12, BC = 15 & AM = 9.6
To find: CN = ?
For this,
Take ∆AMB and ∆CNB,
∠AMB = ∠CNB [∵, ∠AMB = ∠CNB = 90°]
∠ABM = ∠CBN [∵, ∠AMB and ∠CBN are same angles of the same triangle ABC]
⇒ By AA corollary, ∆AMB ∼ ∆CNB for the correspondence AMB ↔ CNB.
By definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
⇒ ![](data:image/png;base64,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)
Or ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ CN = 12
Thus, answer is 12.
Question 9.Fill in the blanks so that the following statements are true:
Areas of two similar triangles are 25 and 16. The ratio of the perimeters of the triangles is _______.
Answer:Given: Area of two similar triangles.
Area of ∆1 = 25
Area of ∆2 = 16
Recall two properties:
First,
Ratio of areas of two similar triangles = Ratio of the squares of the corresponding sides
Second,
Ratio of perimeter of two similar triangles = Ratio of corresponding sides
Using the first property, we can write that
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Taking under root of both sides, we get
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZoAAABBCAMAAADfVpIJAAAAAXNSR0IArs4c6QAAAMBQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpCQOpC2OpDbZgAAZgA6ZgBmZjoAZjo6ZmY6ZpC2ZpDbZrbbZrb/kDoAkDo6kDpmkGYAkGY6kJxmkLaQkLbbkNvbkNv/tmYAtmY6tmZmtpA6tpBmtrZmttvbttv/tv//25A625Bm27Zm27aQ29uQ29v/2/+22//b2////7Zm/7aQ/9uQ/9u2//+2///bfRM1FgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAG/0lEQVR4Xu1c6WLaRhBGOLaVxqlpQ4ub2LhOm1ikRyybtEUV6P3fqjPf7AUIjBA22zD7I+jYOb+d3dlZOZ2ONvWAekA9oB5QDzyfB4qktj2fAipplQeKU/VNpB7I30SqmKql0EQ7BjKNmlixya5i1ezg9VJoYh0C1fVtrKodul4KTbQjYPaDRk2k4Mx+vItUs4NXqxaae42kCAZG+XrS6VTjQZIcMx5lyhW1rkITCzRZctmZ9o9oaitT3eZEgApUKE4oajIuceYJoaLQxAIMQeMKz4VCEw8siBoHTaYTWlzQuMKzTGVlepZKQqBt3x6w0FRDVKCra0oIhpqh7RsWlm/OBKphcNhZpnpQEAE2pvCcnVKiZtusr4fS0UCTcwpdUKwgiJpFTV7/3Yc+3cYDwYiQwnP5kv6tGJaMlpnpkHO1TZuWrjf1VMN+4thMACZophf0e/65ARdUerTt3gPtzwT0Y6ndowKO7c8E2hxgV+8RqjttczzzZ6vU0peWTauPj5C0hoYZVEPMh2c18yCqP6saZ+y7/tZqgWfxbFs0WLrWXOMG32d979YrBRgg2x6nc25AarFe16eopTqeRv6zQQPBzaBZb39raDDqZcOK0rVrG4zXTQxpOtk5niJ/Ay2aSljRf4txtp4EZwJt2ogdINAUvG5UD98kSa/TGfEUd/SHzMAlndS9dtOdvSv42A5oTq/p+K5HmvCr3s0VsSLH0rEevbUMeWa+58CkPt13IRFzWOIp8u+K7ke6wnpm6WAt9+/yIYhoZpj/skIGqfPxt5TNCiShr3/2LUUNVg/hhRfVCCeT4mNjpFlhWDCTeL3kfTC2W+77Za0yUUN+o6tJxZsjM14xMMqUC3N2s+Tv7AivhieT6oYM4L0uLVxEkoMFXXmGVHm9HL+8ZfIxcXVE8JeTsBg1tAKKPpZOkOH+/StP55jXyyh4IX1giL0kIrm1h1yXnerGnan4F9nRbefezCWBkcYllmTOnkbQmE1P/bm0pM6AZuwqbzLtYkLBJU7qXPT6O+vGWZ9tpkUrQ1EC66mFxngS1mDwcx+W54jw0ElYgsauAJYODCEopDPMV8kAV3ZuIMmMOt6nO7XNM6Mt3GKrXk5fcYkjWbDHQ4Oloq6i4LusvcoxISFF8wvuAjRQSlzpfnHn3DhKzv50rzy1e+8tNmzYHkO0imew1jC5iDeQLOtiBo78+L5WhmhC1IEt6Mv38ixExbww0JjtgeXltZnXK7AHjmqZu5oqDesw7R/fMUrjC5pf56JGVLe17eDOpwG0QBF1YKOPGsfQug0DiWdHIfLQQEJdGiAucHSLw4TpFqBZlCFc86O7QPvHoUHZ68ElR0bfRWhq7JGIfnzHVz+hTS+whEmVxjiFPMNLg9jpJ7THo4ZYjFMzRwXUdq0JRqQLPmjPRI2iBr3rIrgmakxfyNguajCbdC8NI6tvXdSEsmz3bffy+fkAs6W4BtBUfAIHQBagwRzt1prgzo5wAEyzhU30wrXGMxTvuYMlSyTynYRVUTN3IGVvPN0cNMsyzFpzGkpyUROo7Wc1jMdwTnL6ch9PsijL47gtNGYgGXJZ6sr0ZMLr6PQnScvedD7dmj0yPYT3uHGOLXcOGs6XBsQi737gAiuoexNceYamd0EfZlX37yRVYqJ6nvPykQsIHVrON59oJFldHHOsnUsyCk6A7yU1mdMeeYNX25cEJNkM5iSnL/qEJDX28HDb/u8EkAz+fCeDNknYRxmlArSL6P3dZ+kj2jjwF4d0+UA/vG2RZu/4D3zhiek1vT/nTxWZ+guig6jffuGNjWXIrGAp7Rm6byeeqJ6nyCcRXUq3IcfQiQ50c/zB62KYr5RRJK/S5AUTOO3BlVcwgoBEkdqJmBu8kKMssdwaKdo4khX2hNCMLzjXhaNHNPqtG9f80tajdS2hjv36PfIGij1Bl+3qFrMBD7J/t/vq0p4JTAfIebgVae+vjYyjWaxlflcv5uuBhjI6MtFm6hs51Xcyhefy/LOwYWSCjGI9t7z7ykZaQ7Fru6PeE1nbLmqkcjVqcma8BA09MNA0+WJjhq+kd93cCrRrxi34mdWqMQeuJzY7M/Yi/GJhoGk0PPKWBbjGph4SwRI02dGvXPzdJA04JD/twVZ/JiBRQ2nw95OgULkHlVSkeMB/c2GhQd5sC7Pqpv15YAU0Nl3bn2Iq2W9MDBr4Sw6Nmn2OjOr3HosPoJHjljKlyrGvY+9TxQOV/XA2kMMh2d1V7/kM+wXX/vjMGtUlbfvyAJeJCRqUF7XF5QF8mqTQxAWKaDPrU1F/+zOBGE36anSisFFo4kSTwuYfjZo4scmT7/T/DooTGjo9fYqqfpzG/s+0wscI2mL0gHzroy1GD7T+O4EYjVKd1ANP4oH/ABlPIvKFTAUvAAAAAElFTkSuQmCC)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAR8AAAAgCAMAAAAc0oyzAAAAAXNSR0IArs4c6QAAAKtQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjo6OjpmOjqQOmY6OmaQOma2OpDbZgAAZjoAZjo6ZjpmZmY6ZpC2ZpDbZrbbZrb/kDoAkDo6kGYAkGY6kGaQkJxmkLbbkNvbkNv/tmYAtmY6tmZmtpA6trZmttvbttv/tv//25A625Bm27Zm27aQ29uQ2/+22////7Zm/7aQ/9uQ/9u2//+2///bMAwFgAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAD/ElEQVRoQ+1YeV/aMBhOypTodCK6A4vbFHAHBRxdId//k+09crVg10L151j7B23S5z3y5D1ShGivloGWgcYZSKSU0ahxtQejMLk5mKU8y0JafsppxfzqPQvzB6M0U22OlW2mji/32Gv9WUonn7xcrU+lrLmt9SWE0MORoPjRMSSalGc/N6lKyxzRcVcknt/05XohuVXqGy/FQyqAi6vXE2jvA5xd97tCLFRufdkJdv5StS45Dfbl+CHLFZbsIXsVEgwEIZJcrFSIBmedsRUk9sjlnGj91daXCAwyPynWEj0/paYGsSVl5zsnenYl5bnLPjtKFUDwNWOnaXQPT5RvAIk+WQOIjwDn5EBqpqKvfIMcN+DVEHC9ZSqj+2+KG2texM+9g/ihkoI/MwbrCfojj5coyLrm3n2U8H4ZU1W3y8QP+ApPSz3GcOBoINozNRCruDNldX5UjB+oYCyLkIXtjITv3wRymTobLE5G7sZgHR8v9S0sL8VKOEeivSnAjjjAcU7f8hP92BfjzkjMTApYXc59K1EwVZEg4mehbKnl7Pb8jDG6XID60QY/tiiMYZXMOVw4oLvTkrElc7PgdR8ZOUd+qGeAWChCXACC1BHELN68IHtUSOGyugjiJTZMWX7G1KDw2lpDqYH5NwV+yBibdHcabas/KMtQw8sWOUM13zx4Is9+oAVWC+KBYcLimOdCaswLw4/ZY6PLe5P3y5oqjR/DGGBQ96p/hAmkF9eQxrn4YZdsPASjp/kh1ZyQeX5IS4EfB4bShz6w2qQzDUz9nR+BmT13LYZ1FfkpmqqTX6kE17H+sP8+v3aMH2O7WvwY8EKBD7vFD53i+LjCF+raFj+hqcr5RdTABhArBX6oFLj6E4yeih9XetC8HQRyufhxYCw9lJSm/nS5BgW5hFTbVpurP7QHwUmVyhjqQkteYtNUxfjhspYpVHgpVh+4MVyKh5E5qMIk+crlwY4cPzksFWnoMTPT4BMcPAD1RTkjbsHUrK4ofsDUjHpoXoTqcxLdidW1Oz+bSo3EBV9KVhe9DSXYL2eqGj8YmUjQGGo0nGB6v/qodgKHloxr0RxuPUOPH8HZw6Y7YmEYQc+mOTiSRB+tAAyO7kC91YIYat7msGTAqyFMXFBzeqvkG5TIidys+ygAp/6o96gk+0Y//AL/jbBuGl3sjZNwfjlT1fh5ZagK3w1bPF5fYVX//R/8H7EbP9Dvgm7Z0Jbv9VXSkA8banbjh7+OJvaU34hzeniKQfm6LlPCajuFH47ywn0l1pbfIpAM4tfHTxMLa0ZH2tMtP09TuX4/1fHgi+vbzZB+OFr4wGCPfYezrgZX0uZXOZmPqtF+2ODOtar+GQb+AHYhoKbB467FAAAAAElFTkSuQmCC)
Now, using second property, we can say
![](data:image/png;base64,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)
Thus, answer is 5/4.
Question 10.Fill in the blanks so that the following statements are true:
Area of ΔABC = 36 and area of ΔPQR = 64. The correspondence ABC ↔ PQR is a similarity. If AB = 12, then PQ = ______.
Answer:Given: Area of ∆ABC = 36,
Area of ∆PQR = 64
AB = 12
The correspondence ABC ↔ PQR is a similarity.
To find: PQ = ?
For this,
Recall the property,
Ratio of areas of two similar triangles = Ratio of squares of the corresponding sides
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ PQ = 2 × 8
⇒ PQ = 16
Thus, answer is 16.
Question 11.Fill in the blanks so that the following statements are true:
In ΔABC, A—M—B, A—N—C and
||
. If AM = 8.4, AN = 6.4, CN = 19.2, then AB = ______.
Answer:We have
![](data:image/jpeg;base64,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)
Given: A – M – B, A – N – C
MN ∥ BC
AM = 8.4
AN = 6.4
CN = 19.2
To find: AB = ?
Take ∆ABC,
We know that, MN ∥ BC. So, by property we can write as:
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ BM = 25.2
We know AM = 8.4 and BM = 25.2.
Now, AB = AM + BM [∵, from the diagram]
⇒ AB = 8.4 + 25.2
⇒ AB = 33.6
Thus, the answer is 33.6.
Question 12.Fill in the blanks so that the following statements are true:
The correspondence ABC ↔ PQR is a similarity in Δ ABC and ΔPQR. AB = 16, AC = 8, PQ = 24, BC = 12, then QR + PR = ____.
Answer:Given: The correspondence ABC ↔ PQR is a similarity in ∆ABC and ∆PQR.
AB = 16,
AC = 8,
PQ = 24 &
BC = 12
To find: QR + PR = ?
By definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
To find (QR + PR), cross multiply it.
⇒ (QR + PR) × AB = (BC + AC) × PQ
Substituting values,
⇒ (QR + PR) × 16 = (12 + 8) × 24
⇒ (QR + PR) × 16 = 20 × 24
⇒ ![](data:image/png;base64,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)
⇒ QR + PR = 30
Thus, the answer is 30.
Question 13.Fill in the blanks so that the following statements are true:
The correspondence ABC ↔ XYZ is a similarity in ΔABC and ΔXYZ. ABC = 72, BC = 6, YZ = 10. Then XYZ = ______.
Answer:Given: The correspondence ABC ↔ XYZ is a similarity in ∆ABC and ∆XYZ.
ABC = 72,
BC = 6 &
YZ = 10
To find: XYZ = ?
So, by property we can say that,
Ratio of areas of two similar triangles = Ratio of squares of the corresponding sides
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Taking reciprocal of both sides, we get
![](data:image/png;base64,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)
Substituting values in the above equation, we get
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ XYZ = 200
Thus, the answer is 200.
Question 14.Fill in the blanks so that the following statements are true:
□ ABCD is trapezium in which
||
. The diagonals intersect in P. If PD = 9, AP = 5, PB = 7.2, then AC = ______.
Answer:
We have
![](data:image/jpeg;base64,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)
Given: ABCD is a trapezium.
AD ∥ BC
PD = 9,
AP = 5 &
PB = 7.2
To find: AC = ?
We have got the trapezium ABCD, in which
AD ∥ BC & P is the intersection point of AC and BD.
Take ∆PAD and ∆PCB,
∠APD = ∠CPB [∵, Vertically opposite angles are equal]
∠PAD = ∠PCB [∵, alternate angles]
∠PDA = ∠PBC [∵, alternate angles]
⇒ By AAA similarity theorem, ∆PAD ∼ ∆PCB for the correspondence PAD ↔ PCB.
Now by definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
⇒ ![](data:image/png;base64,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)
Reciprocating it, we get
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ PC = 4
From the diagram, note that
AC = AP + PC
⇒ AC = 5 + 4
⇒ AC = 9
Thus, the answer is 9.
Question 15.Fill in the blanks so that the following statements are true:
In ΔABC, m∠B = 90 and
is an altitude. The correspondence BDA ↔ _____ between ΔBDA and ΔBDC is a similarity.
Answer:
We have
![](data:image/jpeg;base64,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)
Given: In ∆ABC,
∠ABC = ∠ADB = ∠BDC = 90°
In ∆BDA and ∆CDB,
∠BDA = ∠CDB = 90°
∠ABD = ∠BCD [measure of both being equal, i.e., (90° - m∠A)]
This can be explained as,
In ∆BDA, by angle sum property of triangle,
∠BDA + ∠DAB + ∠ABD = 180°
⇒ 90° + ∠A + ∠ABD = 180° [∵, ∠BDA = 90° & ∠DAB = ∠A]
⇒ ∠ABD = 180° - 90° - ∠A
⇒ ∠ABD = 90° - ∠A …(i)
Similarly, in ∆ABC by angle sum property of triangle,
∠ABC + ∠BAC + ∠BCA = 180°
⇒ 90° + ∠A + ∠BCD = 180° [∵, ∠ABC = 90°, ∠BAC = ∠A & ∠BCA = ∠BCD (from the figure)]
⇒ ∠BCD = 180° - 90° - ∠A
⇒ ∠BCD = 90° - ∠A …(ii)
By equations (i) and (ii), we get
∠ABD = ∠BCD
Hence, by AA corollary the correspondence BDA ↔ CDB is similarity between ∆BDA and ∆BDC.
Thus, the answer is CDB.
Question 16.Fill in the blanks so that the following statements are true:
The correspondence ABC ↔ ZXY is a similarity in ΔABC and ΔXYZ. If AB = 12, BC = 9, CA = 7.5 and ZX = 10, then YZ + XY = ____.
Answer:Given: The correspondence ABC ↔ ZXY is a similarity in ∆ABC and ∆XYZ.
AB = 12,
BC = 9,
CA = 7.5 &
ZX = 10
To find: YZ + XY = ?
Now by definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Reciprocating both sides,
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Thus, the answer is 13.75.
Question 17.ABC ↔ DEF is a similarity in ΔABC and ΔADEF, m∠A = 40, m∠E + m∠F =
A. 40
B. 80
C. 140
D. 180
Answer:Given: m ∠A = 40°
The correspondence ABC ↔ DEF is a similarity between ∆ABC and ∆DEF.
To find: m ∠E + m ∠F = ?
Now by definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
⇒ m ∠A = m ∠D
⇒ m ∠A = m ∠D = 40°
⇒ m ∠D = 40°
In ∆DEF,
m ∠D + m ∠E + m ∠F = 180°
⇒ 40° + m ∠E + m ∠F = 180°
⇒ m ∠E + m ∠F = 180° - 40°
⇒ m ∠E + m ∠F = 140°
∴ Answer is 140.
Thus, option (c) is correct.
Question 18.In ΔABC, M ∈
, N ∈
such that
||
,……….is not true.
A. AN ⋅ MB = AM ⋅ NC
B. AM ⋅ MB = AN ⋅ NC
C. AB ⋅ AN = AM ⋅ AC
D. AB ⋅ NC = AC ⋅ MB
Answer:We have the diagram as
![](data:image/jpeg;base64,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)
Given: In ∆ABC, MN ∥ BC.
By property of triangles, we can say that
,
&
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAE8AAAAqCAMAAADF0/ZdAAAAAXNSR0IArs4c6QAAAI1QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGZmAGa2OgAAOgBmOjqQOmaQOma2OpC2OpDbZgAAZgA6ZgBmZjoAZmY6ZpCQZpC2Zra2ZrbbZrb/kDoAkDo6kGYAkNu2kNvbkNv/tmYAtmY6tpA6ttv/tv/btv//25A627aQ2/+22////7Zm/9uQ/9vb//+2///bZvGmBgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACB0lEQVRIS+1WaVPCMBBNyiH1Kggq1IOIUKW1+f8/z7ebtE0LmsAXdIbM0HRC3tsjm+4T4o+Pcizlgn3MpEyETiWN6Gbr8xvAaCkYQPiPKVDXQGF9SGDMCU8jbFpJPD0DANpUjokuk7dbvSID5WRGk8gvpsSnU7OJbfw6ACTPdIpHERNKPzHfyxhEOn2kqeYL8G/ymsIq82XsEo9yslS9jciHnw3f1/Ng43OPgDlnfIEfGBq+HH6rpLR8dB7Ddy8d8QkVLYnPTQ+WkbPiclPx0Xlgn5eQ+JC3PXwi680SGDn0PGAzk/PdeGEG/rh8bkJ+cpT8Q+YGU5yHEw8vKwR5lH8i53Kmg6nqheuRa66pZ8X/e+rPAJUpwmiOW8BR2vtGL/V96899bDWwiIlWv8VS9h+8l8BLe95wzkA3A4p7APcB72ei3rrz4snrfuBJDuOQeE/i4MmNaup8ptmHqQPjsYXhrZYH5o/cVla4OmjBHHlg+NR9zB/kcHXQgjnygNfLu3UVsFEbfnXQgjnygPmyEVomdXj2L0gduLBuN6SWbNsAH0iIOmDjFtZVT8UVyRpWSPQMUgfYXMO6fFCSGBTwQedRwzrxckdHyXAfNV42csmc/77hwNpyJ7f6lEINVpNwoIE58qDiQJHAKVPPIerAmjawRh6wbiYBTVNvbe5bgDpowUD4X+TBN+vtPgA7b/jWAAAAAElFTkSuQmCC)
Cross multiplication of the above equations,
⇒ AM × NC = AN × MB
Or AM . NC = AN . MB is true.
AM × AC = AN × AB
Or AM . AC = AN . AB is true.
MB × AC = NC × AB
Or MB . AC = NC . AB is true.
So, AM . MB = AN . NC is not true.
Thus, option (b) is correct.
Question 19.In ΔABC, B—M—C and A—N—C,
||
. If NC : NA = 1 : 3 and CM = 4, then BC = ………..
A. 12
B. 16
C. 8
D. 1/2
Answer:We have the diagram as,
![](data:image/jpeg;base64,/9j/4AAQSkZJRgABAQEAYABgAAD/2wBDAAoHBwgHBgoICAgLCgoLDhgQDg0NDh0VFhEYIx8lJCIfIiEmKzcvJik0KSEiMEExNDk7Pj4+JS5ESUM8SDc9Pjv/2wBDAQoLCw4NDhwQEBw7KCIoOzs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozv/wAARCAClALgDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD2aiiigAooooAKKKKACiiigAoorF8VeIk8NaSLvyDcSySCOKPdtBY5OSewABpxTk7ITaSuyxqJV7+0t53KW0iuSNxUOwxgE/TJx7VY08wG3K20zyxo7KGck4wegJ6gdK5/wn4rj8Wpc2l7YxxTQBWZM70dTnBGR2I6V1CIsaBEUKqjAAGAK0mnH3JKzREGpe8tmOooorI0CiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACue8YWdvq9lb6PIpMtzMGRlODEF5Zx+HH410GccmsHT2/tC9uNWblH/c23tGp5P4nn8q1pqz5uxnU25e5naBpNr4Q1ZrdGeSDUQFW4lILK65whxxg5JHvXYVlX9lHqFnJbSEgMMqw6ow6Ee4NO0PUJLu2e3usLe2reXOPX0YexHNVU99c/Xr/mKHuvl6dDTooorA1CiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAMnX7h/s0dhA2J71vLBH8Kfxt+A/nU0UUcEKQxLtjjUKo9AKoWTf2jqVxqh5jGYLb/dB+ZvxP6CtKt2uVKJind8wVl6mslhcx61bqWaEbLlF/wCWkXf8V6itSjqMEZB6iiLsxtXRahljnhSaJg8bqGVh0INPrA0qQ6TqLaRIf9GmzJZse3do/wAOo9q36znHlfkXGXMgoooqCgooooAKKKKACiiigAooooAKKKKACiiigAooooAKy9euZI7RLO3bFzet5UZH8I/ib8Bn9K1KwbV/7R1W41I8xRZt7b6A/O34nj8K0prXmfQzm9Ldy7BBHbW8cES7Y41CqPYVJRRTAKKKKAKmp2P2+0Mav5cyESQyDqjjoas6PqP9pWQeRfLuImMc8f8Accdfw7inVlXrHSNRXV4wfs8mI71R6fwyfh39qtLnXL9xLfK+b7zoqKQEMoZSCCMgjvS1zmwUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAY+r6qvkXFjZyf6ZuWI5BHl7hncT6AZP4U7ThbLptsLM7rfyx5ZxjI9apNayXWoy3Vs8bwzT+XKwcf6sKobHvuXH4mr1hC9tp9vBIArxphgDnHJrpaioWRgm3K7LFFFFZlhRRRQAU10SWNo5FDI4IZT0INOooAoaHO1jcPolwxPljfaO38cXp9V6fTFblYmq2Ul1Ak1sQt5bN5kDe/dT7EcVf0zUI9UsEuYwVLcOh6ow6qfoadRXXOvn/AF5ig7e6Fvq1ldW8E8M2+OeQxoQD94ZyD6dDVyucttGvbWTTWjVRGHDXUe4fKwVgGHrnOD+Bro6mpGKfusqDk17wUUUVmWFFFFABRRRQAUUUUAFFFFABXnPxP1bVLG9sreG6mtLKSItvicpvkB+6WHoMHHv7V6NWAAmrazNcSKslrZ5hhVgCrP8Axtj24H51tQajPmaukZVbuPKnZswvBt5rx8NW4NhHNGrMIZZZtjOmeCRj689+tbn2rW/+gXbf+BX/ANatOiiTu2wirJIzPtWt/wDQLtv/AAK/+tR9q1v/AKBdt/4Ff/WrToqRmZ9q1v8A6Bdt/wCBX/1qPtWt/wDQLtv/AAK/+tWnRQBmfatb/wCgXbf+BX/1qPtWt/8AQLtv/Ar/AOtWnRQBmfatb/6Bdt/4Ff8A1q8x17WNb0/xNfObibTrjzt0cEMh2HgbSB0fPGeOa9hqG5tLe8iaK4hSRWUrkqMgH0PatqU1CV2rozqQc1ZOxds3neyge5QJO0amVR0Vscj86mrG0G7lTzNJvH3XNoBtc/8ALWL+Fv6GtmueUXF2Noy5lcKKKKkoKKKKACiiigAooooAKKKKAM/Wr17LTz5HNzMwigHq56H8Ov4VHZWiWNnFbR8iNcE/3j3P4mqwb+0tdkuOsFhmKL0Mh+8fwHH51oVu1yxUTHd3CiiipKCiiigAooooAKKKKACiiigDO1a2mxFqFmM3doSyr/z0T+JPxH61rWV5DqFnFdQNmOVcj29j71FWXbv/AGJrHlHix1B8r6RTdx9G/nVW5426om/LK/RnQ0UUVgbBRRRQAUUUUAFFFFABVDWb5rDTnkiG6eQiOFf7zngf4/hV+sJn/tLXmk62+nfIno0pHJ/Acfia0ppN3eyIm7KyLFhaLYWUVsp3FB8zf3mPJP4mrFFFU227slKysFFFFIYUUUUAFFFFABRRRQAUUUUAFQXtnFf2clrN92QdR1U9iPcGp6KE2ndCauV9Dv5bmGS0vDi9tDsl/wBsdnHsRWrXP6pHLaTR6xaqWlthtmQf8tYu4+o6ityCeO5gjnhcPHIoZWHcGipFfEtmOD+yySiiisjQKKKKACiimSSLFE8jnCopY/QUAVdXvjp+nSTIN0pwkS/3nPAH51W0+zFjYx2+7c4GZG/vMeWP51XaUarq1tKVdIIIjNEkiFS7Hjd9AD+taNb25Y8pjfmdwoooqSgooooAKKKKACiiigAooooAKKKKACiiigArN01/7H1Q6Y5xaXRMloT0RurR/wBRWlVXUrFdQs2h3GOQEPFIOsbjoRVRa2ezJknutzXorO0XUW1CzImXZdQN5dxH/dYd/oeopmu6hJZQRRwTRxTzMQjSdAAMn+QH41nyS5uXqXzrl5jUoqCyukvbKG6j+7KgYfjRUtWdik7k9VNTmjisJfMcJ5i+WpPdm4A/M1brxGe88RT+LAGkun1NLz93AxbYPm4G3psx39Oa2o0+dvW1tTOrU5EtNz0/TrKS3lidoZIAlt5Ugkn8ze3GMcnAGD6da0qoeV4jP/LPTV/4G5/pR9n8Rn+LTR+Dmqkru7aJWisky/RVD7L4iP8Ay8aaP+2bn+tH2LxCf+X2wH0gY/1qeVfzId32Zfoqh/Z/iA9dTsx9LY//ABVH9m66eur24+lqP8aOWP8AMvx/yC77P8C/RVH+ytaPXWox9LRf8aP7H1Y9deYfS1Si0f5l+P8AkF5dvy/zL1FUf7F1M9dfm/CBBR/Yd/31+6/CNB/Si0f5vz/yC8u35F6iqX9g3Z+9r18foEH9KP8AhHpT97W9RP0dR/7LRaH835h73Yu0VT/4Rz11fUj/ANtgP6Uf8I3H31PUj/28/wD1qPc7h73YuYowfSqf/CM2x632on63TUf8IvYnrcXx+t0/+NHud/wD3+xcwfQ0uD6GqX/CK6b3a6P1uX/xo/4RTSe8cx+tw/8AjRen3f3f8Edp9vx/4Bn65exeG7iPXnJETFYLmNRkyg/dIHdh/Krug67pHiSaS9sXk8+3Xy3jlUqyAnOcdOcdR6VQ1/wFY6pp3lWTG1uY3DxyOzOuehBBPQg0vgvwfL4aNzcXdyk9xcBVxECFRRk9+SSTVSdN073fN+hEVUU7W0/U6S2torSMxwghC7PgnOCTk4/E0VNRXO3c6NgpNo3bsDOMZoopALRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAf//Z)
Given: In ∆ABC,
MN ∥ AB
NC:NA = 1:3
⇒ ![](data:image/png;base64,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)
CM = 4
To find: BC = ?
In ∆ABC,
Since, MN ∥ AB
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ MB = 4 × 3 [∵,
⇒
]
⇒ MB = 12
In B – M – C,
BC = MB + CM
⇒ BC = 12 + 4
⇒ BC = 16
Thus, option (b) is correct.
Question 20.In ΔXYZ and ΔPQR, XYZ ↔ PQR is similarity. XY = 12, YZ = 8, ZX = 16, PR = 8. So, PQ + QR = ………….
A. 20
B. 10
C. 15
D. 9
Answer:For ∆XYZ and ∆PQR, the correspondence XYZ ↔ PQR has similarity.
Also, given that
XY = 12,
YZ = 8,
ZX = 16 &
PR = 8
To find: PQ + QR = ?
Now by definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Or ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ PQ + QR = 10
Thus, option (b) is correct.
Question 21.The bisector of ∠P in ΔPQR intersects
in D. If QD: RD = 4 : 7 and PR = 14, then PQ = ………..
A. 8
B. 4
C. 12
D. 16
Answer:We have
![](data:image/jpeg;base64,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)
Given: PD is the bisector of ∠P.
QD:RD = 4:7
PR = 14
To find: PQ = ?
In ∆PQR, the bisector of P intersects the line QR at D. By property of triangles,
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGUAAAAgCAMAAAA8N0YiAAAAAXNSR0IArs4c6QAAAI1QTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOjoAOjpmOjqQOmaQOma2OpDbZgAAZgA6ZjoAZjo6ZjpmZmZmZmaQZpDbZrbbZrb/kDoAkDo6kDqQkGY6kGaQkNv/tmYAtpA6tpBmttv/tv//25A627Zm27aQ29uQ29u22////7Zm/9uQ/9u2//+2///bFw7TIgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABq0lEQVRIS+2U7XaCMAyGW6fYfShzX1i3yaadGwV6/5e3NKUFEaFH0F/mDz2a5GnepCHkav4KpGzp73yqp1rdXoAiIn5+ipyp81Py563i0eepcnvGCaot8PTu4YaKKY64u41OtFtQOo78Uu6e1tZRjNzxMPaX3WwJycMppMejGEVEfdG5ByZbUB2BlrI2ivFRHChE0CV46/yKd8eQ9GEjLCUPX1wEdBuTZsXX3ddQJFQQG2d99DBHiefSURSfaEwezmoZXC2Km9ulTGM7zVLkfVJSQIkgMU3YN6TsWJDkITjgRao+MQ6HtpqQBSV/XJMKRWP+FlNMVDUcstFM36CJcrQmQ1Ec5AWKcGsE8pk82ooL2u5jwAmKSVtpM8Xd0fRFm+t+db11KaYDDxQri7GYklIMV+weQusEuBnbo9ju1xtTaXZM8VX6berKgy+PCKkPEHYDtllBVt+M0knLtihl/gBPOn7DH6A3dv6yV5MqLb5H1EhhpFuFGuZPSadJZu44gJld3PDKfxiFxzOQqfeEqJVPD/oCZX2n9U3YFH+hUrx2b8/6rqX0FPCc4f81UCTeAZnTbAAAAABJRU5ErkJggg==)
⇒ PQ = 2 × 4
⇒ PQ = 8
Thus, option (a) is correct.
Question 22.The lengths of the sides
,
,
of ΔABC are in the ratio 3 : 4 : 5. Correspondence ABC ↔ PQR is similarity. If PR = 12, the perimeter of ΔPQR is ……….
A. 12
B. 36
C. 24
D. 27
Answer:Given: In ∆ABC,
BC:CA:AB = 3:4:5
From ∆ABC and ∆PQR, the correspondence ABC ↔ PQR is similarity.
In ∆PQR,
PR = 12
To find: Perimeter of ∆PQR = ?
In ∆ABC, since we have BC:CA:AB = 3:4:5
Let the lengths of BC, CA and AB be 3t, 4t and 5t respectively. (where, t > 0)
Perimeter of ∆ABC = 3t + 4t + 5t
⇒ Perimeter of ∆ABC = 12t, t > 0
Also, from given we have,
In ∆ABC and ∆PQR, the correspondence ABC ↔ PQR is similarity.
So, from property we can say that,
Ratio of perimeters of ∆ABC and ∆PQR = Ratio of their corresponding sides
⇒ ![](data:image/png;base64,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)
Substituting the given values, we get
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ Perimeter of ∆PQR = 36
Thus, option (b) is correct.
Question 23.The correspondence ABC ↔ YZX in ΔABC and ΔXYZ is similarity. m∠B + m∠C = 120. So, m∠Y = ………
A. 70
B. 55
C. 110
D. 60
Answer:Given that,
From ∆ABC and ∆XYZ, the correspondence ABC ↔ YZX is similarity.
Also, m ∠B + m ∠C = 120°
To find: m ∠Y = ?
In ∆ABC, by angle sum property of triangles we can say that,
m ∠A + m ∠B + m ∠C = 180°
⇒ m ∠A + (m ∠B + m ∠C) = 180°
⇒ m ∠A + 120° = 180°
⇒ m ∠A = 180° - 120°
⇒ m ∠A = 60° …(i)
Now, from ∆ABC and ∆XYZ, the correspondence ABC ↔ YZX is similarity.
Now by definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
⇒ ∠A ≅ ∠Y
∴ m ∠A = m ∠Y
From equation (i), we get
m ∠A = m ∠Y = 60°
∴ m ∠Y = 60°
Thus, option (d) is correct.
Question 24.Correspondence ABC ↔ DEF of
ABC and ΔDEF is similarity. If AB + BC = 10 and DE + EF = 12 and AC = 6, then DF………..
A. 6
B. 5
C. 7.2
D. 16
Answer:Given: AB + BC = 10,
DE + EF = 12 &
AC = 6
The correspondence ABC ↔ DEF is similarity between ∆ABC and ∆DEF.
To find: DF = ?
From ∆ABC and ∆DEF, the correspondence ABC ↔ DEF is a similarity.
Now by definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
By reciprocating on both sides,
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ DF = 7.2
Thus, option (c) is correct.
Question 25.The lengths of the sides of ΔDEF are 4, 6, 8. ΔDEF ~ ΔPQR for correspondence DEF ↔ QPR. If the perimeter of ΔPQR = 36, then the length of the smallest side of ΔPQR is ………..
A. 6
B. 2
C. 4
D. 8
Answer:Given: Lengths of sides of ∆DEF are 4, 6 and 8.
∆DEF ∼ ∆PQR for the correspondence DEF ↔ PQR.
Perimeter of ∆PQR = 36
To find: length of the smallest side of ∆PQR = ?
Since, sides of ∆DEF are 4, 6 and 8.
⇒ Perimeter of ∆DEF = 4 + 6 + 8
⇒ Perimeter of ∆DEF = 18
Since, ∆DEF ∼ ∆PQR for the correspondence DEF ↔ PQR.
∴ ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAARQAAAAjCAMAAABipgXqAAAAAXNSR0IArs4c6QAAAKVQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjo6OjpmOjqQOmaQOma2OpC2OpDbZgAAZjoAZjo6ZpC2ZpDbZrbbZrb/kDoAkDo6kDpmkGYAkGY6kLyQkLbbkLb/kNv/tmYAtmY6tmZmtpA6tpBmttv/tv//25A625Bm27Zm27aQ29u229vb2////7Zm/7aQ/9uQ/9u2//+2///b3XBCeQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAEFElEQVRoQ+2ZC3uaMBiFEzqtdpc6tat1W9dp11bqNhiX///Tdk4uCEqMpaxzz8jzoCAJeXPyfQGOQnSlU+AgBZKhlDK43Kob9qODWlcqPQ7llfohHOBDX3gU6R0px+ZbV3GX9COa1VUK5UBw21sMuKteBXEPYXyyEmszmqI/hyhqtK4S91brBU9mF69X+OKFvw+h7iN2RDgWcbAQ+dIjSj4fRY5KS/TOrVK2kCz4Tj3VaAvRTcgz+Xx8UGDUd2WaFkLG7+ccOS+MMLnSO/wBonhLNnGqxvHvTIsDqX76thDdhFaUDGE7ErHs3w/7a9mP1F5wcyeDr5h8dfIOWdCPyhUjDDKdSnkeoaU08fZtpZBUl5SbOwwQinLr0yWfBzdsXO49vUb2aUG4aQCBVJCXGolSq0M0ixQQ08zUcyO6CXnmMViQPsUshcEieRdRUuyJJbeBGhpPclqqFXk8iNIJec06lIxEwvwxXQ6UXNSL3/5gya9lD8qVes8mM3FLIC2KAUjOADsu8skcAoJA+ecNtBvRTUhUUOhlcGyGpkSx26A4SVGqFXF8xiFAWStKeAVuarATKfjBGymc12tIWOpdpd5GFAOAkPqAELHpYw7RTANtoN2IbkKT8epSKHpo26KYk0qUSkXboxEQTfNPiBJWrGZsrNPooMJYKIuyfqNyxESKBRA/prhLbdYUdVgWxYC6Ed2E8Ym+ZUwQdPeLGlFuTyN7ctmPflYr6obpfJM+MdJdTSy7TKdmJ1/qu4+3JG8f1OpsRUHvbBf2EGWnETcDgC4oHpFMKBkts8kIQBtoN6KTEMGoJzCZymDG5FApEtxwD6k15mZO8vfLSkXTUOJxBPXU7XKpnkqwMjEfXyHC7fNJ0dNeYdIL9dhU7j2by979DswvXB9KKSQUfUhwEQ/l+Z20oG7EL80IvfPaVegU6BToFDgaBfSSrJflo4FSIAXYS+14hv9SGHv6Oa4J6mjsE80xpk83O50CnQKNFfgDHm3JouVrCV6q5Dnf8njvxUuSpzzLo32KRWuIahFb92hLFi2932Q4E+kcL53aEfX5ns/zaJ9k0SqiesTWPdqSRave7bW1r61avxncjkd7kEVbEO0itu7RlixaaJBNGBv8VPPifXZt6tE2sGitKDWIbXu0ZYsW3q/uEb6psmh7D74lRTTzaJtYtEqUesS2PdrCotXeb2UalMHtLU082iYWrZ6mWsSWPdqqRcsYoelr15Rw+1+3WoUaebRPt2itaVyD2LJHW7Jo9atvLGdaGVqt2eTEk0ANPVrj2OLuc7hFC1HqEXmzbtWjLVm0xrxf4wFF/6VMR978deXMoYYebQOLli5yPaI3wduogKUkSv1LbBtdNb3GX0DEQy3+WD3q8g8gHrV+/yfcb+nQ/fX4o8vDAAAAAElFTkSuQmCC)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ Smallest side of ∆PQR = 4 × 2
⇒ Smallest side of ∆PQR = 8
Thus, option (d) is correct.
Question 26.The bisector of ∠B intersects
in D. If BA = 12 and BC = 16, AD = 9, then AC = ……
A. 15
B. 21
C. 18
D. 8
Answer:We have
![](data:image/jpeg;base64,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)
Given: ∠ABD = ∠CBD
BA = 12,
BC = 16 &
AD = 9
To find: AC = ?
∵, In ∆ABC bisector of ∠B intersects AC at D.
By theorem of proportionality, we have
![](data:image/png;base64,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)
By reciprocating on both sides,
⇒ ![](data:image/png;base64,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)
Substituting the values, we get
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ CD = 12
We can express AC as,
AC = AD + CD
⇒ AC = 9 + 12
⇒ AC = 21
Thus, option (b) is correct.
Question 27.In Δ ABC,
||
, P ∈
, Q ∈
. If 4AP = AB and AQ = 4, then AC =
A. 12
B. 4
C. 8
D. 16
Answer:We have,
![](data:image/jpeg;base64,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)
Given: In ∆ABC,
PQ ∥ BC,
4 AP = AB &
AQ = 4
To find: AC = ?
We know, 4 AP = AB
⇒
…(i)
In ∆ABC, we know that PQ ∥ BC.
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ AC = 4 × 4
⇒ AC = 16
Thus, the option (d) is correct.
Question 28.In ΔABC, the correspondence ABC ↔ BAC is similarity …….. of the following is true.
A. ∠B ≅ ∠C
B. ∠C ≅ ∠A
C. ∠A ≅ ∠ B
D. ∠A ≅ ∠B ≅ ∠C
Answer:Given: In ∆ABC, the correspondence ABC ↔ BAC is similarity.
Now by definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
∠A ≅ ∠B, ∠B ≅ ∠A and ∠C ≅ ∠C are all true.
Thus, option (c) is true.
Question 29.In ΔABC, A—P—B, A—Q—C and
||
. If PQ = 5, AP = 4, AB = 12, then BC = …….
A. 9.6
B. 20
C. 15
D. 5
Answer:We have
![](data:image/jpeg;base64,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)
In ∆ABC,
PQ ∥ BC
PQ = 5,
AP = 4 &
AB = 12
In ∆ABC and ∆APQ, we have
∠A = ∠A [∵, ∠BAC and ∠PAQ are equal angles as they are same angles]
∠ABC = ∠APQ [∵, Alternate angles are equal]
∠ACB = ∠AQP [∵, Alternate angles are equal]
⇒ By AAA-corollary of similarity, ∆ABC ∼ ∆APQ for the correspondence ABC ↔ APQ.
Now by definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ BC = 5 × 3
⇒ BC = 15
Thus, option (c) is correct.
Question 30.In Δ PQR, P—M—Q and P—N—R. If PQ = 18, PM = 12, PR = 9 and NR = …, then
|| ![](data:image/png;base64,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)
A. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB4AAAA0CAYAAACdB4jyAAAAAXNSR0ICQMB9xQAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAAFwSURBVFjD7ZhBbsJADEV9AC6A1OmOAxCLLct0VHEAkogdYoVmwwGG7CJVvUB3LHuGcgLu0BvkDi0RCkRRMrGDBaidhXeWX/zH8egPpGkK9wjwYA++C9gkk7kCyAHgB0DlCuP3VZYNqjnW6PElpx4q18aOWeBtgrPGYk96X4WvQ7Vphh4D4x2r4yxbDbRSn5PEzC/d69eyM0y2s3Me4ltk7KheI0bYlXlkcCGfjuy0TYWyoDXRqAlq7WIYQPC1sHYoMlwnWfHbVfD8gQ6Z2eBCPhWuN5Q8l8ws8Gl6g0NXtxSZyeBy2Np+Da7MZLAJcdklHUdmErgYKMq5cmTu3lxFp6FZ1mV/ecaPpuJUmZ3gPhuJKnMr2AmtLJC+Mvtr0YM9+A+CXUvj2vBSe7AHPw6YYuTEwVQjJwqmGjlxMNXI3Wy4qEZOHEw1cqJgqpETBXOMnCiYY+TEwBwjJ7e5mEZO8A2E/7R0FbiPkfPX4v8G/wLkJfXOXe/PWAAAAABJRU5ErkJggg==)
B. 3
C. 24
D. 6
Answer:We have
![](data:image/jpeg;base64,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)
Given: P – M – Q and P – N – R
In ∆PQR, MN ∥ QR.
PQ = 18,
PM = 12 &
PR = 9
To find: NR = ?
In ∆PQR, when MN ∥ QR.
⇒ ![](data:image/png;base64,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)
Substituting the given values in the above equation,
![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ PN = 6
Observe the diagram,
PR = PN + NR
⇒ NR = PR – PN
⇒ NR = 9 – 6
⇒ NR = 3
Thus, option (b) is correct.
In ΔABC, such that
are positive real numbers. A line passing through P and parallel to
intersects
in Q. Prove that (m + n)2 (area of ΔAPB) = m2(area of ΔABC)
Answer:
We have
Given: In ∆ABC, such that
, where m and n are positive real numbers.
QP ∥ BC
Prove that: (m + n)2 (area of ∆APB) = m2 (area of ∆ABC)
Proof: Since
[by invertendo componendo invertendo rule]
⇒
In ∆APQ and ∆ABC,
∠APQ ≅ ∠ABC [∵, corresponding angles]
∠AQP ≅ ∠ACB [∵, corresponding angles]
⇒ The correspondence APQ ↔ ABC is a similarity.
Remember the property, areas of similar triangles are proportional to the squares of the corresponding sides.
∴
⇒
⇒
⇒
By cross-multiplication, we get
(m + n)2 (Area of ∆APB) = m2 (Area of ∆ABC)
Thus, proved.
Question 2.
D, E and F are the mid-points of,
and
respectively in ΔABC. Prove that the area of □ BDEF = 1/2 area of ΔABC.
Answer:
We have
Given: D is the midpoint of , E is the midpoint of
& F is the midpoint of
.
⇒ AF = FB, AE = EC & BD = DC
To Prove: Area of □BDEF = 1/2 Area of ∆ABC
Proof: Since, given is that D, E and F are midpoints of respectively.
We know, AE = EC.
⇒ AE = FD [∵, EC = FD]
⇒ [∵,
] …(i)
Also, we know that, AF = FB.
⇒ AF = ED [∵, FB = ED]
⇒ [∵,
] …(ii)
Now, in ∆AFE and ∆DEF, we have
AF ≅ ED, AE ≅ FD & FE ≅ EF
⇒ By SSS theorem of congruence, we can say that the congruency AFE ↔ DEF is a similarity.
Similarly, in ∆AFE and ∆FBD, we have
AF ≅ FB, AE ≅ FD & FE ≅ BD
⇒ By SSS theorem of congruence, we can say that the congruency AFE ↔ FBD is a similarity.
Similarly, the correspondence AEF ↔ EDC is a similarity.
So, ∆AFE, ∆DEF, ∆FBD and ∆EDC are all congruent triangles.
⇒ ∆AFE = ∆DEF = ∆FBD = ∆EDC …(iii)
Now,
BDEF = ∆DEF + ∆FBD
⇒ BDEF = ∆AFE + ∆AFE [from equation (iii)]
⇒ BDEF = 2 ∆AFE …(iv)
And
∆ABC = ∆AFE + ∆DEF + ∆FBD + ∆EDC
⇒ ∆ABC = ∆AFE + ∆AFE + ∆AFE + ∆AFE [from equation (iii)]
⇒ ∆ABC = 4 ∆AFE
⇒ ∆ABC = 2 (2 ∆AFE) …(v)
Comparing equations (iv) and (v), we get
∆ABC = 2 (BDEF)
⇒ BDEF = 1/2 ∆ABC
⇒ Area of □BDEF = 1/2 (Area of ∆ABC)
Thus, proved.
Question 3.
Can two similar triangles have same area? If yes, in which case they have the same area?
Answer:
Yes, two similar triangles can have same area.
Note, when two triangles are congruent then all corresponding sides as well as corresponding angles of one triangle are equal to those of the other triangle.
⇒ If two triangles are congruent, they are similar too.
And congruent triangles have same area too.
Hence, the similar triangles need to be congruent to have similar areas.
Question 4.
The correspondence ABC ↔ DEF is similarity in ΔABC and ΔDEF. is an altitude of ΔABC and
is an altitude of ΔDEF. Prove that AB X DN = AM X DE.
Answer:
We have
Given: The correspondence ABC ↔ DEF is similarity in ∆ABC and ∆DEF.
is an altitude of ∆ABC and
is an altitude of ∆DEF.
To Prove: AB × DN = AM × DE
Proof: Since, the correspondence ABC ↔ DEF is a similarity.
By definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
So, we can write ∠B ≅ ∠E. …(i)
In ∆ABM and ∆DEN,
From result (i),
∠B ≅ ∠E
Also, ∠M ≅ ∠N [since, they are right angles of ∆ABM and ∆DEN respectively]
⇒ By AA-corollary, the correspondence ABM ↔ DEN is a similarity in ∆ABC and ∆DEF]
Then, again by definition of similarity of correspondences in triangles we can say that,
By cross-multiplication, we get
⇒ AB × DN = AM × DE
Hence, proved.
Question 5.
Explain with reasons, whether the following statements are true or false:
(1) AAA criterion of similarity of triangles cannot be the criterion for congruence of triangles.
(2) SAS criterion for congruence of triangles cannot be a criterion for similarity of triangles.
(3) Two congruent triangles have the same area.
(4) Two similar triangles always have the same area.
(5) Area of similar triangles are proportional to the squares of measures of their corresponding angles.
Answer:
(1). Similarity of triangles and congruence of triangles are closely related, but has a fine line of difference.
Similar triangles are the same shape, but not necessarily the same size. The triangles are congruent if, in addition to this, their corresponding sides are of equal length.
By AAA criterion of similarity of triangles, we get the shapes of triangles equal but, for congruence of triangles their sizes must be equal too.
∴ AAA criterion of similarity of triangles cannot be the criterion for congruence of triangles.
Hence, the statement is true.
(2). For congruence of triangles, the corresponding sides are of equal length while for similarity of triangles, measure of the corresponding sides of two triangles must be proportional.
SAS criterion for congruence of triangles not only makes sure that the triangles have equal length of corresponding sides but also makes sure that the triangles have same shape.
∴ SAS criterion for congruence of triangles can be a criterion for similarity of triangles.
Hence, the statement is false.
(3). Congruency of triangles ensure that the triangles have same shape as well as equal lengths of corresponding sides.
⇒ The shape and size of two congruent triangles are equal.
∴ Two congruent triangles have the same area.
Hence, the statement is true.
(4). Similarity of triangles indicate that they have the same shape but their sizes are not same.
Area of triangles are equal when their size are equal, shape of triangle cannot determine its area.
∴ Two similar triangles doesn’t always have the same area.
Hence, the statement is false.
(5). Area of similar triangles are proportional to the squares of their corresponding sides.
Since, the measure of corresponding angles of similar triangles are always equal.
∴ Area of similar triangles are not proportional to the squares of their corresponding angles.
Hence, the statement is false.
Question 6.
Fill in the blanks so that the following statements are true:and
are the altitudes of ΔABC. If AB = 12, AC = 9.9, AD = 8.1 and BE = 7.2, perimeter of ΔABC = _____.
Answer:
We have
Given that: AD and BE are altitudes of ∆ABC. ⇒ ∠ADC = ∠
In ∆ADC and ∆BEC,
∠ADC = ∠BEC [∵, ∠ADC = ∠BEC = 90° as AD and BE ]
∠ACD = ∠BCE [∵, ∠ACD and ∠BCE are same angles of the same triangles, so they obviously are equal]
⇒ By AA-corollary, we can say that ∆ADC ∼ ∆BEC for the correspondence of ADC ↔ BEC.
By definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
⇒
⇒
⇒
⇒ BC = 8.8
So, perimeter of ∆ABC is given by
Perimeter of ∆ABC = AB + BC + AC
⇒ Perimeter of ∆ABC = 12 + 8.8 + 9.9
⇒ Perimeter of ∆ABC = 30.7
Thus, perimeter of ∆ABC = 30.7.
Question 7.
Fill in the blanks so that the following statements are true:
D, E, F are respectively the mid-points of ,
,
of ΔPQR. The correspondence DEF ↔ _____ is similarity.
Answer:
We have
Given: D is the midpoint of PQ ⇒ PD = DQ = 1/2 PQ
But PD = DQ = FE [∵, F and E are also midpoints of PR and QR respectively]
So, EF = 1/2 PQ
Or …(i)
E is the midpoint of QR ⇒ QE = ER = 1/2 RQ
But QE = ER = DF [∵, D and F are also midpoints of PQ and PR respectively]
So, DF = 1/2 RQ
Or …(ii)
F is the midpoint of PR ⇒ PF = FR = 1/2 RP
But PF = FR = DE [∵, D and E are midpoints of PQ and QR respectively]
So, DE = 1/2 RP
Or …(iii)
Comparing equations (i), (ii) & (iii), we get
We can also arrange it as,
∴ The correspondence DEF ↔ RPQ is a similarity by SSS theorem.
Thus, answer is RPQ.
Question 8.
Fill in the blanks so that the following statements are true:
In ΔABC, and
are altitudes. If AB = 12, BC = 15 and AM = 9.6, then CN = _____.
Answer:
We have
Given: AM ⊥ BC and CN ⊥ AB.
AB = 12, BC = 15 & AM = 9.6
To find: CN = ?
For this,
Take ∆AMB and ∆CNB,
∠AMB = ∠CNB [∵, ∠AMB = ∠CNB = 90°]
∠ABM = ∠CBN [∵, ∠AMB and ∠CBN are same angles of the same triangle ABC]
⇒ By AA corollary, ∆AMB ∼ ∆CNB for the correspondence AMB ↔ CNB.
By definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
⇒
Or
⇒
⇒
⇒
⇒ CN = 12
Thus, answer is 12.
Question 9.
Fill in the blanks so that the following statements are true:
Areas of two similar triangles are 25 and 16. The ratio of the perimeters of the triangles is _______.
Answer:
Given: Area of two similar triangles.
Area of ∆1 = 25
Area of ∆2 = 16
Recall two properties:
First,
Ratio of areas of two similar triangles = Ratio of the squares of the corresponding sides
Second,
Ratio of perimeter of two similar triangles = Ratio of corresponding sides
Using the first property, we can write that
⇒
Taking under root of both sides, we get
⇒
⇒
Now, using second property, we can say
Thus, answer is 5/4.
Question 10.
Fill in the blanks so that the following statements are true:
Area of ΔABC = 36 and area of ΔPQR = 64. The correspondence ABC ↔ PQR is a similarity. If AB = 12, then PQ = ______.
Answer:
Given: Area of ∆ABC = 36,
Area of ∆PQR = 64
AB = 12
The correspondence ABC ↔ PQR is a similarity.
To find: PQ = ?
For this,
Recall the property,
Ratio of areas of two similar triangles = Ratio of squares of the corresponding sides
⇒
⇒
⇒
⇒
⇒
⇒
⇒ PQ = 2 × 8
⇒ PQ = 16
Thus, answer is 16.
Question 11.
Fill in the blanks so that the following statements are true:
In ΔABC, A—M—B, A—N—C and ||
. If AM = 8.4, AN = 6.4, CN = 19.2, then AB = ______.
Answer:
We have
Given: A – M – B, A – N – C
MN ∥ BC
AM = 8.4
AN = 6.4
CN = 19.2
To find: AB = ?
Take ∆ABC,
We know that, MN ∥ BC. So, by property we can write as:
⇒
⇒
⇒
⇒ BM = 25.2
We know AM = 8.4 and BM = 25.2.
Now, AB = AM + BM [∵, from the diagram]
⇒ AB = 8.4 + 25.2
⇒ AB = 33.6
Thus, the answer is 33.6.
Question 12.
Fill in the blanks so that the following statements are true:
The correspondence ABC ↔ PQR is a similarity in Δ ABC and ΔPQR. AB = 16, AC = 8, PQ = 24, BC = 12, then QR + PR = ____.
Answer:
Given: The correspondence ABC ↔ PQR is a similarity in ∆ABC and ∆PQR.
AB = 16,
AC = 8,
PQ = 24 &
BC = 12
To find: QR + PR = ?
By definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
⇒
⇒
⇒
To find (QR + PR), cross multiply it.
⇒ (QR + PR) × AB = (BC + AC) × PQ
Substituting values,
⇒ (QR + PR) × 16 = (12 + 8) × 24
⇒ (QR + PR) × 16 = 20 × 24
⇒
⇒ QR + PR = 30
Thus, the answer is 30.
Question 13.
Fill in the blanks so that the following statements are true:
The correspondence ABC ↔ XYZ is a similarity in ΔABC and ΔXYZ. ABC = 72, BC = 6, YZ = 10. Then XYZ = ______.
Answer:
Given: The correspondence ABC ↔ XYZ is a similarity in ∆ABC and ∆XYZ.
ABC = 72,
BC = 6 &
YZ = 10
To find: XYZ = ?
So, by property we can say that,
Ratio of areas of two similar triangles = Ratio of squares of the corresponding sides
⇒
⇒
Taking reciprocal of both sides, we get
Substituting values in the above equation, we get
⇒
⇒
⇒
⇒ XYZ = 200
Thus, the answer is 200.
Question 14.
Fill in the blanks so that the following statements are true:
□ ABCD is trapezium in which ||
. The diagonals intersect in P. If PD = 9, AP = 5, PB = 7.2, then AC = ______.
Answer:
We have
Given: ABCD is a trapezium.
AD ∥ BC
PD = 9,
AP = 5 &
PB = 7.2
To find: AC = ?
We have got the trapezium ABCD, in which
AD ∥ BC & P is the intersection point of AC and BD.
Take ∆PAD and ∆PCB,
∠APD = ∠CPB [∵, Vertically opposite angles are equal]
∠PAD = ∠PCB [∵, alternate angles]
∠PDA = ∠PBC [∵, alternate angles]
⇒ By AAA similarity theorem, ∆PAD ∼ ∆PCB for the correspondence PAD ↔ PCB.
Now by definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
⇒
Reciprocating it, we get
⇒
⇒
⇒ PC = 4
From the diagram, note that
AC = AP + PC
⇒ AC = 5 + 4
⇒ AC = 9
Thus, the answer is 9.
Question 15.
Fill in the blanks so that the following statements are true:
In ΔABC, m∠B = 90 and is an altitude. The correspondence BDA ↔ _____ between ΔBDA and ΔBDC is a similarity.
Answer:
We have
Given: In ∆ABC,
∠ABC = ∠ADB = ∠BDC = 90°
In ∆BDA and ∆CDB,
∠BDA = ∠CDB = 90°
∠ABD = ∠BCD [measure of both being equal, i.e., (90° - m∠A)]
This can be explained as,
In ∆BDA, by angle sum property of triangle,
∠BDA + ∠DAB + ∠ABD = 180°
⇒ 90° + ∠A + ∠ABD = 180° [∵, ∠BDA = 90° & ∠DAB = ∠A]
⇒ ∠ABD = 180° - 90° - ∠A
⇒ ∠ABD = 90° - ∠A …(i)
Similarly, in ∆ABC by angle sum property of triangle,
∠ABC + ∠BAC + ∠BCA = 180°
⇒ 90° + ∠A + ∠BCD = 180° [∵, ∠ABC = 90°, ∠BAC = ∠A & ∠BCA = ∠BCD (from the figure)]
⇒ ∠BCD = 180° - 90° - ∠A
⇒ ∠BCD = 90° - ∠A …(ii)
By equations (i) and (ii), we get
∠ABD = ∠BCD
Hence, by AA corollary the correspondence BDA ↔ CDB is similarity between ∆BDA and ∆BDC.
Thus, the answer is CDB.
Question 16.
Fill in the blanks so that the following statements are true:
The correspondence ABC ↔ ZXY is a similarity in ΔABC and ΔXYZ. If AB = 12, BC = 9, CA = 7.5 and ZX = 10, then YZ + XY = ____.
Answer:
Given: The correspondence ABC ↔ ZXY is a similarity in ∆ABC and ∆XYZ.
AB = 12,
BC = 9,
CA = 7.5 &
ZX = 10
To find: YZ + XY = ?
Now by definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
⇒
⇒
⇒
Reciprocating both sides,
⇒
⇒
⇒
⇒
Thus, the answer is 13.75.
Question 17.
ABC ↔ DEF is a similarity in ΔABC and ΔADEF, m∠A = 40, m∠E + m∠F =
A. 40
B. 80
C. 140
D. 180
Answer:
Given: m ∠A = 40°
The correspondence ABC ↔ DEF is a similarity between ∆ABC and ∆DEF.
To find: m ∠E + m ∠F = ?
Now by definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
⇒ m ∠A = m ∠D
⇒ m ∠A = m ∠D = 40°
⇒ m ∠D = 40°
In ∆DEF,
m ∠D + m ∠E + m ∠F = 180°
⇒ 40° + m ∠E + m ∠F = 180°
⇒ m ∠E + m ∠F = 180° - 40°
⇒ m ∠E + m ∠F = 140°
∴ Answer is 140.
Thus, option (c) is correct.
Question 18.
In ΔABC, M ∈ , N ∈
such that
||
,……….is not true.
A. AN ⋅ MB = AM ⋅ NC
B. AM ⋅ MB = AN ⋅ NC
C. AB ⋅ AN = AM ⋅ AC
D. AB ⋅ NC = AC ⋅ MB
Answer:
We have the diagram as
Given: In ∆ABC, MN ∥ BC.
By property of triangles, we can say that
,
&
Cross multiplication of the above equations,
⇒ AM × NC = AN × MB
Or AM . NC = AN . MB is true.
AM × AC = AN × AB
Or AM . AC = AN . AB is true.
MB × AC = NC × AB
Or MB . AC = NC . AB is true.
So, AM . MB = AN . NC is not true.
Thus, option (b) is correct.
Question 19.
In ΔABC, B—M—C and A—N—C, ||
. If NC : NA = 1 : 3 and CM = 4, then BC = ………..
A. 12
B. 16
C. 8
D. 1/2
Answer:
We have the diagram as,
Given: In ∆ABC,
MN ∥ AB
NC:NA = 1:3
⇒
CM = 4
To find: BC = ?
In ∆ABC,
Since, MN ∥ AB
⇒
⇒
⇒ MB = 4 × 3 [∵, ⇒
]
⇒ MB = 12
In B – M – C,
BC = MB + CM
⇒ BC = 12 + 4
⇒ BC = 16
Thus, option (b) is correct.
Question 20.
In ΔXYZ and ΔPQR, XYZ ↔ PQR is similarity. XY = 12, YZ = 8, ZX = 16, PR = 8. So, PQ + QR = ………….
A. 20
B. 10
C. 15
D. 9
Answer:
For ∆XYZ and ∆PQR, the correspondence XYZ ↔ PQR has similarity.
Also, given that
XY = 12,
YZ = 8,
ZX = 16 &
PR = 8
To find: PQ + QR = ?
Now by definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
⇒
⇒
Or
⇒
⇒
⇒
⇒
⇒ PQ + QR = 10
Thus, option (b) is correct.
Question 21.
The bisector of ∠P in ΔPQR intersects in D. If QD: RD = 4 : 7 and PR = 14, then PQ = ………..
A. 8
B. 4
C. 12
D. 16
Answer:
We have
Given: PD is the bisector of ∠P.
QD:RD = 4:7
PR = 14
To find: PQ = ?
In ∆PQR, the bisector of P intersects the line QR at D. By property of triangles,
⇒
⇒
⇒
⇒ PQ = 2 × 4
⇒ PQ = 8
Thus, option (a) is correct.
Question 22.
The lengths of the sides,
,
of ΔABC are in the ratio 3 : 4 : 5. Correspondence ABC ↔ PQR is similarity. If PR = 12, the perimeter of ΔPQR is ……….
A. 12
B. 36
C. 24
D. 27
Answer:
Given: In ∆ABC,
BC:CA:AB = 3:4:5
From ∆ABC and ∆PQR, the correspondence ABC ↔ PQR is similarity.
In ∆PQR,
PR = 12
To find: Perimeter of ∆PQR = ?
In ∆ABC, since we have BC:CA:AB = 3:4:5
Let the lengths of BC, CA and AB be 3t, 4t and 5t respectively. (where, t > 0)
Perimeter of ∆ABC = 3t + 4t + 5t
⇒ Perimeter of ∆ABC = 12t, t > 0
Also, from given we have,
In ∆ABC and ∆PQR, the correspondence ABC ↔ PQR is similarity.
So, from property we can say that,
Ratio of perimeters of ∆ABC and ∆PQR = Ratio of their corresponding sides
⇒
Substituting the given values, we get
⇒
⇒ Perimeter of ∆PQR = 36
Thus, option (b) is correct.
Question 23.
The correspondence ABC ↔ YZX in ΔABC and ΔXYZ is similarity. m∠B + m∠C = 120. So, m∠Y = ………
A. 70
B. 55
C. 110
D. 60
Answer:
Given that,
From ∆ABC and ∆XYZ, the correspondence ABC ↔ YZX is similarity.
Also, m ∠B + m ∠C = 120°
To find: m ∠Y = ?
In ∆ABC, by angle sum property of triangles we can say that,
m ∠A + m ∠B + m ∠C = 180°
⇒ m ∠A + (m ∠B + m ∠C) = 180°
⇒ m ∠A + 120° = 180°
⇒ m ∠A = 180° - 120°
⇒ m ∠A = 60° …(i)
Now, from ∆ABC and ∆XYZ, the correspondence ABC ↔ YZX is similarity.
Now by definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
⇒ ∠A ≅ ∠Y
∴ m ∠A = m ∠Y
From equation (i), we get
m ∠A = m ∠Y = 60°
∴ m ∠Y = 60°
Thus, option (d) is correct.
Question 24.
Correspondence ABC ↔ DEF of ABC and ΔDEF is similarity. If AB + BC = 10 and DE + EF = 12 and AC = 6, then DF………..
A. 6
B. 5
C. 7.2
D. 16
Answer:
Given: AB + BC = 10,
DE + EF = 12 &
AC = 6
The correspondence ABC ↔ DEF is similarity between ∆ABC and ∆DEF.
To find: DF = ?
From ∆ABC and ∆DEF, the correspondence ABC ↔ DEF is a similarity.
Now by definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
⇒
⇒
⇒
By reciprocating on both sides,
⇒
⇒
⇒
⇒ DF = 7.2
Thus, option (c) is correct.
Question 25.
The lengths of the sides of ΔDEF are 4, 6, 8. ΔDEF ~ ΔPQR for correspondence DEF ↔ QPR. If the perimeter of ΔPQR = 36, then the length of the smallest side of ΔPQR is ………..
A. 6
B. 2
C. 4
D. 8
Answer:
Given: Lengths of sides of ∆DEF are 4, 6 and 8.
∆DEF ∼ ∆PQR for the correspondence DEF ↔ PQR.
Perimeter of ∆PQR = 36
To find: length of the smallest side of ∆PQR = ?
Since, sides of ∆DEF are 4, 6 and 8.
⇒ Perimeter of ∆DEF = 4 + 6 + 8
⇒ Perimeter of ∆DEF = 18
Since, ∆DEF ∼ ∆PQR for the correspondence DEF ↔ PQR.
∴
⇒
⇒
⇒
⇒ Smallest side of ∆PQR = 4 × 2
⇒ Smallest side of ∆PQR = 8
Thus, option (d) is correct.
Question 26.
The bisector of ∠B intersects in D. If BA = 12 and BC = 16, AD = 9, then AC = ……
A. 15
B. 21
C. 18
D. 8
Answer:
We have
Given: ∠ABD = ∠CBD
BA = 12,
BC = 16 &
AD = 9
To find: AC = ?
∵, In ∆ABC bisector of ∠B intersects AC at D.
By theorem of proportionality, we have
By reciprocating on both sides,
⇒
Substituting the values, we get
⇒
⇒
⇒ CD = 12
We can express AC as,
AC = AD + CD
⇒ AC = 9 + 12
⇒ AC = 21
Thus, option (b) is correct.
Question 27.
In Δ ABC, ||
, P ∈
, Q ∈
. If 4AP = AB and AQ = 4, then AC =
A. 12
B. 4
C. 8
D. 16
Answer:
We have,
Given: In ∆ABC,
PQ ∥ BC,
4 AP = AB &
AQ = 4
To find: AC = ?
We know, 4 AP = AB
⇒ …(i)
In ∆ABC, we know that PQ ∥ BC.
⇒
⇒
⇒
⇒ AC = 4 × 4
⇒ AC = 16
Thus, the option (d) is correct.
Question 28.
In ΔABC, the correspondence ABC ↔ BAC is similarity …….. of the following is true.
A. ∠B ≅ ∠C
B. ∠C ≅ ∠A
C. ∠A ≅ ∠ B
D. ∠A ≅ ∠B ≅ ∠C
Answer:
Given: In ∆ABC, the correspondence ABC ↔ BAC is similarity.
Now by definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
∠A ≅ ∠B, ∠B ≅ ∠A and ∠C ≅ ∠C are all true.
Thus, option (c) is true.
Question 29.
In ΔABC, A—P—B, A—Q—C and ||
. If PQ = 5, AP = 4, AB = 12, then BC = …….
A. 9.6
B. 20
C. 15
D. 5
Answer:
We have
In ∆ABC,
PQ ∥ BC
PQ = 5,
AP = 4 &
AB = 12
In ∆ABC and ∆APQ, we have
∠A = ∠A [∵, ∠BAC and ∠PAQ are equal angles as they are same angles]
∠ABC = ∠APQ [∵, Alternate angles are equal]
∠ACB = ∠AQP [∵, Alternate angles are equal]
⇒ By AAA-corollary of similarity, ∆ABC ∼ ∆APQ for the correspondence ABC ↔ APQ.
Now by definition, for a given correspondence between the vertices of two triangles, if the corresponding angles of the triangles are congruent and the lengths of the corresponding sides are in proportion, then the given correspondence is a similarity between two triangles.
⇒
⇒
⇒ BC = 5 × 3
⇒ BC = 15
Thus, option (c) is correct.
Question 30.
In Δ PQR, P—M—Q and P—N—R. If PQ = 18, PM = 12, PR = 9 and NR = …, then ||
A.
B. 3
C. 24
D. 6
Answer:
We have
Given: P – M – Q and P – N – R
In ∆PQR, MN ∥ QR.
PQ = 18,
PM = 12 &
PR = 9
To find: NR = ?
In ∆PQR, when MN ∥ QR.
⇒
Substituting the given values in the above equation,
⇒
⇒ PN = 6
Observe the diagram,
PR = PN + NR
⇒ NR = PR – PN
⇒ NR = 9 – 6
⇒ NR = 3
Thus, option (b) is correct.