Class 10th Mathematics Gujarat Board Solution
Exercise 7.1- ∠B is a right angle in ΔABC. bar bd perpendicular bar ac and D bar ac . If AD =…
- 5, 12, 13 are the lengths of the sides of a triangle. Show that the triangle is…
- In ΔPQR, bar qm is the altitude to hypotenuse bar pr ? If PM = 8, RM = 12, find…
- In ΔABC, m∠B = 90, bar bm perpendicular bar ac , M bar ac . If AM — MC = 7, AB^2…
- ∠A is right angle in ΔABC. AD is an altitude of the triangle. If AB = √5, BD =…
- m∠B = 90 in ΔABC. bar bm is altitude to bar ac . If AM = BM = 8, find AC.…
- m∠B = 90 in ΔABC. bar bm is altitude to bar ac . If BM = 15, AC = 34, find AB.…
- m∠B = 90 in ΔABC. bar bm is altitude to bar ac . If BM = 2√30, MC = 6, find AC.…
- m∠B = 90 in ΔABC. bar bm is altitude to bar ac . If AB = √10, AM = 2.5, find…
- In ΔPQR m∠Q = 90, PQ = x, QR = y and bar qd perpendicular bar pr . D in bar pr .…
- ∠Q is a right angle in ΔPQR and bar qm perpendicular bar pr , M in bar pr . If…
- âPQRS is a rectangle. If PQ + QR = 7 and PR + QS = 10, then find the area of â…
- The diagonals of a convex â ABCD intersect at right angles. Prove that AB^2 +…
- In ΔPQR, m∠Q = 90, M in bar qr and N in bar pq . Prove that PM^2 + RN^2 = PR^2…
- The sides of a triangle have lengths a^2 + b^2 , 2ab, a^2 — b^2 , where a b and…
- In ΔABC, m∠B = 90 and bar be is a median. Prove that AB^2 + BC^2 + AC^2 = 8BE^2…
- AB = AC and ∠A is right angle in ΔABC. If BC = √2 a, then find the area of the…
Exercise 7.2- In rectangle ABCD, AB + BC = 23, AC + BD = 34. Find the area of the rectangle.…
- In ΔABC m∠A = m∠B + m∠C, AB = 7, BC = 25. Find the perimeter of ΔABC.…
- A staircase of length 6.5 meters touches a wall at height of 6 meter. Find the…
- In ΔABC AB = 7, AC = 5, AD = 5. Find bar bc , if the mid - point of BC is D.…
- In equilateral ΔABC, D in bar bc such that BD : DC = 1 : 2. Prove that 3AD = √7…
- In ΔABC, AB = 17, BC = 15, AC = 8, find the length of the median on the largest…
- bar ad is a median of ΔABC. AB^2 + AC^2 = 148 and AD = 7. Find BC.…
- In rectangle ABCD, AC = 25 and CD = 7. Find perimeter of the rectangle.…
- In rhombus XYZW, XZ = 14 and YW = 48. Find XY.
- In ΔPQR, m∠Q : m∠R : m∠P = 1 : 2 : 1. If PQ = 2√6, find PR.
Exercise 7- bar ad , bar be , bar cf are the medians of ΔABC. If BE = 12, CF = 9 and AB^2 +…
- bar ad is the altitude of Δ ABC such that B—D—C. If AD^2 = BD ⋅ DC, prove that…
- In ΔABC, bar ad perpendicular bar bc , B—D—C. If AB^2 = BD ⋅ BC, prove that ∠…
- In ΔABC, bar ad perpendicular bar bc , B—D—C. If AC^2 = CD ⋅ BC, prove that ∠BAC…
- bar ad is a median of ΔABC. If BD = AD, prove that ∠A is a right angle in ΔABC.…
- In figure 7.25, AC is the length of a pole standing vertical on the ground. The…
- In ΔABC, AB AC, D is the mid-point of bar bc . bar am perpendicular vector bc…
- In ΔABC, bar bd perpendicular bar ac , D e bar ac and ∠B is right angle. If AC =…
- In ΔPQR, if m∠P + m∠Q = m∠R. PR = 7, QR = 24, then PQ =A. 31 B. 25 C. 17 D. 15…
- In ΔABC, bar ad is an altitude and ∠A is right angle. If AB = root 20 , BD = 4,…
- In ΔABC, AB^2 + AC^2 = 50. The length of the median AD = 3. So, BC =A. 4 B. 24…
- In ΔABC, m∠B = 90, AB = BC. Then AB: AC =A. 1:3 B. 1:2 C. 1 : √2 D. √2:1…
- In ΔABC, m∠B = 90 and AC = 10. The length of the median BM =A. 5 B. 5√2 C. 6 D.…
- In ΔABC, AB = BC = dc/root 2 . m∠BA. Is acute B. Is obtuse C. Is right angle D.…
- In ΔABC, if ab/1 = ac/2 = bc/root 3 then m ∠C = ……A. 90 B. 30 C. 60 D. 45…
- In ΔXYZ, m∠X : m∠Y : m∠Z = 1 : 2 : 3. If XY = 15, YZ = …A. 15 root 3/2 B. 17 C.…
- In Δ ABC, ∠B is a right angle and bar bd is an altitude. If AD = BD = 5, then…
- In Δ ABC, bar ad is median. If AB^2 + AC^2 = 130 and AD = 7, then BD =A. 4 B. 8…
- The diagonal of a square is 5√2. The length of the side of the square isA. 10…
- The length of a diagonal of a rectangle is 13. If one of the side of the…
- The length of a median of an equilateral triangle is √3. Length of the side of…
- The perimeter of an equilateral triangle is 6. The length of the altitude of…
- In ΔABC, m∠A = 90. bar ad is a median. If AD = 6, AB = 10, then AC =A. 8 B. 7.5…
- In Δ PQR, m∠Q = 90 and PQ = QR. bar qm perpendicular bar pr , in bar pr . If QM…
- In Δ ABC, m∠A = 90, bar ad is an altitude. So AB^2 = ……A. BD.BC B. BD.DC C.…
- In Δ ABC, m∠A = 90, bar ad is an altitude. Therefore BD.DC = ……A. AB^2 B. BC^2…
- ∠B is a right angle in ΔABC. bar bd perpendicular bar ac and D bar ac . If AD =…
- 5, 12, 13 are the lengths of the sides of a triangle. Show that the triangle is…
- In ΔPQR, bar qm is the altitude to hypotenuse bar pr ? If PM = 8, RM = 12, find…
- In ΔABC, m∠B = 90, bar bm perpendicular bar ac , M bar ac . If AM — MC = 7, AB^2…
- ∠A is right angle in ΔABC. AD is an altitude of the triangle. If AB = √5, BD =…
- m∠B = 90 in ΔABC. bar bm is altitude to bar ac . If AM = BM = 8, find AC.…
- m∠B = 90 in ΔABC. bar bm is altitude to bar ac . If BM = 15, AC = 34, find AB.…
- m∠B = 90 in ΔABC. bar bm is altitude to bar ac . If BM = 2√30, MC = 6, find AC.…
- m∠B = 90 in ΔABC. bar bm is altitude to bar ac . If AB = √10, AM = 2.5, find…
- In ΔPQR m∠Q = 90, PQ = x, QR = y and bar qd perpendicular bar pr . D in bar pr .…
- ∠Q is a right angle in ΔPQR and bar qm perpendicular bar pr , M in bar pr . If…
- âPQRS is a rectangle. If PQ + QR = 7 and PR + QS = 10, then find the area of â…
- The diagonals of a convex â ABCD intersect at right angles. Prove that AB^2 +…
- In ΔPQR, m∠Q = 90, M in bar qr and N in bar pq . Prove that PM^2 + RN^2 = PR^2…
- The sides of a triangle have lengths a^2 + b^2 , 2ab, a^2 — b^2 , where a b and…
- In ΔABC, m∠B = 90 and bar be is a median. Prove that AB^2 + BC^2 + AC^2 = 8BE^2…
- AB = AC and ∠A is right angle in ΔABC. If BC = √2 a, then find the area of the…
- In rectangle ABCD, AB + BC = 23, AC + BD = 34. Find the area of the rectangle.…
- In ΔABC m∠A = m∠B + m∠C, AB = 7, BC = 25. Find the perimeter of ΔABC.…
- A staircase of length 6.5 meters touches a wall at height of 6 meter. Find the…
- In ΔABC AB = 7, AC = 5, AD = 5. Find bar bc , if the mid - point of BC is D.…
- In equilateral ΔABC, D in bar bc such that BD : DC = 1 : 2. Prove that 3AD = √7…
- In ΔABC, AB = 17, BC = 15, AC = 8, find the length of the median on the largest…
- bar ad is a median of ΔABC. AB^2 + AC^2 = 148 and AD = 7. Find BC.…
- In rectangle ABCD, AC = 25 and CD = 7. Find perimeter of the rectangle.…
- In rhombus XYZW, XZ = 14 and YW = 48. Find XY.
- In ΔPQR, m∠Q : m∠R : m∠P = 1 : 2 : 1. If PQ = 2√6, find PR.
- bar ad , bar be , bar cf are the medians of ΔABC. If BE = 12, CF = 9 and AB^2 +…
- bar ad is the altitude of Δ ABC such that B—D—C. If AD^2 = BD ⋅ DC, prove that…
- In ΔABC, bar ad perpendicular bar bc , B—D—C. If AB^2 = BD ⋅ BC, prove that ∠…
- In ΔABC, bar ad perpendicular bar bc , B—D—C. If AC^2 = CD ⋅ BC, prove that ∠BAC…
- bar ad is a median of ΔABC. If BD = AD, prove that ∠A is a right angle in ΔABC.…
- In figure 7.25, AC is the length of a pole standing vertical on the ground. The…
- In ΔABC, AB AC, D is the mid-point of bar bc . bar am perpendicular vector bc…
- In ΔABC, bar bd perpendicular bar ac , D e bar ac and ∠B is right angle. If AC =…
- In ΔPQR, if m∠P + m∠Q = m∠R. PR = 7, QR = 24, then PQ =A. 31 B. 25 C. 17 D. 15…
- In ΔABC, bar ad is an altitude and ∠A is right angle. If AB = root 20 , BD = 4,…
- In ΔABC, AB^2 + AC^2 = 50. The length of the median AD = 3. So, BC =A. 4 B. 24…
- In ΔABC, m∠B = 90, AB = BC. Then AB: AC =A. 1:3 B. 1:2 C. 1 : √2 D. √2:1…
- In ΔABC, m∠B = 90 and AC = 10. The length of the median BM =A. 5 B. 5√2 C. 6 D.…
- In ΔABC, AB = BC = dc/root 2 . m∠BA. Is acute B. Is obtuse C. Is right angle D.…
- In ΔABC, if ab/1 = ac/2 = bc/root 3 then m ∠C = ……A. 90 B. 30 C. 60 D. 45…
- In ΔXYZ, m∠X : m∠Y : m∠Z = 1 : 2 : 3. If XY = 15, YZ = …A. 15 root 3/2 B. 17 C.…
- In Δ ABC, ∠B is a right angle and bar bd is an altitude. If AD = BD = 5, then…
- In Δ ABC, bar ad is median. If AB^2 + AC^2 = 130 and AD = 7, then BD =A. 4 B. 8…
- The diagonal of a square is 5√2. The length of the side of the square isA. 10…
- The length of a diagonal of a rectangle is 13. If one of the side of the…
- The length of a median of an equilateral triangle is √3. Length of the side of…
- The perimeter of an equilateral triangle is 6. The length of the altitude of…
- In ΔABC, m∠A = 90. bar ad is a median. If AD = 6, AB = 10, then AC =A. 8 B. 7.5…
- In Δ PQR, m∠Q = 90 and PQ = QR. bar qm perpendicular bar pr , in bar pr . If QM…
- In Δ ABC, m∠A = 90, bar ad is an altitude. So AB^2 = ……A. BD.BC B. BD.DC C.…
- In Δ ABC, m∠A = 90, bar ad is an altitude. Therefore BD.DC = ……A. AB^2 B. BC^2…
Exercise 7.1
Question 1.∠B is a right angle in ΔABC.
and D
. If AD = 4DC, prove that BD = 2DC.
Answer:![](data:image/png;base64,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)
m∠B = 90
![](data:image/png;base64,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)
AD = 4DC (Given) …Equation (i)
∴ BD2 = AD × DC
⇒ BD2 = 4DC2 (From Equation (i))
Taking Square Root both sides we get
∴ BD = 2DC
Question 2.5, 12, 13 are the lengths of the sides of a triangle. Show that the triangle is right angled. Find the length of altitude on the hypotenuse.
Answer:
![](data:image/jpeg;base64,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)
Hypotenuse (AB) = 13
Perpendicular (AC) = 12
Base (BC) = 5
∴ AC2 + BC2 = 52 + 122
⇒ AC2 + BC2 = 25 + 144
⇒ AC2 + BC2 = 169 = AB2
Hence Pythagoras theorem is valid
m∠C = 90
Let AD = x, BD = 13 – x
Applying Pythagoras theorem
∴ Altitude CD2 = 122 – x2 = 52 – (13 – x) 2
⇒ 144 – x2 = 25 – 169 – x2 + 26x
⇒ 26x = 288
⇒ x = 11.07 units
So Altitude CD = √21.43 = 4.63 units
Question 3.In ΔPQR,
is the altitude to hypotenuse
? If PM = 8, RM = 12, find PQ, QR and QM.
Answer:![](data:image/png;base64,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)
m∠Q = 90
RM = 12
MP = 8
∴ RP = (RM + MP) = 20
Let QM be x
Applying Pythagoras theorem in ΔQMP, we get
QP2 = x2 + 82 = x2 + 64
Applying Pythagoras theorem in ΔQMR, we get
QR2 = x2 + 122 = x2 + 144
Applying Pythagoras theorem in ΔPQR, we get
RP2 = (x2 + 64) + (x2 + 144)
⇒ 400 = 208 + 2x2
⇒ x2 = 96
⇒ x = 4√6
⇒ QM = 4√6
PQ2 = x2 + 64
⇒ PQ2 = 160
⇒ PQ = 4√10
QR2 = x2 + 144
⇒ QR2 = 240
⇒ QR = 4√15
Question 4.In ΔABC, m∠B = 90,
, M
. If AM — MC = 7, AB2 — BC2 = 175, find AC.
Answer:
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m
B = 90
![](data:image/png;base64,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)
AM — MC = 7
Applying Pythagoras theorem in ΔABM and ΔCBM
AB2 = BM2 + AM2 …Equation (i)
BC2 = BM2 + MC2 …Equation (ii)
Subtracting Equation (ii) from Equation (i) we get
AB2 – BC2 = AM2 – MC2
⇒ (AM – MC)(AM + MC) = AB2 – BC2
Putting the values we get :
7 × (AM + MC) = 175
⇒ AM + MC = 25
⇒ AC = 25
Question 5.∠A is right angle in ΔABC. AD is an altitude of the triangle. If AB = √5, BD = 2, find the length of the hypotenuse of the triangle.
Answer:![](data:image/png;base64,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)
m∠A = 90
AB = √5
BD = 2
Applying Pythagoras theorem in ΔADB we get
AD2 = AB2 – BD2
⇒ AD2 = 5 – 4 = 1
⇒ AD = 1
Let AC = x
Applying Pythagoras theorem in ΔADC we get
DC2 = AC2 – AD2
⇒ DC2 = x2 – 1
⇒ DC = √(x2 – 1)
Applying Pythagoras theorem in ΔABC we get
AB2 + AC2 = BC2
⇒ 5 + x2 = (2 + √(x2 – 1))2
⇒ 5 + x2 = 4 + x2 – 1 + 4√(x2 – 1)
![](data:image/png;base64,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)
Hypotenuse BC = BD + DC
⇒ Hypotenuse BC ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAB4AAAAgCAMAAAAynjhNAAAAAXNSR0IArs4c6QAAAEtQTFRFAAAAAAAAAAA6OgAAOgA6Oma2OpDbZgAAZgA6ZjoAZrbbZrb/kDoAkDo6kLbbkNv/tmYAtmY625A627Zm29u2/7Zm/9uQ//+2///b2hz91gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAfElEQVQ4T7WTyQ6AIAxEwX1HQZb//1JL9GTSaTSh1zKvdBiUKlFWa10ZlmxnOFRqE7yFgFDDAWnpOHlajQLqdNBeUwnHvjB3MuAu4PEL+AjkMX/gMrXgiXNEcYjDphy2KTR8Uunelg1LXsrBJFrY9dT1bFJjn98M/xTZ1guvhwL/mOXtCAAAAABJRU5ErkJggg==)
Question 6.m∠B = 90 in ΔABC.
is altitude to
.
If AM = BM = 8, find AC.
Answer:![](data:image/jpeg;base64,/9j/4AAQSkZJRgABAQEAYABgAAD/4RDaRXhpZgAATU0AKgAAAAgABAE7AAIAAAAFAAAISodpAAQAAAABAAAIUJydAAEAAAAKAAAQyOocAAcAAAgMAAAAPgAAAAAc6gAAAAgAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAEFuaWsAAAAFkAMAAgAAABQAABCekAQAAgAAABQAABCykpEAAgAAAAMzMQAAkpIAAgAAAAMzMQAA6hwABwAACAwAAAiSAAAAABzqAAAACAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA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Applying Pythagoras theorem in ΔABM
AB2 = AM2 + BM2
⇒ AB2 = 64 + 64 = 128
Let CM = x
Applying Pythagoras theorem in ΔCBM
⇒ BC2 = BM2 + CM2
⇒ BC2 = x2 + 64
Applying Pythagoras theorem in ΔABC
AC2 = AB2 + BC2
(x + 8)2 = x2 + 64 + 128
⇒ x2 + 64 + 16x = x2 + 192
⇒ 16x = 128
⇒ x = 8 units
AC = AM + CM = 16 units
Question 7.m∠B = 90 in ΔABC.
is altitude to
.
If BM = 15, AC = 34, find AB.
Answer:![](data:image/jpeg;base64,/9j/4AAQSkZJRgABAQEAYABgAAD/4RDaRXhpZgAATU0AKgAAAAgABAE7AAIAAAAFAAAISodpAAQAAAABAAAIUJydAAEAAAAKAAAQyOocAAcAAAgMAAAAPgAAAAAc6gAAAAgAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAEFuaWsAAAAFkAMAAgAAABQAABCekAQAAgAAABQAABCykpEAAgAAAAMzMQAAkpIAAgAAAAMzMQAA6hwABwAACAwAAAiSAAAAABzqAAAACAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA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Let AM = x, CM = 34 – x
BM2 = AM × MC
⇒ x2 – 34x = – 225
⇒ x2 – 34x + 225 = 0
⇒ x2 – 25x – 9x + 225 = 0
⇒ x(x – 25) – 9(x – 25) = 0
⇒ (x – 9)(x – 25) = 0
⇒ x = 9, 25
AB2 = 152 + 92
⇒ AB2 = 306
AB = √306 units
Question 8.m∠B = 90 in ΔABC.
is altitude to
.
If BM = 2√30, MC = 6, find AC.
Answer:![](data:image/jpeg;base64,/9j/4AAQSkZJRgABAQEAYABgAAD/4RDaRXhpZgAATU0AKgAAAAgABAE7AAIAAAAFAAAISodpAAQAAAABAAAIUJydAAEAAAAKAAAQyOocAAcAAAgMAAAAPgAAAAAc6gAAAAgAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAEFuaWsAAAAFkAMAAgAAABQAABCekAQAAgAAABQAABCykpEAAgAAAAMzMQAAkpIAAgAAAAMzMQAA6hwABwAACAwAAAiSAAAAABzqAAAACAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA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BM = 2![](data:image/png;base64,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)
MC = 6
Applying Pythagoras theorem :
BM2 = CM × AM
⇒ 120 = 6 × AM
⇒ AM = 20
AC = AM + MC = 26 units
Question 9.m∠B = 90 in ΔABC.
is altitude to
.
If AB = √10, AM = 2.5, find MC.
Answer:![](data:image/jpeg;base64,/9j/4AAQSkZJRgABAQEAYABgAAD/4RDaRXhpZgAATU0AKgAAAAgABAE7AAIAAAAFAAAISodpAAQAAAABAAAIUJydAAEAAAAKAAAQyOocAAcAAAgMAAAAPgAAAAAc6gAAAAgAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAEFuaWsAAAAFkAMAAgAAABQAABCekAQAAgAAABQAABCykpEAAgAAAAMzMQAAkpIAAgAAAAMzMQAA6hwABwAACAwAAAiSAAAAABzqAAAACAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA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AB = ![](data:image/png;base64,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)
AM = 2.5
Applying Pythagoras theorem in ΔABM
AB2 = AM2 + BM2
⇒ BM2 = 10 – 6.25 = 3.75
Let CM = x
AC = x + 2.5
Applying Pythagoras theorem in ΔCBM
⇒ BC2 = BM2 + CM2
⇒ BC2 = x2 + 3.75 …Equation (i)
Applying Pythagoras theorem in ΔABC
AC2 = AB2 + BC2
(x + 2.5)2 = BC2 + 10
⇒ x2 + 6.25 + 5x = BC2 + 10
⇒ BC2 = x2 – 3.75 + 5x …Equation (ii)
Equating Equation (i) and Equation (ii)
x2 + 3.75 = x2 – 3.75 + 5x
⇒ 5x = 7.5
⇒ x = 1.5
So CM = 1.5
Question 10.In ΔPQR m∠Q = 90, PQ = x, QR = y and
. D
. Find PD, QD, RD in terms of x and y.
Answer:m
Q = 90
Applying Pythagoras theorem
PR = √(x2 + y2)
PQ2 = PD × PR
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGwAAAAtCAMAAAB8vH42AAAAAXNSR0IArs4c6QAAAHVQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjpmOmaQOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDoAkLbbkNv/tmYAtpA6tpBmttvbttv/tv//25A627Zm27aQ29u22////7Zm/9uQ/9u2//+2///bKKzGcAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAB60lEQVRYR+1X21qDMAxu0Q2cE3VD8cSQ4/s/ok3T0lK6jw0avTFXuSD5c/wbGPuXNRX4fuJ8v8bBFbbd4ws7Ra9XWKz8tLkLDtZnHGT3CaGVoni3BxVk8bAyWo95l8YCJbn5YqyIDqx/4whyomhZnwkwVvAjaxKA6TPZq4ICSzgHsFrkk+NEgMpqgVUfg9dxyKzPoJRMJBizLoVGEoGVybbq0m0FYLKJRCLHMdpXAuQ3wFQidhmJElMDAt6HAQnfKx08DsgwhgIT54RErHnIuVxqysQ4x8EQ+/yecL4Jzohna9SIHSApoNdpzeOqfZ7HMwSOWrT7mDeafHFKuFi6eTEELrU2UwQ+b7ngC0PgqCkCX+BpatLgi2fEELhaHuDUUFI7V4Kb2UB3ywGR/lHGaBIMCFxzUJ/Zg5yfMbs0lCKy130gcD+Y16kJ3KeNTFww3SLVM/1qXBr79LuZMiLxILviVaFlXRn9A2KBqUdjeWLGcjL6hsDVUo86GgLSXjNN4Iqu7p09DAu30JtLquo+Xuhtzswh1TKh/KdwqQfuYzJxSVUexVQyyYy8jIZUyxEXhM7QJtUJvYcHs0m1TTd0l+NApZpUxdETOh3Ln7mKpRbynPA9JDoVpFfK6xHmA5900AAtd24Awqr+kesfu+8lDQ4mcyMAAAAASUVORK5CYII=)
![](data:image/png;base64,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)
QR2 = RD × PR
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAJ0AAAA2CAMAAADqI2vJAAAAAXNSR0IArs4c6QAAAI1QTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6Ojo6OjpmOjqQOmY6OmaQOma2OpCQOpDbZgAAZgA6ZgBmZjoAZjo6ZjpmZmYAZrb/kDoAkDo6kDpmkGYAkLbbkNvbkNv/tmYAtmY6ttvbttv/tv//25A627Zm27aQ29u22////7Zm/9uQ/9u2//+2///bJ/8eWwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACcklEQVRoQ+1ZaVODMBBN0BavqngWTzxSLZT8/5/nbpJaIAGbNqnrjPsp0wnhscfbzStj//bnPPB5xfkpVdT1xSObJc9U4QGu6oAyOnFO2HUzsmkHThPUwMmc89G8SjmUQwngyimp0Mp8NAdQAK7OOBgtdEzsvUNMx6RctgJTpVMm76gSiczHrMTo0jQIbUEs2xqOqrOzG0g9qlZwyh2ixKolayVVOkEuCUUn2HbAjt4Y08vk6HXLiMj89CGU6+oMTvpIcURUy0W+bULDV05CcR0yJ7QdbIZ6KfPfG2YrjGHTNKQSHaaXrEpDxcU/RXB0sNEJ/NWgq7NdNyE9ymhrw1OQPlIkT4NOD0BLK3qe8/fLek+IpNkCVaVqwE50zjNXnxpi1XpFF92YLbJ9pHaqkS25rlZTFc0eudvIOqtCs4hBVxBiFEXBwCJQCYaNW2m5XirH2oVFgfAKnjzpTjbpEOIPbxbb10Wsb4NsIHt9UI3lOFRLjeHB9iDnKYh5bvfH37rceApintv9wdWX3QndEsRkPnQ7i6qf2WlnCWLD6KLqZ3g4dJPRHIhFucgWxDromhqQa7t//PqfeMHxAZuLweAQxLq+W2lAsfUznXb4Ph0hlyBmRXalAUXWzwyfVOmJEhEtQcw5QnxrQLH1M2HKUY3WPWb5blca0LKN1Te3/Xd/u2aja0CLa/Sa4RMAqScdp9noYmtAYnKlxFcFSRZQEepStyY64KC+zYH4pER6U20MCCwB3/G+NzrYOLoGhCwi7zcUmuJrQFCmG01PATWggTSAOtioR4bUgAbgieRwE1U9pAY06DzS+ibZv0sC0dKfOOYLIkM485HgY3QAAAAASUVORK5CYII=)
QD2 = PD × RD
⇒ QD2![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 11.∠Q is a right angle in ΔPQR and
, M
. If PQ = 4QR, then prove that PM = 16RM.
Answer:m
Q = 90
![](data:image/png;base64,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)
PQ = 4QR
PQ2 = PR × PM …Equation(i)
QR2 = PR × RM …Equation(ii)
Dividing Equation (ii) by Equation(i) 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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)
Hence PM = 16RM
Question 12.âPQRS is a rectangle. If PQ + QR = 7 and PR + QS = 10, then find the area of â PQRS.
Answer:PQ + QR = 7
PR + QS = 10
Since Diagonals of a rectangle are of equal length, so
PR = QS = 5
Let PQ be x, QR = 7 – x
Applying Pythagoras theorem :
x2 + (7 – x)2 = 52
⇒ 2x2 – 14x + 24 = 0
⇒ x2 – 7x + 12 = 0
⇒ x2 – 4x – 3x + 12 = 0
⇒ x(x – 4) – 3(x – 4) = 0
⇒ (x – 3)(x – 4) = 0
x = 3, 4
∴ The two sides are 3 and 4
Area of the Rectangle = (3 × 4) = 12 sq. units
Question 13.The diagonals of a convex â ABCD intersect at right angles. Prove that AB2 + CD2 = AD2 + BC2.
Answer:![](data:image/jpeg;base64,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)
Using Pythagoras theorem :
AO2 + OB2 = AB2 …Equation(i)
DO2 + OC2 = CD2 …Equation(ii)
AO2 + OD2 = AD2 …Equation(iii)
BO2 + OC2 = BC2 …Equation(iv)
Adding Equation(i) and Equation(ii)
AB2 + CD2 = AO2 + OB2 + DO2 + OC2 …Equation(v)
Adding Equation(iii) and Equation(iv)
AD2 + BC2 = AO2 + OB2 + DO2 + OC2 …Equation(vi)
The RHS of equation (v) and (vi) are similar
So we can say that
AB2 + CD2 = AD2 + BC2
Question 14.In ΔPQR, m∠Q = 90, M
and N
. Prove that PM2 + RN2 = PR2 + MN2.
Answer:![](data:image/jpeg;base64,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)
m
Q = 90
Applying Pythagoras theorem
PM2 = QM2 + PQ2
RN2 = QN2 + QR2
Adding the above two equations we get
PM2 + RN2 = QM2 + PQ2 + QN2 + QR2
⇒ PM2 + RN2 = (QM2 + QN2) + (PQ2 + QR2)
⇒ PM2 + RN2 = MN2 + PR2
Question 15.The sides of a triangle have lengths a2 + b2, 2ab, a2 — b2, where a > b and a, b ϵ R + . Prove that the angle opposite to the side having length a2 + b2 is a right angle.
Answer:Let a2 + b2 = p, 2ab = q, a2 — b2 = r
Since a, b ϵ R + , hence
p is the largest side of the triangle
p2 = (a2 + b2)2 = a4 + b4 + 2a2b2
q2 + r2 = 4a2b2 + (a2 – b2)2
⇒ q2 + r2 = a4 + b4 + 2a2b2
So p2 = q2 + r2
Hence the Pythagoras theorem is established
Hence the angle opposite to the side p = a2 + b2 is a right angle
Question 16.In ΔABC, m∠B = 90 and
is a median. Prove that AB2 + BC2 + AC2 = 8BE2.
Answer:![](data:image/png;base64,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)
m
B = 90
is a median
∴ AE = EC
⇒ AC = 2AE
Applying Pythagoras Theorem :
AB2 + BC2 = AC2 …Equation (i)
Applying Appoloneous theorem
AB2 + BC2 = 2(AE2 + BE2)
⇒ AC2 = 2(AE2 + BE2)
Now since AC = 2AE
⇒ (2AE)2 = 2(AE2 + BE2)
⇒ 2AE2 = 2BE2
⇒ AE = BE …Equation (ii)
∴ AB2 + BC2 + AC2 = 2AC2 (From Equation (i))
⇒ AB2 + BC2 + AC2 = 2 × (2AE)2 = 8AE2
From Equation (ii) we can say that
⇒ AB2 + BC2 + AC2 = 8BE2
Question 17.AB = AC and ∠A is right angle in ΔABC. If BC = √2 a, then find the area of the triangle. (a ∈ R, a > 0)
Answer:![](data:image/png;base64,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)
m
A = 90
Let AB = AC = x
BC =
a
Applying Pythagoras Theorem :
∴ x2 + x2 = 2a2
⇒ 2x2 = 2a2
⇒ x2 = a2
⇒ x = a
∴ Area of the triangle ![](data:image/png;base64,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)
⇒ Area of the triangle ![](data:image/png;base64,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)
∠B is a right angle in ΔABC. and D
. If AD = 4DC, prove that BD = 2DC.
Answer:
m∠B = 90
AD = 4DC (Given) …Equation (i)
∴ BD2 = AD × DC
⇒ BD2 = 4DC2 (From Equation (i))
Taking Square Root both sides we get
∴ BD = 2DC
Question 2.
5, 12, 13 are the lengths of the sides of a triangle. Show that the triangle is right angled. Find the length of altitude on the hypotenuse.
Answer:
Hypotenuse (AB) = 13
Perpendicular (AC) = 12
Base (BC) = 5
∴ AC2 + BC2 = 52 + 122
⇒ AC2 + BC2 = 25 + 144
⇒ AC2 + BC2 = 169 = AB2
Hence Pythagoras theorem is valid
m∠C = 90
Let AD = x, BD = 13 – x
Applying Pythagoras theorem
∴ Altitude CD2 = 122 – x2 = 52 – (13 – x) 2
⇒ 144 – x2 = 25 – 169 – x2 + 26x
⇒ 26x = 288
⇒ x = 11.07 units
So Altitude CD = √21.43 = 4.63 units
Question 3.
In ΔPQR, is the altitude to hypotenuse
? If PM = 8, RM = 12, find PQ, QR and QM.
Answer:
m∠Q = 90
RM = 12
MP = 8
∴ RP = (RM + MP) = 20
Let QM be x
Applying Pythagoras theorem in ΔQMP, we get
QP2 = x2 + 82 = x2 + 64
Applying Pythagoras theorem in ΔQMR, we get
QR2 = x2 + 122 = x2 + 144
Applying Pythagoras theorem in ΔPQR, we get
RP2 = (x2 + 64) + (x2 + 144)
⇒ 400 = 208 + 2x2
⇒ x2 = 96
⇒ x = 4√6
⇒ QM = 4√6
PQ2 = x2 + 64
⇒ PQ2 = 160
⇒ PQ = 4√10
QR2 = x2 + 144
⇒ QR2 = 240
⇒ QR = 4√15
Question 4.
In ΔABC, m∠B = 90, , M
. If AM — MC = 7, AB2 — BC2 = 175, find AC.
Answer:
mB = 90
AM — MC = 7
Applying Pythagoras theorem in ΔABM and ΔCBM
AB2 = BM2 + AM2 …Equation (i)
BC2 = BM2 + MC2 …Equation (ii)
Subtracting Equation (ii) from Equation (i) we get
AB2 – BC2 = AM2 – MC2
⇒ (AM – MC)(AM + MC) = AB2 – BC2
Putting the values we get :
7 × (AM + MC) = 175
⇒ AM + MC = 25
⇒ AC = 25
Question 5.
∠A is right angle in ΔABC. AD is an altitude of the triangle. If AB = √5, BD = 2, find the length of the hypotenuse of the triangle.
Answer:
m∠A = 90
AB = √5
BD = 2
Applying Pythagoras theorem in ΔADB we get
AD2 = AB2 – BD2
⇒ AD2 = 5 – 4 = 1
⇒ AD = 1
Let AC = x
Applying Pythagoras theorem in ΔADC we get
DC2 = AC2 – AD2
⇒ DC2 = x2 – 1
⇒ DC = √(x2 – 1)
Applying Pythagoras theorem in ΔABC we get
AB2 + AC2 = BC2
⇒ 5 + x2 = (2 + √(x2 – 1))2
⇒ 5 + x2 = 4 + x2 – 1 + 4√(x2 – 1)
Hypotenuse BC = BD + DC
⇒ Hypotenuse BC
Question 6.
m∠B = 90 in ΔABC. is altitude to
.
If AM = BM = 8, find AC.
Answer:
Applying Pythagoras theorem in ΔABM
AB2 = AM2 + BM2
⇒ AB2 = 64 + 64 = 128
Let CM = x
Applying Pythagoras theorem in ΔCBM
⇒ BC2 = BM2 + CM2
⇒ BC2 = x2 + 64
Applying Pythagoras theorem in ΔABC
AC2 = AB2 + BC2
(x + 8)2 = x2 + 64 + 128
⇒ x2 + 64 + 16x = x2 + 192
⇒ 16x = 128
⇒ x = 8 units
AC = AM + CM = 16 units
Question 7.
m∠B = 90 in ΔABC. is altitude to
.
If BM = 15, AC = 34, find AB.
Answer:
Let AM = x, CM = 34 – x
BM2 = AM × MC
⇒ x2 – 34x = – 225
⇒ x2 – 34x + 225 = 0
⇒ x2 – 25x – 9x + 225 = 0
⇒ x(x – 25) – 9(x – 25) = 0
⇒ (x – 9)(x – 25) = 0
⇒ x = 9, 25
AB2 = 152 + 92
⇒ AB2 = 306
AB = √306 units
Question 8.
m∠B = 90 in ΔABC. is altitude to
.
If BM = 2√30, MC = 6, find AC.
Answer:
BM = 2
MC = 6
Applying Pythagoras theorem :
BM2 = CM × AM
⇒ 120 = 6 × AM
⇒ AM = 20
AC = AM + MC = 26 units
Question 9.
m∠B = 90 in ΔABC. is altitude to
.
If AB = √10, AM = 2.5, find MC.
Answer:
AB =
AM = 2.5
Applying Pythagoras theorem in ΔABM
AB2 = AM2 + BM2
⇒ BM2 = 10 – 6.25 = 3.75
Let CM = x
AC = x + 2.5
Applying Pythagoras theorem in ΔCBM
⇒ BC2 = BM2 + CM2
⇒ BC2 = x2 + 3.75 …Equation (i)
Applying Pythagoras theorem in ΔABC
AC2 = AB2 + BC2
(x + 2.5)2 = BC2 + 10
⇒ x2 + 6.25 + 5x = BC2 + 10
⇒ BC2 = x2 – 3.75 + 5x …Equation (ii)
Equating Equation (i) and Equation (ii)
x2 + 3.75 = x2 – 3.75 + 5x
⇒ 5x = 7.5
⇒ x = 1.5
So CM = 1.5
Question 10.
In ΔPQR m∠Q = 90, PQ = x, QR = y and . D
. Find PD, QD, RD in terms of x and y.
Answer:
mQ = 90
Applying Pythagoras theorem
PR = √(x2 + y2)
PQ2 = PD × PR
QR2 = RD × PR
QD2 = PD × RD
⇒ QD2
Question 11.
∠Q is a right angle in ΔPQR and , M
. If PQ = 4QR, then prove that PM = 16RM.
Answer:
mQ = 90
PQ = 4QR
PQ2 = PR × PM …Equation(i)
QR2 = PR × RM …Equation(ii)
Dividing Equation (ii) by Equation(i) we get
Hence PM = 16RM
Question 12.
âPQRS is a rectangle. If PQ + QR = 7 and PR + QS = 10, then find the area of â PQRS.
Answer:
PQ + QR = 7
PR + QS = 10
Since Diagonals of a rectangle are of equal length, so
PR = QS = 5
Let PQ be x, QR = 7 – x
Applying Pythagoras theorem :
x2 + (7 – x)2 = 52
⇒ 2x2 – 14x + 24 = 0
⇒ x2 – 7x + 12 = 0
⇒ x2 – 4x – 3x + 12 = 0
⇒ x(x – 4) – 3(x – 4) = 0
⇒ (x – 3)(x – 4) = 0
x = 3, 4
∴ The two sides are 3 and 4
Area of the Rectangle = (3 × 4) = 12 sq. units
Question 13.
The diagonals of a convex â ABCD intersect at right angles. Prove that AB2 + CD2 = AD2 + BC2.
Answer:
Using Pythagoras theorem :
AO2 + OB2 = AB2 …Equation(i)
DO2 + OC2 = CD2 …Equation(ii)
AO2 + OD2 = AD2 …Equation(iii)
BO2 + OC2 = BC2 …Equation(iv)
Adding Equation(i) and Equation(ii)
AB2 + CD2 = AO2 + OB2 + DO2 + OC2 …Equation(v)
Adding Equation(iii) and Equation(iv)
AD2 + BC2 = AO2 + OB2 + DO2 + OC2 …Equation(vi)
The RHS of equation (v) and (vi) are similar
So we can say that
AB2 + CD2 = AD2 + BC2
Question 14.
In ΔPQR, m∠Q = 90, M and N
. Prove that PM2 + RN2 = PR2 + MN2.
Answer:
mQ = 90
Applying Pythagoras theorem
PM2 = QM2 + PQ2
RN2 = QN2 + QR2
Adding the above two equations we get
PM2 + RN2 = QM2 + PQ2 + QN2 + QR2
⇒ PM2 + RN2 = (QM2 + QN2) + (PQ2 + QR2)
⇒ PM2 + RN2 = MN2 + PR2
Question 15.
The sides of a triangle have lengths a2 + b2, 2ab, a2 — b2, where a > b and a, b ϵ R + . Prove that the angle opposite to the side having length a2 + b2 is a right angle.
Answer:
Let a2 + b2 = p, 2ab = q, a2 — b2 = r
Since a, b ϵ R + , hence
p is the largest side of the triangle
p2 = (a2 + b2)2 = a4 + b4 + 2a2b2
q2 + r2 = 4a2b2 + (a2 – b2)2
⇒ q2 + r2 = a4 + b4 + 2a2b2
So p2 = q2 + r2
Hence the Pythagoras theorem is established
Hence the angle opposite to the side p = a2 + b2 is a right angle
Question 16.
In ΔABC, m∠B = 90 and is a median. Prove that AB2 + BC2 + AC2 = 8BE2.
Answer:
mB = 90
is a median
∴ AE = EC
⇒ AC = 2AE
Applying Pythagoras Theorem :
AB2 + BC2 = AC2 …Equation (i)
Applying Appoloneous theorem
AB2 + BC2 = 2(AE2 + BE2)
⇒ AC2 = 2(AE2 + BE2)
Now since AC = 2AE
⇒ (2AE)2 = 2(AE2 + BE2)
⇒ 2AE2 = 2BE2
⇒ AE = BE …Equation (ii)
∴ AB2 + BC2 + AC2 = 2AC2 (From Equation (i))
⇒ AB2 + BC2 + AC2 = 2 × (2AE)2 = 8AE2
From Equation (ii) we can say that
⇒ AB2 + BC2 + AC2 = 8BE2
Question 17.
AB = AC and ∠A is right angle in ΔABC. If BC = √2 a, then find the area of the triangle. (a ∈ R, a > 0)
Answer:
mA = 90
Let AB = AC = x
BC = a
Applying Pythagoras Theorem :
∴ x2 + x2 = 2a2
⇒ 2x2 = 2a2
⇒ x2 = a2
⇒ x = a
∴ Area of the triangle
⇒ Area of the triangle
Exercise 7.2
Question 1.In rectangle ABCD, AB + BC = 23, AC + BD = 34. Find the area of the rectangle.
Answer:AB + BC = 23
AC + BD = 34
Since Diagonals of a rectangle are of equal length, so
AC = BD = 17
Let AB be x, BC = 23 – x
Applying Pythagoras theorem :
x2 + (23 – x)2 = 172
⇒ 2x2 – 46x + 240 = 0
⇒ x2 – 23x + 120 = 0
⇒ x2 – 15x – 8x + 120 = 0
⇒ x(x – 15) – 8(x – 15) = 0
⇒ (x – 8)(x – 15) = 0
x = 8, 15
∴ The two sides are 8 and 15
Area of the Rectangle = (8 × 15) = 120 sq. units
Question 2.In ΔABC m∠A = m∠B + m∠C, AB = 7, BC = 25. Find the perimeter of ΔABC.
Answer:Let m∠A = x
Sum of interior angles = m
A = m
B + m
C
⇒ m
A + m
B + m
C = 180 (Sum of interior angle of triangle is 180)
⇒ 2x = 180
⇒ x = 90
So ∠A is right angled
Applying Pythagoras theorem
AC2 = BC2 – AB2
⇒ AC2 = 252 – 72
⇒ AC2 = 576
⇒ AC = 24
Perimeter of ΔABC = (25 + 24 + 7) = 56 units.
Question 3.A staircase of length 6.5 meters touches a wall at height of 6 meter. Find the distance of base of the staircase from the wall.
Answer:Length of staircase = 6.5 metres
Height of Wall = 6 metres
Let the distance of base be d
Applying Pythagoras theorem
d2 = 6.52 – 62
⇒ d2 = 6.25
⇒ d = 2.5 metres
Distance of base = 2.5 m
Question 4.In ΔABC AB = 7, AC = 5, AD = 5. Find
, if the mid – point of BC is D.
Answer:![](data:image/jpeg;base64,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)
D is the mid – point of BC
AB = 7
AC = 5
AD = 5
Let BD = x
Applying Appoloneus theorem :
AB2 + AC2 = 2(AD2 + BD2)
⇒ 72 + 52 = 2(52 + x2)
⇒ 74 = 50 + 2x2
⇒ 2x2 = 24
⇒ x2 = 12
⇒ x = 2√3 units
BC = 2 × BD = 4√3 units
Question 5.In equilateral ΔABC, D
such that BD : DC = 1 : 2. Prove that 3AD = √7 AB.
Answer:BD : DC = 1 : 2
Let BD = x, DC = 2x –
⇒ AB = BC = AC = 3x
Let M be the mid point of BC
BM ![](data:image/png;base64,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)
∴ DM![](data:image/png;base64,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)
⇒ DM ![](data:image/png;base64,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)
Applying Pythagoras theorem :
AB2 = BM2 + AM2
⇒ AM2![](data:image/png;base64,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)
⇒ AM2![](data:image/png;base64,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)
Applying Pythagoras theorem :
AD2 = AM2 + DM2
⇒ AD2![](data:image/png;base64,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)
⇒ AD2 = 7x2
⇒ AD = √7x
So
AB : AD = 3x : √7x
⇒ AB : AD = 3 : √7
So 3AD =
AB
Question 6.In ΔABC, AB = 17, BC = 15, AC = 8, find the length of the median on the largest side.
Answer:AB = 17
BC = 15
AC = 8
Let the median meet the side AB at point D
AD = 9.5
∴ CD is the median on the largest side
Applying Appoloneus theorem :
AC2 + BC2 = 2(AD2 + CD2)
⇒ 82 + 152 = 2(9.52 + CD2)
⇒ 144.5 = 90.25 + CD2
⇒ CD2 = 54.25
⇒ CD = 7.36 units
Question 7.
is a median of ΔABC. AB2 + AC2 = 148 and AD = 7. Find BC.
Answer:AB2 + AC2 = 148
AD = 7
Applying Apploneus theorem :
AB2 + AC2 = 2(AD2 + BD2)
⇒ 148 = 2(49 + BD2)
⇒ BD2 = 74 – 49 = 25
⇒ BD = 5 units
BC = 2 × BD = 10 units
Question 8.In rectangle ABCD, AC = 25 and CD = 7. Find perimeter of the rectangle.
Answer:AC = 25
CD = 7
Applying Pythagoras theorem :
AD2 = AC2 – CD2
⇒ AD2 = 252 – 72 = 576
⇒ AD = 24 units
Perimeter of rectangle = 2(AD + CD) = 62 units
Question 9.In rhombus XYZW, XZ = 14 and YW = 48. Find XY.
Answer:![](data:image/jpeg;base64,/9j/4AAQSkZJRgABAQEAYABgAAD/2wBDAAoHBwgHBgoICAgLCgoLDhgQDg0NDh0VFhEYIx8lJCIfIiEmKzcvJik0KSEiMEExNDk7Pj4+JS5ESUM8SDc9Pjv/2wBDAQoLCw4NDhwQEBw7KCIoOzs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozv/wAARCAB1AQQDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD2aiqd1q1hZXtrZXVysU94SsCNkeYR1APTPtRp+q2OqrM1hcrcLBKYpGTOA46jPf8ACgC5RRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRSGuBXxD4oPis+E9sX2oXH2j7d5I2fYcf3c/f3fL6UAd/RSUtABRRRQAUUUUAFFFFABRRRQAUUUUAcl4/wBIvvEdha6JZWgxPMskl+5GLMLzuXkEuegx6mrfga2vtP8ADMGmX+mrYzWJMPyEFJgDxIuD/F1Oec5roaMUAeE+J/H3xI8Ma5Jpt9cQK7NmFo7VSkqk8FeMn6daNG+IfjLWH8r/AISvRbC4zjyr6HyjnPTOwr+tet+K/CmneLtIawvk2sMtBOo+eF/Uf1HevOtOWxGrDwb8R9Nt57vhbDVGG03KdFBkGDn0OevB56gGwtj8XnUMuuaGynoQmQf/ACHS/wBnfGD/AKDWif8AfH/2umyfDHVtElM3gzxTdWCjGLS6JkiP9PzU03/hNPHfhnC+JvC3263Xrd6cc8epAyPz20ASf2d8YP8AoNaJ/wB8f/a6P7O+MH/Qa0T/AL4/+11v6J8SfCmvFY7fVY4J2OPJuh5TZ9OeCfoTXUBgQGBBBGQR3oA84/s74wf9BrRP++P/ALXR/Z3xg/6DWif98f8A2uvSKKAPN/7O+MH/AEGtE/74/wDtdH9nfGD/AKDWif8AfH/2uvSKKAPN/wCzvjB/0GtE/wC+P/tdH9nfGD/oNaJ/3x/9rr0iigDzf+zvjB/0GtE/74/+10f2d8YP+g1on/fH/wBrr0iigDzf+zvjB/0GtE/74/8AtdH9nfGD/oNaJ/3x/wDa69IooA83/s74wf8AQa0T/vj/AO11V1I/FXSNOuNQvvEGhxW9uheRinYdh+75J6AdzXpl3d29jay3V1MkMEKl5JHOAoHUmvLYIrr4va+9xcPLB4T0+XEMQBQ3jjuf88A46kmgCn4T+KHiG2ntLzxhGRo2pZjt7xYFRUcHGTjt1H4ZHQ16jHo+nya2PEEYL3b2otxKshKmLO4YHTr3pdQ0DS9U0RtGu7ONrFkCCFRgIB02+hHYivPNL1TUvhXq0eg69K9z4cuXxY6gRn7P/st7eo/EcZAAPVKWmpIkqLJG6ujAMrKcgg9CDTqACiiigAooooAKKKKACiiigAooooAKKKKACsPxZ4T03xdpDWF+mGHzQzqPnhb1H9R3rcooA828J+K9R8O6uvg3xk225X5bDUGPyXKdACT37A/geevo/WsXxZ4T07xdpD2F+u1h80E6j54X9R/Ud65Lwp4r1Hw7q6+DfGT4uV4sNQY/Jcr0ALHv2B/A89QDqNd8B+GfEZZ9R0qFpiP9fH+7k/NcZ/GuU/4Vt4i8PHzPB/iy4hRTkWl988Z9uhH/AI7XpYOaWgDzU/EPxR4acReL/CsvlDGb3T/nj+uOR+orptC+IPhjxEyR2OqxCd+kE37uTPoAev4ZroyM1zOt/Dnwrr25rrSoopm/5bW37p8+vHB/EGgDp6K81fwP408NPu8J+KWuLZcEWWpfMPcA4I/LbRF8S9d0B1i8ZeFbm0TIDXloN8f1xyPyb1oA9KorC0Txp4d8Qqv9matbyyH/AJZM2yT/AL5bBrdoAKKKKACmSypDE0sjqkaKWd2OAoHUk9hTicDmvK9d1e++J2qy+GPDknlaJAw/tHUccPg52r6jI49fp1AI764uvi34jXTrFpovClhJm6nB2/anHQD+g7Dk84r1GysrbTrOGzs4Egt4VCxxoMBRUelaVZ6Lp0On6fCsFtAu1EUfmT6k9SauUAFUdY0ex17TJtO1KBZ7aZcMp7ehB7EdjV6igDyrStV1H4V6tHoOvSvc+HblsWGoMM+R/st6D1H4jjIHqaOkiLJGyujAFWU5BB7iqesaPY69pk2m6lAs9tMMMp6j0IPYjsa840rVdR+Ferx6Dr0r3Xh24cix1AjPkf7Deg9R+I4yAAeq0U2N1ljWRGVkcAqynIIPcGnUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFYfizwnp3i/SGsL9NrD5oJ1Hzwv6j+o71uUUAebeFPFeo+HtXXwb4yfbcjAsdQY/Jcp0AJPf0P4Hnr6QDWJ4s8J6d4v0hrC+Tawy0E6j54X9R/Ud65Lwp4r1Hw7q6+DfGT7bleLDUGPyXKdFBY9+wP4HnqAek0UgNLQAUjKrKVYAqRggjrS0UAcfrfwu8Ka47TPp/2O4Y58+zbyzn1x939KxF8IePvC7BvDviVdVtU6WWo9wB0BOf0K16VXEfEa5EU2hQXt5NZ6PPeMt/NFK0eAFJQFhyAT/KgDOj+K1xo84tvGHhu90t87fPiUyRN7j2+hNdjovizQfEKg6VqlvcMRnyw+JB9VPP6Vi/DqWXUvCDreO97ai6mjtnufnMsAbCE561w3jrQtB1HxZZeH/COnxQ640u+5uLZmSO2UeoXgHucdPqaAOg8V+I9U8W67L4J8KSeUFGNS1AHiJO6qR+R7k8eprtvDfh6x8L6JBpWnqRFEMlm+9Ix6sfc//WrzvTvBXjrwB5z+Gbyw1W2lYPLBPEI5HIHqT/7N+FaVr8WhYXCWfi7Qb7RZidpl2F4j7+uPpmgD0eis7SvEGka5CJdL1G3u1xkiKQEj6jqPxrQzQAtFFFABVHWNHsde0ubTdRgWe2mGGU9R6EHsR2NXqKAPKtK1XUfhXq8ega/K914duHxYagRnyP8AZb29R26jjIHqcciSxrJGwZGGVZTkEdiKp6xo9jr2lzabqNus9vMMMp6g9iD2I7GvONK1XUfhXq8ega/M9z4duHxYagRnyP8AZb29R26jjIAB6rRTUkSVFkjYOjAFWU5BB6EGnUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFYXizwnp3i7SGsL5NrDLQTqPnhf1Ht6jvW7RQB5v4T8V6l4e1dfBvjJttyOLDUGPyXK9ACT37A/geevo4NYfizwnpvi7SGsL9NrD5oJ1Hzwt6j29R3rlPCfizUfD2rr4N8ZNtuV+Wx1Bj8lynQAk9/Q/geeoB6RRSA0tABTJYo5kKSxq6HqrDIP4U+uN8deOT4eEWk6VD9u169+W2tlG7Zno7D+Q749KAKnjvxjNYMnhbwwpn168ARFhxi1U/xHsDjp6Dk9s63gfwXaeENK2j99qNwA95dMctI/UgH+6DnH5nmq3gHwX/AMI1aS3upMtzrd8xkurnqRnnYD6Z6nufoK68DFABio57aC6haG5hjmiYYZJFDKfwNS0UAcHq/wAIvDd/O13pwuNHu87lls5Nqg/7vQfhiqKad8TvChzZ39t4mslH+qufkmwPQnnP/Aj9K9KpOtAHntn8X9NhufsPiXTL3QrsHDCaMume+CBnH4d67XTNZ0zWYPO02/t7tPWGQNj6jt+NS32nWWp25t7+0huoT1SaMOP1ri9R+EehPcfbNEuLzQ7wcrJaSnaD/unt7AigDvKWvMs/FTwr1Fr4ns0Hb5ZsAfgSf++qvaZ8XtFlufsWuWl3oV2OCl3Gdmf97GfzAoA7+qOsaPYa9pk2nalbrPbzLhlPUHsQexHY1NZahZ6lbrcWN1DdQt0eFw4/MVYoA8q0rVdS+Fmrx6Br8r3Ph25fFhqBGfI/2W9vUduo4yB6mkiSorxsrowBVlOQQehFU9Y0ew17TJtO1KBZ7aZcMp7HsQexHY15xpWq6l8LNWj0DXpXufDty2LDUGGfI/2W9B6jt1HGQAD1Wimo6SIroyujAFWU5BHqKdQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAVheLPCeneLtIexv12svzQTqPnhf1H9R3rdooA838KeK9R8Pauvg3xk+25Xixv2PyXKdACx7+h/A89fRwaw/FnhPTvF+kNYX67WX5oJ1Hzwv6j29R3rgbX4g6p4CtrvQPF0Elxe2kJbT7pclbteign/2b2IPI5AOz8ceM4vCWmx+VD9q1K7fyrO1ByXY9yOu0HHTqSBVDwL4NvNOuZ/EfiKc3WvX6/vCcEW6n+BffoDjgdB743gnSrS6vh4x8WatZXGsXID28DTpttEPQAZ4bB6dvrmvQf7c0j/oK2X/gQn+NAF6lqh/bmkf9BWy/8C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XZ = 14
⇒ XO = 7
YW = 48
⇒ YO = 24
Since Diagonals of a Rhombus bisect at Right angles,
So we apply Pythagoras theorem in ΔXOY
XY2 = XO2 + YO2
⇒ XY2 = 49 + 576 = 625
⇒ XY = 25 units
Question 10.In ΔPQR, m∠Q : m∠R : m∠P = 1 : 2 : 1. If PQ = 2√6, find PR.
Answer:Given: Ratio of angles of triangles = m
Q : m
R : m
P = 1 : 2 : 1
Let m
Q = x, m
R = 2 x, m
P = x
Sum of angles of triangle = 180°
m
Q + m
R + m
P = 180°
⇒ x + 2x + x = 180°
⇒ 4x = 180
⇒ x = 45
So by putting value of x we get,
⇒ m
R = 90
⇒ m
Q = 45°
⇒ m
P = 45°
It is an isosceles right angled triangle at R.
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Let PR = QR = d
According to pythagoras theorem, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides
Applying Pythagoras theorem
PR2 + QR2 = PQ2
2d2 = PQ2
⇒ 2d2 = 24
⇒ d2 = 12
⇒ d = 2√3
So PR = 2√3 units
and QR = 2√3 units
In rectangle ABCD, AB + BC = 23, AC + BD = 34. Find the area of the rectangle.
Answer:
AB + BC = 23
AC + BD = 34
Since Diagonals of a rectangle are of equal length, so
AC = BD = 17
Let AB be x, BC = 23 – x
Applying Pythagoras theorem :
x2 + (23 – x)2 = 172
⇒ 2x2 – 46x + 240 = 0
⇒ x2 – 23x + 120 = 0
⇒ x2 – 15x – 8x + 120 = 0
⇒ x(x – 15) – 8(x – 15) = 0
⇒ (x – 8)(x – 15) = 0
x = 8, 15
∴ The two sides are 8 and 15
Area of the Rectangle = (8 × 15) = 120 sq. units
Question 2.
In ΔABC m∠A = m∠B + m∠C, AB = 7, BC = 25. Find the perimeter of ΔABC.
Answer:
Let m∠A = x
Sum of interior angles = mA = m
B + m
C
⇒ mA + m
B + m
C = 180 (Sum of interior angle of triangle is 180)
⇒ 2x = 180
⇒ x = 90
So ∠A is right angled
Applying Pythagoras theorem
AC2 = BC2 – AB2
⇒ AC2 = 252 – 72
⇒ AC2 = 576
⇒ AC = 24
Perimeter of ΔABC = (25 + 24 + 7) = 56 units.
Question 3.
A staircase of length 6.5 meters touches a wall at height of 6 meter. Find the distance of base of the staircase from the wall.
Answer:
Length of staircase = 6.5 metres
Height of Wall = 6 metres
Let the distance of base be d
Applying Pythagoras theorem
d2 = 6.52 – 62
⇒ d2 = 6.25
⇒ d = 2.5 metres
Distance of base = 2.5 m
Question 4.
In ΔABC AB = 7, AC = 5, AD = 5. Find , if the mid – point of BC is D.
Answer:
D is the mid – point of BC
AB = 7
AC = 5
AD = 5
Let BD = x
Applying Appoloneus theorem :
AB2 + AC2 = 2(AD2 + BD2)
⇒ 72 + 52 = 2(52 + x2)
⇒ 74 = 50 + 2x2
⇒ 2x2 = 24
⇒ x2 = 12
⇒ x = 2√3 units
BC = 2 × BD = 4√3 units
Question 5.
In equilateral ΔABC, D such that BD : DC = 1 : 2. Prove that 3AD = √7 AB.
Answer:
BD : DC = 1 : 2
Let BD = x, DC = 2x –
⇒ AB = BC = AC = 3x
Let M be the mid point of BC
BM
∴ DM
⇒ DM
Applying Pythagoras theorem :
AB2 = BM2 + AM2
⇒ AM2
⇒ AM2
Applying Pythagoras theorem :
AD2 = AM2 + DM2
⇒ AD2
⇒ AD2 = 7x2
⇒ AD = √7x
So
AB : AD = 3x : √7x
⇒ AB : AD = 3 : √7
So 3AD = AB
Question 6.
In ΔABC, AB = 17, BC = 15, AC = 8, find the length of the median on the largest side.
Answer:
AB = 17
BC = 15
AC = 8
Let the median meet the side AB at point D
AD = 9.5
∴ CD is the median on the largest side
Applying Appoloneus theorem :
AC2 + BC2 = 2(AD2 + CD2)
⇒ 82 + 152 = 2(9.52 + CD2)
⇒ 144.5 = 90.25 + CD2
⇒ CD2 = 54.25
⇒ CD = 7.36 units
Question 7.
is a median of ΔABC. AB2 + AC2 = 148 and AD = 7. Find BC.
Answer:
AB2 + AC2 = 148
AD = 7
Applying Apploneus theorem :
AB2 + AC2 = 2(AD2 + BD2)
⇒ 148 = 2(49 + BD2)
⇒ BD2 = 74 – 49 = 25
⇒ BD = 5 units
BC = 2 × BD = 10 units
Question 8.
In rectangle ABCD, AC = 25 and CD = 7. Find perimeter of the rectangle.
Answer:
AC = 25
CD = 7
Applying Pythagoras theorem :
AD2 = AC2 – CD2
⇒ AD2 = 252 – 72 = 576
⇒ AD = 24 units
Perimeter of rectangle = 2(AD + CD) = 62 units
Question 9.
In rhombus XYZW, XZ = 14 and YW = 48. Find XY.
Answer:
XZ = 14
⇒ XO = 7
YW = 48
⇒ YO = 24
Since Diagonals of a Rhombus bisect at Right angles,
So we apply Pythagoras theorem in ΔXOY
XY2 = XO2 + YO2
⇒ XY2 = 49 + 576 = 625
⇒ XY = 25 units
Question 10.
In ΔPQR, m∠Q : m∠R : m∠P = 1 : 2 : 1. If PQ = 2√6, find PR.
Answer:
Given: Ratio of angles of triangles = mQ : m
R : m
P = 1 : 2 : 1
Let mQ = x, m
R = 2 x, m
P = x
Sum of angles of triangle = 180°
mQ + m
R + m
P = 180°
⇒ x + 2x + x = 180°
⇒ 4x = 180
⇒ x = 45
So by putting value of x we get,⇒ mR = 90
⇒ mQ = 45°
⇒ mP = 45°
It is an isosceles right angled triangle at R.
Let PR = QR = d
According to pythagoras theorem, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sidesApplying Pythagoras theorem
PR2 + QR2 = PQ2
2d2 = PQ2
⇒ 2d2 = 24
⇒ d2 = 12
⇒ d = 2√3
So PR = 2√3 units
and QR = 2√3 units
Exercise 7
Question 1.
,
,
are the medians of ΔABC. If BE = 12, CF = 9 and AB2 + BC2 + AC2 = 600, BC = 10, find AD.
Answer:From the given info,
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)
As AB2 + BC2 + AC2 = 600
and BC = 10,
⇒ AB2 + AC2 = 500 ...... (1)
As AD is median,
![](data:image/png;base64,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)
⇒ BD = 5 ...... (2)
In Δ ABC, using the theorem of Apolloneous, we know that,
For AD to be the median, we have,
AB2 + AC2 = 2(AD2 + BD2)
Substituting values for (1) and (2) in above equation,
⇒ 500 = 2 (AD2 + 52)
⇒ 250 = AD2 + 25
⇒ AD2 = 225
⇒ AD2 = 152
⇒ AD = 15
Question 2.
is the altitude of Δ ABC such that B—D—C. If AD2 = BD ⋅ DC, prove that ∠BAC is right angle.
Answer:![](data:image/png;base64,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)
Since AD is the altitude to BC,
⇒ ∠BDA = ∠CDA = 90°
So, Δ ABD and Δ ACD are right angle triangles.
Using Pythagoras theorem,
AB2 = BD2 + AD2 ...... (1)
And, AC2 = AD2 + DC2 ...... (2)
It is given that AD2 = BD.DC …… (3)
From equation (1) and (3), we get,
AB2 = BD2 + BD.DC
⇒ AB2 = BD(BD + DC) = BD. BC
![](data:image/png;base64,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)
Now, from equation (2) and (3),
AC2 = BD.DC + DC2
⇒ AC2 = DC (BD + DC)
⇒ AC2 = DC(BC)
![](data:image/png;base64,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)
Also, BC = BD + DC,
Substituting the value of BD and DC in above equation, we get,
![](data:image/png;base64,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)
⇒ BC2 = AB2 + AC2
So, by the converse of Pythagoras Theorem, we can say that ∠BAC is a right angle.
Question 3.In ΔABC,
, B—D—C. If AB2 = BD ⋅ BC, prove that ∠ BAC is a right angle.
Answer:![](data:image/png;base64,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)
Since AD is the altitude to BC,
⇒ ∠BDA = ∠CDA = 90°
So, Δ ABD and Δ ACD are right angle triangles.
Using Pythagoras theorem,
AB2 = BD2 + AD2 ...... (1)
And, AC2 = AD2 + DC2 ...... (2)
It is given that AB2 = BD.BC
![](data:image/png;base64,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)
From equation (1) and (2), we get,
AB2 – BD2 = AC2 – DC2
⇒ AB2 – BD2 = AC2 – (BC – BD)2
⇒ BC2 + BD2 – 2BC.BD = AC2 + BD2 – AB2
⇒ BC2 – 2BC (BD) = AC2 – AB2
Substituting the value of BD above, we get,
⇒ BC2 – 2(AB)2 = AC2 – AB2
⇒ BC2 = AB2 + AC2
So, by the converse of Pythagoras Theorem, we can say that ∠BAC is a right angle.
Question 4.In ΔABC,
, B—D—C. If AC2 = CD ⋅ BC, prove that ∠BAC is a right angle.
Answer:![](data:image/png;base64,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)
Since AD is the altitude to BC,
⇒ ∠BDA = ∠CDA = 90°
So, Δ ABD and Δ ACD are right angle triangles.
Using Pythagoras theorem,
AB2 = BD2 + AD2 ...... (1)
And, AC2 = AD2 + DC2 ...... (2)
It is given that AC2 = CD.BC
![](data:image/png;base64,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)
From equation (1) and (2), we get,
AB2 – BD2 = AC2 – CD2
⇒ AB2 – (BC – CD)2 = AC2 – CD2
⇒ AB2 – BC2 – CD2 + 2.BC.CD = AC2 – CD2
⇒ AB2 – BC2 + 2.BC.CD = AC2
Substituting the value of CD above, we get,
⇒ AB2 – BC2 + 2AC2 = AC2
⇒ AB2 + AC2 = BC2
So, by the converse of Pythagoras Theorem, we can say that ∠BAC is a right angle.
Question 5.
is a median of ΔABC. If BD = AD, prove that ∠A is a right angle in ΔABC.
Answer:![](data:image/png;base64,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)
In Δ ABC, using Apolloneous Theorem, we know that,
For AD to be the median, we have,
AB2 + AC2 = 2(AD2 + BD2)
As it is given that AD = BD,
⇒ AB2 + AC2 = 4.BD2
Since AD is the median to line BC, we have,
![](data:image/png;base64,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)
Substituting the value of BD in above equation, we get,
AB2 + AC2 = BC2
So, by the converse of Pythagoras Theorem, we can say that ∠BAC is a right angle.
Question 6.In figure 7.25, AC is the length of a pole standing vertical on the ground. The pole is bent at point B, so that the top of the pole touches the ground at a point 15 meters away from the base of the pole. If the length of the pole is 25, find the length of the upper part of the pole.
![](data:image/jpeg;base64,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)
Answer:Let the length of upper part of pole be x,
![](data:image/png;base64,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)
The length of AB becomes 25 – x.
Applying Pythagoras theorem in ΔABC,
(AC’)2 = AB2 + (BC’)2
⇒ x2 = (25 – x)2 + 152
⇒ x2 = 625 + x2 – 50x + 225
⇒ 50x = 850
⇒ x = 17 m
∴ The length of the upper part of the pole is 17 meters.
Question 7.In ΔABC, AB > AC, D is the mid–point of
.
such that B—M—C. Prove that AB2 — AC2 = 2BC . DM.
Answer:From the given information, we construct the figure as:
![](data:image/png;base64,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)
As AM is perpendicular BC, ΔACM and ΔABM are right triangle.
By using Pythagoras Theorem, we get,
AC2 = CM2 + AM2 ...... (1)
Also, AB2 = BM2 + AM2 ...... (2)
Eliminating AM2 from equations (1) and (2), we get,
AC2 – CM2 = AB2 – BM2
⇒ AC2 – AB2 = CM2 – BM2
By using the identity a2 – b2 = (a + b)(a – b) on above equation,
⇒ AC2 – AB2 = (CM + BM)(CM – BM)
⇒ AC2 – AB2 = BC(CM – (BC – CM))
⇒ AB2 – AC2 = BC(2CM – BC)
As D is mid point of BC,
⇒ AB2 – AC2 = BC(2CM – 2BD)
⇒ AB2 – AC2 = 2BC(CM – BD)
⇒ AB2 – AC2 = 2BC(CM – CD)
⇒ AB2 – AC2 = 2BC(–DM)
⇒ AB2 — AC2 = 2BC.DM
Hence, proved.
Question 8.In ΔABC,
, D
and ∠B is right angle. If AC = 5CD, prove that BD = 2CD.
Answer:![](data:image/png;base64,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)
ΔABC is a right triangle, right angle at B and BD is perpendicular from B vertex to hypotenuse AC.
So, as we know that,
If an altitude is drawn to hypotenuse of a right angled triangle, then the length of altitude is the geometric mean of lengths of segments of hypotenuse formed by the altitude.
⇒ BD2 = AD.CD
⇒ BD2 = (AC – CD).CD
As, AC = 5CD,
⇒ BD2 = (5CD – CD).CD
⇒ BD2 = 4CD2
⇒ BD2 = (2CD)2
⇒ BD = 2CD
Hence, proved.
Question 9.In ΔPQR, if m∠P + m∠Q = m∠R. PR = 7, QR = 24, then PQ =
A. 31
B. 25
C. 17
D. 15
Answer:![](data:image/png;base64,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)
In ΔABC, we know that sum of all interior angles in triangle is equal to 180°
⇒ ∠P + ∠Q + ∠R = 180
Also, given that, ∠P + ∠Q = ∠R,
⇒ 2∠R = 180
⇒ ∠R = 90°
Using Pythagoras Theorem,
PQ2 = PR2 + QR2
⇒ PQ2 = 72 + (24)2
⇒ PQ2 = 49 + 576
⇒ PQ2 = 625
⇒ PQ = 25
∴ Option (b) is correct.
Question 10.In ΔABC,
is an altitude and ∠A is right angle. If AB =
, BD = 4, then CD =
A. 5
B. 3
C. √5
D. 1
Answer:![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAALsAAACwCAYAAABXVx+pAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAB9/SURBVHhe7Z0HYFTF1sf/m+ymJyQkhJ4QmoLSBaQjvSki+p7l+Z6IHVEsdBJ6s4PwQPksyEMRFcGOAgI+UOAhVapAAkkgVCnp2ex3zty9m5tQtmR3s2VGl2x25869c+Z3T/5zZu6M3kQJMkkL+IEF9H5QR1lFaQFhAbfDbiw2Ia+wGPxTp9MhMiRQNoW0gFss4HbY53x/HF9sOY2q0UEC+AbVw5ByT11Ehbr9UtxiYHkSz7GA2wlLPZOHQW3i8WjXGgL2lM+O4r11mXihf4LnWEVeiU9awO2wGwJ1iIs0oHKEQRg0qUooiozFPmlcWSnPsoDbYQ82BGDuDyewbu95XMo14sS5PHw1splnWUVejU9awO2wFxaZMKBFHB7oUA351FH9avsZAf/MB+ojgDqsMkkLuMoCbofdWFyMWxOi0KhmuKgTy5qh7+wj8E0IDZKwu6qhZbkVEHokZ45PN2chO88oPPvXv59B7+ZxBHqAbA9pAZdawO2e/b62VbF27zmknskVFRt6R00Mah3v0krKwqUF2AJuh71zo2jwSyZpAXdbwO2wu7uC8nzSAqoFXAr7mh++xudfrcGV3EJ0adsYQ4YMhT44VFpfWqBCLOAy2Oe8+TpeWPhfoG5/BAaFYum/N+Gnjc9g8aI5CA2PqpDKypP6twVcAvuhfbsw8f2NCOswEuGhQUCxEcUJTfHZ+vno+Z8lePzJYf5tdVn7CrGAS2DftWs3LobVR3xYGEwFl0XFAnTZ0CXejl93/I7HK6Sq8qT+bgGXwB4dXQkBRbkwoSR2rgugUxVcQXxMpL/bXNa/gizgEtjbd+iI22IWY+vhbajSsDV0hH32pfMwHfkO3Ua+VEFVlaf1dwu4BPbwqMpY9MpoDHlhGvZu3gp9SASirhxEo9ZtsPZCQzS/VID4KNLyMkkLuNECLoGdr79pyzb45ZvFGP/2SuxJ+wuznnoMl0Lr4vWvDmH6ihwkD64rpvrKJC3gLgu4DHauQFhkDBp3HIjL1S6hZr06uDW0GHk9a2Le6hOY+sVRjKcnlOKjJPDuamx/P49LYWfj5udmozAvG3l5ucihTmrnRjGk4IH5BPz0FUcx4Z4kVJGSxt85dEv9XQ67tha8aEdOgRFdGHh6/+8fCfgvjxHwUtK4pbX9/CRuhZ1trQLftXEMrS6gePhpDPygJKnh/RxGV1ff7bBrgVc8vImATxcefrwE3tXt7dflVwjspT18ZQJeh/kkaWYS8ONIw8eaH8b265aRlXe6BSoMdi3wd9wSwwKHNDx5+BWs4ZMsqw84vcayQL+1QIXCXhp48vD0wQICfhpFacYPqotYGYf3WzBdUfEKh70s8Ez8gp/SMWOlouHV9WVcUXlZpn9ZwCNgV4HPpbBkt1tZ0jDwJxTg705CjNTw/kWli2rrMbCXePhiAr5E0nCUZpz08C5qfv8q1qNg10qa7gw8hSVZ0sxcmYpxd9eRHt6/2HR6bT0Odi3wPZrEigovXJOBGatSSdLUQXS4nEvjdAr8pECPhF2r4Rl4WskdC8nDKwNPdQl4j71sP8HGO6vp0dTQitZiLk3PJpXJuia8Y47SjKNOqwTeO4GryKv2aNivkjQE/8I1pOFXUaeVgK8U5vGXX5FtK89dxgJeQQtPHssjD9+jqRKleYeAn0HAc1hS7tghmbbVAl4BO1eGJU1eQTF6NaVOK3v4tTzwlIqxA+tID29ra/t5Pq+BXQWeB556NeNOq+LhZ1OUZjRFaSrJPZn8HGXr1fcq2LXA9ybgOTHws2ikdayUNNZb289zeB3sV3l4EvSL1mWIgaex5OGlhvdzom9Qfa+EvUTDG8VGBhyHX7SOPDxJmjEDudMq91aVyF9tAa+FXQt8n+YchwcWUad1FkVpJPAS9WtZwKth10Zp+jTnTitJmrUZmP0VR2kSERHi9dWT1DrRAj5BgxKWNKIvSxp6/3+k4WevSsOouxIRKaM0TsTFu4vyCdi1Hr4fbTvJSQD/dRpG31UHkSFSw3s3ps65ep+BXQHeJDx8vxYkaej9ez9n4hWSNGNo4Ck8WALvHGS8txSfgl3r4fu3rEK/6gj4DNFpHX1XEml4Cbz3olr+K/c52EtLGtXDZwgPP4okjQS+/NB4awk+CbsqafILgQEtFQ3PHv7VrxXgpaTxVlzLd90+C7tFw9M28QNI0vBcmvcZePLwIyXw5aPGS4/2adgtkqbQiDtbKWHJD9ZnAAz8neThpYb3Umwdu2yfh71Ew5twFwHP6YP1mfQvSRqK0oQFyU6rY+h431F+AbtW0rCH5/QhefhXv0ojD5+IMBmW9D5yHbhiv4Fd9fD5RSaLpPlwA0kaShJ4B8jxwkP8CnYBPM0tyC8C7rqNPbwJH25gSUPAU6c1LKhkK0svbEt5yVYs4HewW4CnsOTA1hSloU7rhxszERCgw8sDEhEqgffZm8YvYVc1PMfhB7aOF2HJxcLDm/BSf6nhfZV2v4WdG9TIkobCkneTh+e0mDw8k/8yhSWlh/c95P0adkunlQaeVOA/IuB1tNkTS5oQKWl8ini/h12VNAUUpVE0vAkfbTwpGvklqeEl7D5lAXNljMX0hoC/26zhl/xyUuzmx8CHGGSUxhfaXHp2TSuyhuc0qE28+LmEPDxLmhf7J0jgfYB2CXuZRhTAUxz+HgKew5LCw1Ov9UWK0gRLD+/VyEvYr9F8DHwBA99WCUsuJeAZ/BelpJGwe7UFrnPxqoe/l4DnxMCzpHlBShqvbW7p2W/QdKqHH8wanjz70v9ylIYGnsjDB+llp9XbqJewW2kxtdOqSBoTPt50ih5tpU5rvwSp4b2Mdgm7DQ2mAn9v26qKpPnvKdBUGozoK4G3wXwek0XCbmNTaIHnzip7eOIdI0jDBwVKSWOjGSs0m4TdDvOrwN93e1URpVnGkoaBJ0kjNbwdhqygrBJ2Ow2vAv83Ap49+ydCwyuSRgJvpzHdnF3C7oDBLZKGPTy5+E82myUNeXiDlDQOWNQ9h0jYHbSzRdK0Y0ljwqebs6gknZA0hkD2+TJ5mgUk7OVoEYukaVdNlLKMgOfJY8/1YUkjgS+HaV1yqIS9nGbVAs+Shj08Y87AGxwE/kJ2kVjB7PTFAnHzxEYY8Pf21ZAYF1LOq/XvwyXsTmh/FXgGkoFf/muJh9c7IGkyz+eLp6ae6lFL7AJ46GQOHp63Bwsfa4zGtcKdcMX+WYSE3UntrgJ/fwdF0iz/jTU8MLxPbbs7rbz0dp0qoRjWq7bl6matCsTc749j4eONnHTF/leMhN2JbV4KeJIf7OEV4O3rtLJ04SU/ruQZLasO81aY45f9iUJ6wMRReeTEqnplURJ2JzebRdJQp7WYnn76fAtreB2e61sbgTzHwMHEx/Ijg1y+QfQKZLLXAhJ2ey1mQ34GknF8gCWNzoTPfzstOprP9q4NWzU859dGdA5lZiMqzCAfArfB/tfLImEvh/FudGgRAc/GZeB1Jh0+M2t4W4DnTu7lXCOOn80TMubYmTy8vfoEnqdRWpkct4CE3XHbWT1SAV4ngDcSwSu2lHj4G0maSmF6sdjqiMUHEUjRHANJmMe71RSPCsrkuAUk7I7bzqYjGXhOD3WsLqTMFww8/T6MJM31gE+gePq3o5vTo4E0NkuHywWbbDK11UwSdqsmKn8GVdI82KE6PeikwxdbKUpD5D/Ts9Z1NXwAfR9ikB3R8lu/pAQJuzOteYOyBPDE7oMdq4otLL/clkUSRY9hPavjTOYxnDp1CrUTkhAdp8TpZXK+BSTszrfpdUssMhLwpMEf6lgNAYGBWL3jJNYuewN7DqXjdEE4qusvYtSQnnj0sSfceFX+cyoJu5vbWgE+AA93qYF1n87Bit8uIvL2Z2EICkLa5b8w9LV3ERUZgXv//qCbr8z3Tydhr4A2Zq+edfwo9h5KQ2TbZ0ib00UYcxAVFYXCW+7H/I9XYdA9gxFoCK6Aq/PdU0rYK6BtOZx47twZXCgMhT4olOYG5IirMBnzERwRg/O0cnZhXraE3cltI2F3skFtKa6wsBiJdW9CnWgj9l4+i8ho2rayKA8BQRHIPbEdTWuHISQy2paiZB47LCBht8NYzsjKcXM9rRmZF2BANcMFbFs7E0Xth8EQHoP84/th2r0USQNepGefAuQMGGcYXFOGhN3JBr1RcQx6WFgQzp+7gFnTJuNydh7GDOmA3UfX4NTpAtSrokedl4djR97NWPhTBp7qWVMC78T2kbA70ZjWQA8l0M8R6DOnpGDXju0YMXIsBg++EznZhcjNuYLIqGgU09SA9386TtOD6SFumkT2ZPdaYuRVpvJbQMJefhtaLUH16Az6jMkp2L3zd7w4ahz6DxyAi5dpuWDy34bQKOTkG2kKgQkPd64uBp6Wm1cteJyAL8fsYKvX5y8ZJOwubmkt6NMnTcCeXTvx0ujx6HdnP/LmRTTnnbf8KElFRtL0FK15hOLw4plW8wMgT9AjetLBl6+xJOzls98Nj1ZBP3v2HKZNHI99e/fg5bET0G9AX2QT6KYyoKuFqSOtQ7rWEAswqct0PNmjpguv1veLlrC7qI1V0M+cOUugJ2P/H3sJ9GT07d8HOTcAvSzwj3Q2e3iWNKThH+8mNbyjTSZhd9RyNzhOBf306dOYThr9wL69GDkuBX369UI2dUb58TpbEnt4XnCJJQ0vLqks06HMbZeaxhYLls4jYbffZlaP4PBiVtYZ8ugTcPjQAYwaPwm9+vQg0Em62Ai6epJCFXiWNJTE2pKUHu8uJY3VhiiTQcJur8WsePTw8CCcOnkaU0mj/3n4EEaNm4ievbvb5dHLnkILPC+1x8DzljePdVNuAJlss4CE3TY7Wc3FDptBP5mZhakp43HkyCGMmTAJ3Xt1Q/YV26XL9U6kAj/kjpoiSvPxJmUXv6EsaWSyyQISdpvMZD0Tg55JoE+eMAZpqUcxNnkKuvXoSqDbL12sAa8CvpSA5ySBt94+nEPCbpudrp+LpwAw6BknMTl5HI6nHsPYlCm4o3sXp3j0sicuIA0fRJ3WR8nDc1pq3uNp6B1S0lhrSgm7NQtZ+Z5BT0/PJOlCoKelYuzEKejarbNLQOdL4YElVdIw8ELS0C5+/PmjEvgbtpaEvRywM+gnjmcI0NOPn8C4lKno0q2Ty0DXXqoK/FDupBLp6raVqscvR7V89lAJu4NNy+HFE2npmEKd0cx0An3iVHTu2sEtoKuXrEoaIWHIw/+HdvHjKI0YeZXpKgtI2B2AgjujaanppNHH4lRmhgC9UxcC3Y4BIwdOe9UhLF0sGp6iMjx5bInYiRvKQJRMpSwgYbcTCAY99VgaJk0YizOnszB+0jR07NLerR5de8laDc+zI1nTfLRRidJI4Es3roTdDtgZ9GNHCfTxY3D27BkkT5qOdp1urzDQr6XhHxMjqyYJ/DXaVcJuI+wM+pE/UzGFpAs/LJ08mUDv2NYjQL9Kw5Ok4dk3S8jDs4b/F82Pl0nG2W1iIDwiCH8eOkqgj8OFC+cwgUHv0BZXaGRUxP48JGklzWPmkdWPaLsa/vyfEng5qGSNU/bohw8eEaBfvPgXefQZaNu+Na5cLrB2aIV9L8KStNYeA8/34ocbaG0OCbyE/UZEskdn0CdTZ/TSpYtCurS5/TYhXTw98XY0PNLKsyN5piUDz4/2/aOT/0oaqdmvQ20EgX7wwBHqjI6mkOJl4dHbtGtFHp1B9xzpcqObjsOS7OH5kT6O0nywXvHw/gq8hP0atLBHP7D/MIUXRyM3OwcpU2fitjYtPVq6XA96ZcOxAPAjfXyL+jPwEvYylLBH37/vkJAuubm5SJk2E61at/BK0C1RmqJisT/TExyWJEnzAUkaHnjiDRL8KUnYNa3NHv2PvQdFeDE/P488+iwCvZlZungvFmKklTx8sEXSlHh4fwJewm5mmD363j37xaSugoICIV1a3sage27UxZ7bj4HPF8AHmDU88P7PpOHJxT9k3qjYnvK8Ma+EnVqNPfre3fvEfHSjsUiA3qJlU58BXQVT8fBmSUOdVt7u6f11GSJK8wBtRe/rye9hZ4++h0Enjc4LFjHozVs28TnQtSArHl5Ha0lylMaE9wh4vhHu93Hg/Rp2Bn33rj/Eo3Tc2pOmzULT5rf4NOgMvarhudP6dK/aIkqzaG2GGIASGxX7aPJb2Bn0XTv3knQZQ3/GAzCROqNNmjWmKQC+odFt4ZU7rUGk4Z8WHl6H934mD093gq96eL+EnUHf8ftu0RkN1OsJ9Bm4tYl/ga7eDPmk4bnT+pQYeGJJwyOtOvytXVVb7hevyuN3sDPov2/fJZa7CAo2IGXKLNxy601+5dG1hFrCkrTn6lM9aovB4XfXpospBn/3MQ3vR7DrEBFhwPb/7RSgB4cEC+nS+JaGfgv6VZ1W2hHkSZI0HKVZRB6epwf7kof3G9gjIw3Ytm2HkC4hIaHUGZ2JmxtJ0LVhSZY0IQT8M+ZOK3t41vD33e4bksbnYWfvFE4efduW7eLh6PCICPLoM3FTowY0e9F/OqO2iGvLwBMBP4yA5/TOmgwRvrmvrfcD79Owq6Bv/W07efSxiIyshEnTZ6HhTfWkdLkO/Qrw1GkVHr6WCEu+Q/s78Rtv9/A+DDt59HADtmzehmm040VUVCVMpDh6Awm6VSevAh9iCMQw0vC8puS7a9LFSOtgL/bwPgm78OgE+q+bttAeRsmoFBMjpEv9hnWldLGKupLBAnwQx+E5SqPDQgKeRfzgNvE2luJZ2XwOdlW6bNr4K2bQrnSVY2MF6PXqJ0nQ7WRPDUuGBgXSSCtLGhNtWXlC3Aj3eCHwPgW7AJ2iLps2/Eo7XiQjNq6KGfQ69LSR7IzaybolO08eY+A5SsNhyQU/saTR4e7WVRwtskKO8xnYGXSOo29cv0lsvxhftSoNGM1E3XqJEnQnoMWd1jAC/tk+tTHvhxOY/+MJUao3Ae8TsKvSZcPPvwjQq9WoITx6naQECboTQFc1fB4DT53WZ3vXxttm4DlaM8hLPLzXw24Bfd0vmDk1BdUJdJ6mK0F3EuWaYlirC+DNHn4+Ab+APDxHaQbe5vmSxqthV6Mu69duxKxpE1GjZi0hXRLr1JIe3fmsW6I0DHw4AT+MPDzLGVXSeDrwXgu7Cvq6Nesxe/ok1KqVQB59BmonStBdxLmlWDUsGR5CwFOndd5qAp5enDwZeK+EXZUu6340g56QSB59BhISa0qP7mrSNeXnF5KHD9ZoeAKeJc2drTxT0ngd7GrUZc2PPwuPnpCYJECvlVBDgu5G0NVTCUnDwFOU5u3vj4uOKwfi72zpecB7FeyqdPnxhzV4ZfpkJNVrgOQp01GzZnXkyDh6BaCujLQy8BEkaZ7rm4A5DDy9OHka8F4Du47+PoaHGSBAnzGZ4uca0HPkgFGFkG4+qdDwLGkYePLwc8m7cyyet54f0DKuIi+t1Lm9AvaAgACEhemx+vuf8NrMqTT031AsMlqjZjUhXXjOtUwVbwEGPiJUj+HmgSeWNNw0/T0EeI+HnUEPJQN+/+1qvD57Gho0vBkTaGuX6jWqIoc8ugS94iHXXkEeAR9J7SU0/A/HycsrksYTgPdo2Fm6hIXr8d3X3xPo09Hw5kYK6NUV0GXyPAuokoaBH94nAXNJv88h4GkBB/RrXrGSxmNhDxCgGxTQZ02nR+ga02ZdU1FVgu55hF/jiljSCOCp0zr3OwKeXnwj9C0n8EW0DPeKradxIDMbtWNDxDOyHA2yJXkk7KpG/2blt3j9len09H9TAXp8fBURdZHSxZamrfg8LGmiCPjnCfi3CHZ+8VwaRz38lTwjhr23H+wI29SvhK1HLuK7nWexYGgjxNFsV2vJ42BXQf965TcE+gw0bdac9hmdgirxcciVGt1ae3rU9xZJQ8GF5/spYcnyeHhexIkcOxY/fYulni//5xB2pF5CzyaxVutuF+wXaFPb5z88KC68VVKU1cLtzaB2Rr9a+TXefGUmmjVvIUCPi4ulTQFIo8uoi70m9Yj8wsMT8ByHFxqeXhyW7NPcOqDaCmw+dJEWYC394Pdr/2gII0+ytyHZBfvaPeex6dBfqEVaydmwq6CvWrEKc96YTYuLtsK4lMn0pFFlGXWxoSE9OYsYeCLgo1XgWdIQ8IxoXzuAL6TBq2jqx5VNgTxHwYZkM+y86CXro1cfaojlv2Yh/Xw+alUOtuEU1rOooK/84ksC/RVaLrq1AD0mNoZ2v5Aa3boFPT+HFvjh/Wjg6bsTQtLQHmfo1cw2D8/6/9Rf+aUqyx3VnPxitEyKtGoEm2HfdfwyCo3Ks4e76f2KrVk0WpZg9QTWMqigf0mgzyXQW7Zqg7HJkxTQpUa3Zj6v+549fAyNhD9PwL9FwL/xbZqQp72aWge+R5PKWLrpFD0sEi+W+uAtMEcsPih+dyrs3+04i33pVzB95THsSruCy7lFGE7zmXm+iqMpIJAGjEL0+Hz555j31mto3aYdxiRPRHR0tATdUaN6wXGKpDFgBGn4N79LI+AVSdPbCvD303La/zt2GYNe34VmiRH4Iz0b9aqG4ZEuNWyqtU2ePafAiN8OX8TDtEtyFQrx1IkLpbUA07Fh/wV0bVzZphNdpbMCA2m9xUAsX7Yc8wn0tu06EOiTUSk6SoLukEW966BcBp4GDF/on4g3CXZ+8UPcPcl7Xy/pSZu/9c+G2Ez9xoOZOejdLI74i7G54jbB/s32s+AT8Z2oJtZOvDRal0Yxdnv3QAadBgI+++RTzJ/zBu0Y3QljJkxEpUoSdJtbzsszajX8CIruvcUe/huSNOTjrYUR2zeMBr/sTTbBztro0TtK/6ngpdCy843IKVDmM9uaVNA//WQZFrz9Jtp17IRR41NQKYpAl51RW83oM/mEhicPL+Lw5N0Z+IDAYHRvHIGzJ1Ppr3wOatdtQLre+qCRNaPYBPtDHa/eeiQhLgQpg+taK7/U9wx6AHUsli39GAvnzUF7AfpEWpouUoJulyV9KzMDX5lCis8R8PPXZGHeqp344I3l2Hr4HIUs9WgUX4xpo59A63adylVxm2AvzxlCwyMRFErzJKJCERwELFm8FO8Q6B06d8XI8cmIiiTQ82R4sTw2dvRYq6EFTQblLf2r/G9J4v1V+dSsJV9cN5+mrATqCw7vWRX97h2N/Tl1ENn0afLyevyYcRB7npqOtUtnotGtLRytLlwK+58H9mD1R5/gQPpFbKjcD5cuXsK7C+ahY9duGDl2Aq2qG+FxHr0UADeg4eqvzJ+U/nFNKCzHCnAsGN0AmjJwlSWOvzYPIvISdddK4tMyX1l+tRxb+kj+mHfgKKaXqRjip3hPX5T8VD9X82p+0sgmD27ySymnTF4q08if0xc8CmoIDsO+335EZnYwYtvej4CiK3RgPqrWa4aTl07i/SXL8epsD4R9xWef4LHJS5BTpSP0odXw0MRliM5ah38NGYIXR42jhUfDkJdXCAOPKlzLM6h21wBhcSJlgSrjbZRDVYzMBV0DXP5IhUCFhBtShYDfC0bMb5TGL/ns6t8VECyfc+PSb7TjpAKKBhjR8JbPNSCYP2MIGDDxk/IyDMoxyk8jHayWwd+JlwUcM0Q0LsKfiWPVPCKfUp6Rz2X+vOT70sdoAeX8KvzqtajglwbanM9clxKbaO2jWJzbQLQU/R8UGokjv+xCfkg1hNCU4GI2ANu8MAe6mFrIOEtLZ5cjucSzZ6QdwXOzliK7ybOIjiO9Tw1TlNQJWeuBenXrI7xSGM6dzRMLZQpDaO9+M1BaD6KFQjSQ2qgWCEoaToVK6zEUOJTjVDCUxi35jM9RAoYZBpFfA4T5dwU8BV4LhBqolPMpgHFzaetn8YpmT8vwWGAw31glN5T2ZlTAoMV0zbcoI2ISkTDxH33Oo+bK72KxXfE7zxBUPxe/8/fmn5bPCSzOF0hf0NCH+T1goP4VD8WXvLT5lLza7+k5GxG1E2WVOc7yO5+DHBz7OPHTfF4+Njg4GGn1OmPIhHfpMT8jggL0ZBuyYFA4TGf+ROM25du20iWwb926BRkB9REfW53+Cl0U965eH4Lwht3x1rKN2KnvhPz8ArHJrtZbaf/UKe+pORkGFQyLZy35Y309paF6C8XHK40vQNARIGY4xOfC2CoEChhKw5d8xw2iQiPeUzl6ah1qC/Gej9E2HPXDzeBoGpbyl4XjWrCojX8VYOp1qhDxectCKj4zA2u+LvW6BeQMpxl2pR4l8Cu2UW1VcsOUw5E6dGirxK74Zc03eGv1uwhpfg8CDUHIPrgWjXX/w6P/nOtQmepBLoFdbzAIr6OViGzMoiIjKkcGoVH1UBQZDcLw3GBaL8CewQKBuPPN4AhvoHgCxXuYoTQ3WomHKGlsFVIBs/kYcU6Nx1NgKGl4Ua75dxUArce0vL/eXVau5pAHswVeoeeMY2NexaqNi1Fo0qNxQhCmjJuFGgn2Rf/KWtMlsLdr1w71ApcgNSsNsdXrkPY0illv+Yd/QMob/0Cv3rYN78qm908LGIJDMSE5BaNzL6EgPw/h0c7Z/MAlsMdVrYW3k4fikTH/RlZqcyAkEoFZ2zDpwRbo1W+gf7agrLXdFjCERoFfzkougZ0vrk//gdjcoAHWrPsZhSRfWjQZjvZdejjrumU50gJ2W8BlsPOV1G3YGE/QSyZpAU+wgEth94QKymuQFnBpNEaaV1rAEy0gPbsntoq8JpdYQMLuErPKQj3RAhJ2T2wVeU0usYCE3SVmlYV6ogUk7J7YKvKaXGIBCbtLzCoL9UQL/D+KLoTPao2WswAAAABJRU5ErkJggg==)
As we know that,
If an altitude is drawn to hypotenuse of a right angled triangle, then the length of each side other than the hypotenuse is the geometric mean of length of hypotenuse and segment of hypotenuse adjacent to the side.
⇒ AB2 = BD.BC
⇒ 20 = 4. BC
⇒ BC = 5
Also, CD = BC – BD
CD = 5 – 4
CD = 1
∴ Option (d) is correct.
Question 11.In ΔABC, AB2 + AC2 = 50. The length of the median AD = 3. So, BC =
A. 4
B. 24
C. 8
D. 16
Answer:![](data:image/png;base64,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)
In Δ ABC, using the theorem of Apolloneous, we know that,
For AD to be the median, we have,
AB2 + AC2 = 2(AD2 + BD2)
⇒ 50 = 2(32 + BD2)
⇒ 50 = 2 (9 + BD2)
⇒ 25 = 9 + BD2
⇒ BD2 = 16
⇒ BD = 4
⇒ BC = 2BD = 8
∴ Option (c) is correct.
Question 12.In ΔABC, m∠B = 90, AB = BC. Then AB: AC =
A. 1:3
B. 1:2
C. 1 : √2
D. √2:1
Answer:![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMgAAADICAYAAACtWK6eAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAABxtSURBVHhe7Z0JfFRVlodPJamEQIJJCHvIAgiEfVEQVHbZRR3tEfcFbJ1haUUWZXqG3/RMK8pOy0iL2qJouwGyK5sLKLagKCpqo2JAFhFkzZ6qmnPufbeqEhJSqbxX9ZbzEKl69d599/7v+d455963xPlwAV5YAVagUgXiWBdWgBWoWgEGhK2DFbiIAgwImwcrYDZAvF4f7D9WANkNEyHBHcMdxAqYVoGoeJBPfjgLw2ftgafHtoMxfZqYVhyuGCsQFUBW7ToOvVrXhw17TsDNvZuAy8UdwQqYU4GIA3L0dDHs/vEsLLyzLUx9+Z+wc/9p6NMmxZzqcK0cr0DEAVn32Qm4tHFd6JiZBP3ap8FrH/3CgDjeDM0rQMQBWbv7V4jHxPzZbYfhyKli2Lz3JJzKL4PUehGvinl7hWtmGgUiapVfHjwPh04WQd/2qfDVofPgjnPh3xhYvfs43N2vmWlE4YqwAkqBiALyyofHoDfmGwvvauvvAfIky7cfhTuuagqxsZyts2maS4GIAXKuqAx2fHcaZt6UU06BUd3T4al3DsJH+8/A1e04WTeXeXBtIgZILI7lLrq7LXRqkVRO9SYpCbB8QidIqefm3mAFTKdAxACpmxAL3bKTKxWgYwVoTKcSV8ixCkQMEMcqzA23tAIGAeKDNSteg20f7AS622Rg3yvgupvGoFCchFvaWhxYed0BKSrIhwfGT4BlO04DZPQWUCza+AbcvuZteObpxZBYr3wO4kDNuckWUkB3QBYunA/LthdBg6H/BbEuj5DC02EwLH97NuTOnwsz/jjTQvJwVZ2ugK6AlJYUwopNn0B897sh1lsEPk+J0Dc21g0Jl42BlVuWwpSHz0N8InsRpxueVdqvLyBFBZBfFgfuhHqYe5T5NfB5PeCukyR+K8ZtGBCrmAfXU1dAEpNToVXjOrDvRB4kpfQAb/F5obDLXRcKD38BHZrXheSUNFadFbCMAroC4nLFwKRxt8KG3z8Op+s3hvqNMsW41fkTh8DzyTNw2cwJSIuuh7SM0FxRayqgu7UOHjYKlv3vLzBj9hw49k0mxOAMehPfIcgeNhw+9faE9XiT1Mhu6dZUi2vtOAV0B4QUvO3OsTBg4DUwaf56KCjxwJwJ/w2lydkwe9W38H+bDgE9iGsUQ+I4Y7Nigw0BhIRolpEJrXuPhrMFHkjHz8luL0wcmoFzIofg6XcO4RAwwPCu7EmsaDROqrNhgJSW+aCkMB9KizxQUlwM+d4Y6JCRBJOGtxCQPIWQ0Cz7CPYkTrI3y7XVMEAqUyK/2APtNUgWbjwIizHcogc2sCexnN04psIRBYRULUBIyJP8YXgmLNQ8CUEyrAuHW46xOgs1NOKAkDZ+TzKMwq2D8NTbh2i2BCFpYCHpuKpOUCAqgChPQveBTERPoiChOZOhDIkT7M4ybYwaIMqT0B2GkzRIFr19UFwRP7QzexLLWJDNKxpVQBQk5EkEJBvy4C8YbpEnGcKQ2Nz0rNG8qAOiwi3yJBRuESD0lxaGxBpGZOdamgIQ5Uk6ZybDRJG4MyR2Njortc00gFwACeYj5EnoWq7BnfgKYCsZlZ3qaipAFCRdspJh0jAZbtGEog//XNOJE3c7GZ5V2mI6QIIhmahdlqIS98EMiVXsyjb1NCUgfki0nOQv6EUWaeHWwI4cbtnG+izQENMCoiDpig+bm6iFWwtEuAUwiCGxgGnZo4qmBkRAglcDC0hEuIWeBP/SFY6DOqTaowe4FaZWwPSA+D0JJu40BKwSd3r15wCGxNTGZYfKWQIQBUm37Poy3EIvsmBjnrhUvj++a4QXVsAoBSwDSACSZJggcpKDMB8vTaF5kr65KUbpw+U6XAFLAaIg6Z4TCLfmrc8DL96ayJ7E4ZZsUPMtB0gAEgq3ZOK+YMNB9CSAnoTDLYPsxLHFWhKQYEgo3KL728mTUE5ydTuGxLHWbEDDLQsIaUG3717Wsj5MGNoCFmuQeHGipB97EgNMxZlFWhoQmjSk23cJkvEIyVOYuMtwy8XvO3SmPeveaksDotQQkLQiSDKFJ5mvhVtXteXRLd0txmEF2gIQEW7hjPvlCMkETNzpIRDz1uWJ527xm3MdZtE6N9c2gFC4pXISEW7hq6XJk9Do1pXsSXQ2G+cUZxtAqMsUJORJCBIKt+aiJ6GFIXGOUevZUlsBoiChnKRnq0vAN8SFkByEuesPiiHgPm04J9HTeJxQlu0AUZ1G4VavS+ujW0FPgo84nbPuIEwZ5UJILnFCv3IbdVLAtoCIcAsT916XoifBz/TahbnrfgLXqGzozZDoZD72L8a2gATnJFcoSERO8hNMuTYbaB0vrEB1CtgakGBIemtAkCeZszYPHh6VxZ6kOuvg38H2gFwACU6OCEjQk0xlT8IIVKOAIwDxQ4Kvg+uNI1leXPG0yEnyEJIs6Nmawy0mpXIFHAOIgASzdRrduhIhoef/EiRPYrg1BcMtzkkYkcoUcBQgwZ5EzIkgMIs3/yw8yRT0JL3YkzAlFRRwHCDBnqQPXoJC4dYSkZPkwTTMSWgWnhdWQCngSECCE3e64teFruRp9CRPrqHEnXMSxiOggGMBCQ63rmybioi4xOupaQh46ugs9CScuDMo4Ixh3ot1tEjccXSLPAl9XrL5EMxGSKaPzoEeLZPZRhyugKM9iOp7BQndOyJGtxASCrcoceecxNmEMCBa//s9CULixYBryaaf0ZP8BNNGZ4tbenlxpgIMSFC/K0j60pNRKNzagpCgJyFIejAkjiSEAanQ7X5ItCejLMHRLcpJpmHi3j2HPYnTKGFAKulxPyT43F/6/Ff0JE+socSdIWFAnKZAFe1Vl6X0o4dju3zw182HMXEnT5KNnoRHt5xiJuxBLtLTypP0y01DT+ISnoRGtx65Phu64usYeLG/AgxINX2sIOkv3kWCnmTLYZi1Wibu3fHFPrzYWwEGJIT+VeFW//by/YhLEBIaAp6O127R2694sa8CDEiIfev3JAgJPf93qUjcf4JHr8uBzllJIZbCm1lNAQakBj2mIBkowi0XPIOQzFp9QIRb7ElqIKSFNmVAathZEhKv9n5EykkwcadwCyHpwol7DdU0/+YMSBh9RJAU4gWOAzrQ6JYPntl6RIRbj1C4lcnhVhiSmnYXBiTMrlGQDOzYQLyWeikm7k9guEVXAXNOEqaoJtyNAalFpxAkRehJBpEnwVsTl27DnGTNAZG4d2JPUgtlzbMrA1LLvqARLbqfZFAnHALGGfelW8mTYE5yXTZDUkttzbA7A6JDL6hwaxCFWzQETJBgTiIgacE5iQ4SR60IBkQn6f2QdGog5kme3SZn3B9FSDoyJDqpHPliGBAdNVc5yTWdG4g7E599NxBuMSQ6Ch3BohgQncUm70GJ+2D0JPRUefIklJPQBY4dMjjc0lluw4tjQAyQWEBS6oEhnXF0C/88t+0IzHqL5kkQEg63DFDcuCIZEIO0JUgKccZ9SOd0DLdc0pNQ4o7zJB1a1DPoqFys3gowIHorGlSehMQDlJN4MUF57l054/4ohlu5zRkSA6XXrWgGRDcpKy9IJe5Du8gh4OffOwKPvyUnE3MzGBKD5a918QxIrSWsvgCZuHthaNd0cVnKczi6NUtcu8WepHr1orsFAxIh/dXo1lAMt+gCx+cx3BLzJBhutWvGniRC3VDjwzAgNZYs/B3k6JYXhnVJF0PALyAkj+Po1qPX5yAkdcMvmPc0TAEGxDBpKy+YknUaAh7eNZCTzMKchOZJ2JNEuDNCOBwDEoJIem+icpLhmJOQJ/nbe4HLUtpyuKW33LUqjwGplXzh76wmE0eQJ0FMXqDRLcxJZmC41aYph1vhK6vvngyIvnrWqDQJiQ9GdMPRLfz8NwHJAQHJpU0YkhqJadDGDIhBwoZarMhJSnwwvFtDmbi/L+dJ6PZd9iShqmjcdgyIcdqGXDJ5kmJM3EcqSNCT0BDwDEzcW7MnCVlHIzZkQIxQNYwy1RDwSC3ckp4Eh4BvQEgac7gVhqS67MKA6CKjPoVQuFWMOcmo7g1Fgcs0SGjGnXMSfTSuaSkMSE0VM3h7Nbo1srscAl72vryfhBL3lo0TDT46F19RAQbEhDYhcpIy8iQSkhc/OAKP0QWOCEkrhiSiPcaARFTu0A/mRUqKSwGuRUjo9l0ZbhEk2QgJ5yShK1m7LRmQ2uln6N4iJ9E8CXkV8iRidOuGHMhpyOGWoeJrhTMgkVC5Fsfw4gPpivECx9E9aDLRBy9tPwqPrZJDwDmNGJJaSBvSrgxISDJFdyM5T+JDSHB0C+OtF99HSDDcosSdITG2bxgQY/XVrXQKt0ow3CJIPOhVXkZPQjkJhVvZHG7ppjOPYhkmpfEFe9CTECQ3XN5QJO7LdxAk8qYrhsQY/dmDGKOrYaV6MN4qKQO4HiGhReQkCAmFW9kN6xh2XKcWzIBYsOcVJNchJBR6vbLjGHqSHxGSlpDFkOjaowyIrnJGrjCCBNCT3NCzkTjoywQJXipPT0thSPTrBwZEPy0jXlIwJPR4oVc+lJD8B4ZbLdI53NKjQxgQPVSMYhkKkn9BT0KXpfwdIfnzKnlZCnuS2ncMA1J7DaNegoAEF4IE3+EjPAk9d4tGtzIbsCepTQcxILVRz0T7+iHp1QhwmgRe/UibJ6FwiyEJu6cYkLClM9+OCpKbEBK6yZ3CLbosRXgSzknC6jAGJCzZzLuTguTGXo1FTvIqhVva7bsZ7Elq3HEMSI0lM/8OCpLfISREyWsf0eiWfB0cQ1Kz/mNAaqaXZbb2Q3IFQiJyEgVJDkKSYJl2RLuiDEi0e8DA4wdDQvMkr+2kcEsOATdPY0hCkZ4BCUUlC2/jh6S39CSv7fxFTCbSgyAy0ngIuLquZUCqU8gGv18ICXoSusARL5Vvlsqe5GJdzIDYAIBQmhAMCb2f5PWP0ZNob7pqxuFWlRIyIKFYl022UZD8a58m2CIXQiITd3r1QnP2JJX2MgNiE+MPtRl+SDAnoXmSNxASej8JXSrfNDU+1GIcsx0D4piuDjRUQXKzgMQHb1K4pSXunJOUNwgGxIGAUJMJErptd0xvDLfQlbzxj1/EExxpdKsph1t+q2BAHAoINbtMuwr4ZsxJKNx6EyGRLxbNgSYpHG6RRgyIgwFRkJARjBGJO8CKfxzXwq0caMqQMCAO50M0nzyJHxJ0JeRJnhA5CXsS9iBMiB+SWPIkVzbB+0l8sBI9iXqPe+NLnBtuMSAMiF8Bkbhj5n4rQuLCPyvQk6gHQTgVEgaEASmnQBk+nS4uxgW3ICQ0477iE/Ik8mkpjRzoSRgQBuQCBUROgp7klquaitGtVQQJ3uM+fXQ2OM2TMCAMSKUKSE8iwy1aCJInEJJHEBIneRIGhAGpUgHhSTDcIkjofpJVu6QnoTsTG9Z3RuLOgDAgF1VAQXLbVeRJfPDWrl/F6NZ0hKSRAyBhQBiQahUQkMS64DbMScDnglW7ZbhFiXt6srva/a28AQNi5d6LYN1FTkKQXC3nSdbsVp4kCxom2zfcYkAiaGRWP5SC5I6rm4oLHd9CSOgCx+mjczAnsacnYUCsbrURrn/Ak+AQMCbuqzHcepKGgDEnsWO4xYBE2MDscDi/J+mLOQkuqz8lT3IAIbFfTsKA2MFio9AGBcntCAnm7bAGR7fIk0zDeRI7eRIGJArGZZdDBuckPnxi9trPEJK1cjIxLckeOQkDYhdrjVI7CBI3jm7diZ6ELnRcI8It9CSYkzSwASQMSJQMy06HLdWGgO+gcAsbtlYl7jbwJAyInSw1im3xexIcAvbixOL6z074cxIrh1sMSBSNym6HJk9C4dbd/ZqJcGvdpwRJHibuWZbNSRgQu1lplNujwq27+jYT8yTr0JPMXpsn5klS6lrP3KxX4ygbAB++egVUuHUXehKCZP3nMtyaem0WpNaz1ugWA1J9f/MWYShQguFWPIZbd/XXwq09v8KcdQBTR2VBioUgYUDC6HzepXoF6FotlZOQJ/GiK9mw5yTM9lFOkg2XWCTcYkCq72veohYKKEju6d8cS3EhJHLGfSpCYoWchAGpRefzrqEpEBjdovtJfLARc5I5OOM+9VrzexIGJLQ+5q1qqYAfkgHNxWTixi9O4IyiHAKun2heMzRvzWrZIby7+RRQifs9mLjTsvFzzEnWAkzBxN2sOQkDYj47sm2NKHEPQNIcE3eAd9CT0PopOARsRk/CgNjWHM3ZsODRrbED5BDwO+hJaCFPUt9ko1sMiDntyPa1CoxuyclE8iTkSqaOyobkRHpKsDkWBsQc/eDIWhAkNJl4L3oSeszppr2/ISN5wpOYBRIGxJGmaZ5GU07ixuecjh2I8yQYb236QoZbD4/KNEVOwoCYx1YcW5PSMvQkBAl6Ekrct+xFSFw+EW4l1YluuMWAONYszdXwYg2ScQgJJSVbvjwJMRhwTR4Z3XCLATGXnTi2NmIIGCFJQE8yDsMt+r75y9+EHtGEhAFxrEmar+EKEjc+Vp5yEkxPYOtXMiehS+XrJkQ+3GJAzGcnjq9RaZlXJO73aZ6EIKH5EvIkeuUk54o8+CDu43DgeCG0bVYPbuzZSDxateLCgDjeHM0pgAi33BhuDaKrgMmTBMKt2kKy7+d8GP/8N/gW3wTIaZwIL+84Kl43t2Rc7gW3BjMg5rQPx9eqXE6CkNA8ybavaZ6EhoDDD7do7mXyS/+E/u1TYeZNrYTOdK/K9XO+gKXbDou3aAUvVQJSjG6OxqSPni4WM530IpU+bVMgt3k9x3ceCxA5BWTiHgP3ESR42G1fnxLzJZNHZkK9MHKS3T+ehZPnSuAhDNfUEoPlPf9ABzhXWBZ6iHXqfBlMf+V7aJ9RT7ii/GIPPPvuEZh5Yw4M65oeOYX4SI5XoARP1gnuGPj9oAyhxTYMt1yICxl5TSH54VgBpOOLfypeGEmPS63skalVehAPug3aYcFdbSAjrY6o2PLtR+GZrYcZEMebbOQFkDmJhIQimq1foSfBpaaQUCJO4VqoS5WAqHyesn1RHP7vVH4ZNElNCLVs3o4V0FUBCvsFJINxngQNlCCh97k/hOFWqEPA7TFFOHm+FE6cKy3nMbZ/exo+3n9a3OUYUg5CpFFYNf65b8UzVs8VeuCbI/mwemoXXRvNhbECoSpAJ22CpA5Cch96EnFZijYE/CCGW3Xj8bW81SwdWiSJiOixVT/CvDvbiq3PFJTBjFf3Q+9LUy7Yu0oPQo+PJFrvxbu/WuJQGF0v8943p2De+jxYfG8uJEf5GpnqhODf7amAGt0iSO5HT0KLmnF/cET1niQWB5vm3t4G7lu6D0bM2gPNMCL67mg+tGpcF0e1WoYOCM1i0mMkB3VKE0k6LVe3S4GOU3fCN4fzoWer+vbsAW6VJRSQniQW7qd5ErRVgoTCrgdHZEFiNZ6ETvjrpnXDfU7C4d+KxWuuB3ZMq7TdF81BCku88D1m/bF0ZFxe//gXMduYkcZ5iCWsyMaVlJ4EIYlHSNCT0ItF6X4SWkKBhCAa3aNhtQpVCQi5sLSkOJj+8veCSErUad18jNvILfHCCphBAeFJ0D4fGNxCTCIqSP4wIhtzkgsvHalpnasEhJ6h+vqDnaGoFF8dRHTgseh1v5ozqelxeHtWwBAFZOLuw5O4Cz2JHALe8vUZSEw8Bi0Ld8LGdzbDb+eKoHv7HBg3bhw0aCSfqBLqUnWIhUemyZT6iaEWVX47F5LkionFvwRX7UkOrxa8lxMUUIl7XQy3HhiSCfHxx2DRnx+BAz8fA1fLYeBKSIbX39gLL668Bd58YSHkduwasiyGXYsVR1cmlxWBr7QMkpPQ8yDZBQUlzErIXRPZDSs9hQWtDHzEE1+FjcVXbV25n/CL/B78/yq2raqMSmWg2Q+5+OQsnfyMH5uluCGn4EM4cPAopA79E7jdaOJeD0DrXrBvzwZ46JE/wdqVr4A7Xk5+V7cYAsjpk8dh3rz5sGLzPijD4eLk79vBv02YBE1aZED+eXNDcoGhVOP8yv8sLaLiLsEGFGQH0nSCfqzU0CqUR6YhjYK2Dn1GWBpT5btcUIq2gi7iI6Ojr/SZ5h18Xi8mxNIY1e80JSC+U83oX9xAbq9917YV+9N67XdVhvru0dZ7cENaR1dzUNm0XqzD72Ibsa7Cd61+ce54WP73jZDQbqSAw1eSLxnwFENqbn/46MMd8MN3X0O7Tj2qY0P8rjsgRw/nwXVjxsKuczhX0u4ONAAXzH93O2zefge8+tJSyGrZGoqKSv2Vu5jxSAui/4LPX5ptBP4JMjRZbPD2co00KNnvFzcudRWCMia1T2Xr5TbSXFXnCwPQDEEaQMCQAr8F1ikjUNtSx4vyhAGof4M/S2NR+8ltgo1JGhedmISRasZV0diCjVCUJ4xSM0ZhiFgPv6EHAJH1Kl8fdRw/RAoIrQ3UNjq+gM2vj6YBSUjEUasrYV70VkUbCPZWQSeQmJgYiI2NE/Ma8Zk4DeENuvgQC4rBEVhPbBIU5J8PCQ5DAJn15FzYdb4zNOp3L47DnRUVqdN3LHy1/UWYPXsOLHhqMSb+1AF4vlGGFGRgFY1LCSo6L8hg6ExCv/nXaZ0hzyxSfA/+qD6X318zoIuclVQ5AeMpb4Tlfw86Ywqj0r5T+X5jV+AEjJ9QIAMOPhuL9kruAgYlvksDU4sCOPg72Y0wJsr//J/lCUOulzZIv+N8mfgrthWf5Tr1m38bXCm3pZxS/kvD/jTcT5Nu8q+2Xvus1sfgd7wQV+4TtK3/d7Ee0Kjl77Js+V3tgzYvjhcoE79TWdq+dAyxr3asOgkJ8NypzvD4+i8gscVl2NElAj5XXDwUnDkJTd2nIKdVm+gAUlxwFt777CdIzL0fhxbO4JlBI7joDKR0vgbWfTwH3Et3QVy9BlBWWnqB8QRcc5Ar1ty2MA4yGi3AUAYkT+KBSFSs187s8rwUiFaFKpoDUWclv+FoRiWNRDMIvyFpRqR1tjIY6jxhOEEGJjsMIB6tSRgddXAlxhFYFzCyi29H5eAZkgxIGIT8TnWh73Q7AhmwNB6sl2ZE9K8yMlU3WS/NOIPrX0X7KraX9lUAkqTBWoZseQZuOHn8PbBqw23w7Z6mkNp+ALY1BgrPnoTC9xfAQ1OGQGpDfMp8iIuuIRbFp76YODF6FZw80anPhevB5caOpCE5HN1y+8oZkOroYMOp9OzhP5toRqDOJuqMpp2B/OUIA0JDEsajGZn/rCbLUIaszo7K8FWdhIEEnUnLfw6cff3rg7YNsR94Mx0VSG/SAlYuWwgTJ/8n7N7xIZTE1YV07zGYNGkITHhwco2OpCsgdZJSoHurNPhy/x5I7j4SvOg5aImpcwmc+vxtuLVbY5h/3+VobAiI5uLlGTuQY9So9rwxK1CFArkdu8GmjSth3949kJ+fD9k5raBx86wa66UrIHT0aQ/9O2z63Xg4ujceknNwpABDmnP7t0KjIytgxqLFOOup+yFr3GjewRkKxMTGQ8duvWrVWN2ttX2n7rB++Tz448wnYN9XH4jKXdXQBf+D6zp0waSJF1bAQgroDgi1vdtlvWH9+rfg2KEf8ZsP5z/oMmIOoyxkF1xVTQFDAFHqSjB4YQWsq4ChgFhXFq45KyAVYEDYEliBiyjAgLB5sAIMCNsAKxCeAuxBwtON93KIAgyIQzqamxmeAgxIeLrxXg5RgAFxSEdzM8NTgAEJTzfeyyEK/D/+TxoY2vcSHwAAAABJRU5ErkJggg==)
In ΔABC, applying Pythagoras Theorem, we get,
AC2 = AB2 + BC2
As AB = BC,
⇒ AC2 = 2AB2
⇒ AB:AC = 1:√2
∴ Option (c) is correct.
Question 13.In ΔABC, m∠B = 90 and AC = 10. The length of the median BM =
A. 5
B. 5√2
C. 6
D. 8
Answer:![](data:image/png;base64,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)
In Δ ABC, using the theorem of Apolloneous, we know that,
For BM to be the median, we have,
AB2 + AC2 = 2(BM2 + CM2)
Also, as Δ ABC is right angled at B,
⇒ AB2 + AC2 = AC2 = 102 = 100
⇒ 100 = 2 (BM2 + 52)
⇒ 50 = BM2 + 25
⇒ BM = 5
∴ Option (a) is correct.
Question 14.In ΔABC, AB = BC =
. m∠B
A. Is acute
B. Is obtuse
C. Is right angle
D. Cannot be obtained
Answer:![](data:image/png;base64,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)
Let us assume,
![](data:image/png;base64,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)
⇒ AB = BC = x and AC = √2x
As we see that, AC2 = AB2 + BC2
So by converse of Pythagoras Theorem, ΔABC is right angled and right angle at B.
∴ Option (c) is correct.
Question 15.In ΔABC, if
then m ∠C = ……
A. 90
B. 30
C. 60
D. 45
Answer:![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAQEAAACTCAYAAACONawBAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAB7wSURBVHhe7V0JYBXV1T5ZyApJSAgBQlY2ZZOCIqgVWcQfUBb1b91al7ZYl4pQwSou1RbUAkUBrdpa/bWKigoooLggshStrSKiWFnDvieACWR9//nuzLzMC4G8zEzeNue2yGNm7p17v3vuN+fee+45sR5OJEkQEARci0Csa1suDRcEBAGFgJCACIIg4HIEhARcLgDSfEFASEBkQBBwOQJCAi4XAGm+ICAkIDIgCLgcASEBlwuANF8QEBIQGRAEXI6AkIDLBUCaLwgICbhABkrLq2nZ+mIqLq0kmIe2S4ujwT3SKToqygWtlyY2hICQQEMIRcD9bQeO0+/mbqThvTIoJiaKFv3nOK36bwk9eGWhEEEE9K/dJggJ2EUwDPJXVHuoa3YyTbuus6ptRVUNjZy2lvaWVFC7lvFh0AKpYlMiICTQlOiGSNnN+Ot/mKcCH39TTLH8e+22Y3RGu2TKaNEsRGoo1QgmAkICwUQ/QO8GCRw8Vkl/XbZLkcDWfWV04ZktKZpkTSBAXRDSrxESCOnucaZyxytqqGNWEr3ym+7eAsfM+Ire/s8BuuLc1s68REoJWwSEBMK26xpX8fLKGqqs8lCz2CjeJaiiI2VVogc0DsKIfVpIIGK7trZhzeNjeDpQQdc9uZ7iY6OphNcH+nZIoeG9W7mg9dLEhhAQEmgIoQi4n5+ZSK+O60HHTlQT/EilJsZSXmZCBLRMmuAEAkICTqAY4mVgMRBEIEkQqA8BIQGRi9Mg4KEt331NUWxZWNClhyAVoQgICURox9pt1ob1a2nigzPo3zs8agGxf2Ezmv7wRCrs3NVu0ZI/xBAQEgixDgmF6uzbvZ2uGDuZNiRfSml9ziYcOJj/39VUNPYe+vCNZ6llq6xQqKbUwSEEhAQcAjKSipm/YCFtqOpGrbueT57jxappmT0G0herttGSxYvp2utviqTmur4tQgKuF4GTAdh3oIS3ENoRVZ3w3oyqriBKaUt7DhwWxCIMASGBCOtQJ5rTp0dHinlnFXliBhFh8POqQE10HMUf20R9elzmxCukjBBCQEgghDojVKpytNV51PzES3Tg05cpsTMTARsXHN/wBmVHH6TkwgtCpZpSD4cQEBJwCMhIKKa6xkN/40NG87/4gSbd/wjtXfMirf72JdYDPNTvwnZU1XkyTXt3P02MiqG+HVMjocnSBkZASEDEQCGAQ0az39tOS786RKPPzqSfDcylhJ/MoJJDJcpOID0jlTbvraDpCzfSlPlb6Z7RBdSvkxBBJIiPkEAk9KLNNsDt2IxF2+lfm47Qzy9sS6POaU3VVZV0pNxDsQktVOmHj7ADktRouuuyfJq5uEgRwe9G5dP5XdJsvl2yBxsBIYFg90CQ37+7uJweW7iNvt9TSr++uD1dzL4Hy/m0IaYGSNXV1d4aQlto1zKBfnupRgSPcr67LsujAeybQFL4IiAkEL59Z7vmm/aW0SM8kA+xw5HxI/LovM5pPC2oJn3811s+7rdhR6UTLs1TRDDt7SKqqSEa2E2IwHaHBKkAIYEgAR/s164tOkZ/YgLAgJ80Mo965rWgMpwy9KNiZawRtE6JUxrB40uKaPoilOOhwd3T/cgtj4QaAkICodYjAajPyg3FNGPxdkpPjqVxw3OpA3sdKmUC8DfhLMEJdlKSyT4KJ7AG8fiS7UwERTx18NDQszL8LUaeCxEEhARCpCMCVY0lXx7kXYAd1LFNIv3mf3KpLXsbRlwCK+k4E0E6E8H4EbmKCP7M0wPmARrGrs0lhQ8CQgLh01e2azr3n3vp7x/vonMKU+lmXgRMS4qlMosEYFTmBE8NWiYzEQzPU1uMmB5gajDiR+K1yHaHBagAIYEAAR3M1xhGQK+v2cfz9gy6aVA7SmA3Y1DpnUjQCFIxtRiWS7OW7qBZ727nxUIPXdYn04nipYwmRkBIoIkBDnbx2Nabs3Q7vbf2EI3h/f+rL2jDxj88p+cAJE45HEc5cGTagjULEMFsJgFMOapZIxh9tngzDrYMNPR+IYGGEArj+/AqjHn6p9+bjIB40l7Jf5wiADM80CyaJ8TQHcNymHh20FNLd6rtw8v7ChGEshgJCYRy79io296ScnpkARsBsS3ALUPZCKgnGwFV1hoB2Sj6lFkNjSA5PlYtOj71/g76C//BdOR/+4kjkqbA3IkyhQScQDHEyti87zhNnb+FjYCqaAJvAfb3wwjIySZgqpEYF023Dc3lgKc76JkPtanBVf3bOPkaKcshBIQEHAIyVIr5io2AYAUIq59JbNvfM7e52gHwxwjIqTZAI6hg0+MEJoJbh+ZQTDTRXz/apaYG15wvROAUzk6VIyTgFJIhUM7K70roz2y917J5M7VAp4yAbG4B2mlWOWsE8c2i+UwCiCCanuNjypga/OzHbe0UK3kdRkBIwGFAg1UcjICwGNchyzACigsqAQAHTSOooTiOe3DzkGy1K/HC8t1q+/D6Aey+TFJIICAkEBLdYK8Sc1ezERAPrrMLWtCvWf3WjICcsQGwVzMtdznvRoAIxg5uT6wQ0Isr9hBqd6MQgRPw2i5DSMA2hMErAKo1wo3PgxEQHwG+6aJsSmD12ykjIKdaBo0A25IIkf6rQe15ahBF/2AiqOJrvxyoaQiSgoeAkEDwsLf1ZhzphUHOe+wJCPvwV/OCWxT/z0kjIFsVrCezQQS/4IEfwyMfGgymBtAQhAicRtv/8oQE/McqZJ6EEdAMXgD8bOMRup49AY1mEsBXtbLaOSvApmqsQQQ3XtSOtw+JXmMtBrsGcGgiRNBUqJ++XCGB4OBu+a272BPQn97eRhv3lKnttyE9M5TJruEJyHLBAcwIIkCQVCwOYtdg3qf7qIo1glvZqAlTBUmBRUBIILB423rbRrb+gyuwg8oTEBsBdWrYE5CtFzZhZmguIILrWJOJjvLQW5/vV6cPQWxYO5AUOASEBAKHta03wRMQCIDHCU1iv37KE1CAjYBsNaCezBoREF3LdgPRrAG8+ZlGBLdfAiLgbQRJAUFASCAgMNt7iWYEVETpbAR0BxsBFbItQDCNgOy1xje3IgImgGv4dCOmAjjuDN+m44YLETiJ8+nKEhIIFNIW37NYGQFtp05ZyXQ7n85rkxpn2xGIxao0WTasB+Bc41XnteGpQRS9ys5PcNbgTiY8WBxKaloEhASaFl9bpWMwPLeMjYA6pKiTgKmJsSpISCQmY2HzJ/2zlEHRK2r7kOhOXvtIFCJo0i4XEmhSeK0VbjYCGtKDPQENbEfx7AkIHnwiecnMIIIr+dhxLDPBiyt3q12D3zIRJMXz4oGkJkFASKBJYLVeKKz9Zr27g5auO+jjCQiHcSKZAAzEFBFUEY3pm6nsBl5csZsXQz10F8c5ECKwLleyJtA02DleaklZFU1/p0gLB8Yr5iocGA+KSj6W66aENlewA5TR3H5MDf5vOZ814GsIgQbPRZKcRUA0AWfxtFyaEQ4MtgC3sPXcYJ4G4OsP4Xdjguvy8spqGtmntTIxxgGpKjaSwvZoCq+NSHIOAUHTOSwtl7SZB/6UBVvpsB4OrD9H+20oHJjll4VRRvAfrCEv7d1K2RHAH8Gj7DLtd6PzhQgc7EchAQfBtFLU2m3HVGBPpLs53HePnMB7ArJS70DlgfHQCZ4aDOc4BoaHoql6aPRUPjItyT4CgqJ9DC2XsOK7YmUEBE9Adw7LUw5BIsUIyDIo9WSERnCCT01echZrBDw1eObDnfRHJoJ7WSNA4BNJ9hAQErCHn+XcMAJ6kj0BdeSBfzt75m2TZj0cmOVKhFFGEAGmSEP5wBSmBk9/wETwFoiggDI4FJok6wgICVjHznJOnKN/Hp6AClPUEVqotRBwSadHAOcmyhinIRz9GHsET+kawWTWCFpxlGRJ1hAQErCGm6VcXiMgPjqLMN438Zn6hGYxIecJyFLjApTJIIKBjB80gic5rsEfeGpw35gCyhQisNQLQgKWYGt8Jpj7ImDnUvYENMbrCYhXv3kbUFLjEDCIYEDXlsqOYA5HOnrozS10/+WFlMVnKyQ1DgEhgcbhZelpeAKa/s42NgI6qhxpjDo7U/ME5FIbAEsg1slkEMGPzwARRNEcdrX20Bub6T4mgnYcbl2S/wgICfiPlaUnlREQG7l8v6dUDweWodR/txoBWQLxFJkUEbBPhfM5whJ2DRAN+aE3ttADVxRSdroQgb9YCwn4i5SF52D9B+OWwz9U0IQRecoTEBa2ILySnEEAUGJbFQZWMVG59ASIQE0NCignI8GZl0R4KUICTdTBMAKCBqA8AY1kI6AghANroqaFZLHQCPp2TGW3a3n0+BJNI7ifNYK8VkIEDXWYkEBDCFm4v+q/JeogUHrzWPYElKdsAcQIyAKQjchiaATYdoX/RYMI7mONoLB1YiNKct+jQgIO9znCgSEeQKc2YgTkMLR+FQey7VOQwj4I8mgmawQP6xoBrDEl1Y+AkICDkqHCgX28m87RPQHhtJsYATkIsJ9FgQh65beg316aSzMXG4uFBdSxTZKfJbjrMSEBB/ob3m/+ZoQDYyOWXwzKVr7xQi0cmANNDZsiQAQ9c1vQBHZGMnNxET04bws9eGUBdW6bHDZtCFRFhQRsIo3An7PZEej7pnBgKBJHYCUFFwEQAU5lTmRnJIjY9HtMDdiO4MxsIQJzzwgJ2JDT4tJKtQD4+WY2AuIgGvAEBCMgaAaSQgMBEAEGPbwSzeATm1gjgEFRtxwhAqOHhAQsymqtJ6BSdQjoYt0TUDiFA7PY9LDLBiI4o10yawR5NB1EwHYE2DXAtq0kIiEBC1JgGAEd+gHhwMQIyAKEAc8CIujE6wGYGsCHwx/e2kKT+RjyWbyA6PYkJNBICUA4MHgCghHQ3SPzqUee7glIZgCNRDLwj8OgCDsEd3G/aUTA/gjG5FNv3lJ0cxISaETvr9xQTDN4yymDPQGNG87hwNgIpfSE+AFoBIRBfxREgH6DC3OsEfyRNYJ7RheqbV23JiEBP3t+ERsBPWXyBNQ2LU6sAP3ELtQeAxHkZybwYmEe/Zm3D6fO14igb0d3EoGQgB8SipBYL7AnIHwtbh6ieQIqi9BwYH7AERGPoP9yW7FGoK8RTF2whad3BdS/c2pEtK8xjRASOA1ayhPQR7toHnsCUuHA2BMQjIAiPRxYYwQonJ+FNWf79AS2LIRGsJ0eWbiV7h6Vr44muykJCZyityEgs/gMwPtfHdY8AXHEXITFcks4MLcMAvQzfA9gagDLwsf46PfEkXkEZyVuSUIC9fR0CXsCmsYWZsoTkDIC0j0BISyOpIhDAK7fsMYDjWAmLxY+trBIRUSG+zI3JCGBOr28hz0BPcJbgJvYIcitQ3N4GpBO5Swk1eIJJKLHA4igNfsnnMBrBI9DI2BfEOjzQd3SI7rdaJyQgKmLMfBBAIdhBMRbgP15bijhwCJ+DHgbeAJEwDEMcOjoCT6GPI39QmJdCNagkZyEBPTe/RKegJgAMO+fBCMghAMTV2CRLPv1tq2MD37BDgSOSZ5YsoONirarqcElZ0UuEQgJsCis/K5EnTJLhxGQhANz3cA3N5i/AWr3B+HN7mRtED4L4ZwEjmGHcTzESEyuJ4HFX3A4MA5gAXPS2y/J4XBgYgQUiYLemDaBCOALIjU5VhEBPEWBDLAujAjJkZZcTQIwAnr+JE9A4gcg0oTcantABClsGDZueI4igtkgAtYIEDcikpIrSQBn/pUnIBgBdc+gGwdxOLBY8QQUSYLtRFsMjaB5Qiz9hoPGIoDsHI4iBSK4nG1HIiW5jgSw2Df7XTYCWndIdeTV57ch0o2AIqVTpR3OIeAlgvgYJoIceup9DoTK00f4jbny3MggAleRALb+4FTi33XCgUEzkCQInAoBRQQcMzIxLoZu43UjRDt6+oMdrBHU0E/780ckzJNrSKDWE1CZCgcGI6ATVR4JBxbmAhyo6oMIYDKOKNIwIkP8Q5wrqeYlpGugTYZxcgUJGEZAh47BE5BuBMTHScUVYBhLbhCqbhBBfLMouoVdyjEPsIv5XexgxkPXXtA2CDVy5pURTwJmI6C7+WBIj7wWVMaOQGQC4IwAua0UjQg8FB8bRb/mY+UxPDX4Ox8zh0bwcz5nEo4pokkAnoCms8VXKzYFvWNYLiEKjXgCCkcxDa06gwgqmAiaMRH8akg2Tw2IXvgEROChG/m4ebiliCUBGAHN4XgAndomqe2dNnw4ROIBhpt4hnZ9K0EEMUwEg7MphucGL63QiADBZ0AU4ZIikgRgBIRwYHAXBZUNll/iCShcRDK86lnJO0sggpsGQiOIoldW71FTg7FMDDiHEg4pokjA7AnoYl79v4E7JgHhwMQVWDjIYtjWsYKJII6JAFMBLBa+tmYvLzp72BUdE0MYMEHEkECtJyCTERAWcSQcWNgOrnCpOD740AhimQhuGAAiiFLWqDh0hMA0mCqEcooIElDhwNgI6HPdCGgk23ZDKxAjoFAWvcirG+QNRPDzAW3VwH/rX/tVSLrb2K4A10M1hT0J7D5cTo+yFxjYAmhGQBnqBBhYWJIgEGgEFBEwAVz34zZqavAmEwFEESbHuB6KKaxJYOOeMpq6YCsVs0/ACRwOrF+nVPEEFIpS5rI64esfy/sDMCCKYQ3g9TX7lGY6jrepsYgYailsSUAzAtrKK7BRWjgwDi6JoBLy/Q81EXNnfUAEkEV4qcYawav/3KuI4E4mAritD6UUliTwCRsBwe0TjIAQDqwA4cCYACQJAqGEgBGh+qf9s9iykOhl3rrGrsF49l6VEBc6RBB2JLBI9wTUKYuNgIaxJyA2AoIGIEkQCEUEDCK4kokAdgT/WLVH+SzE9DUxRIjAMRLYxQt0+49WUGe20Evms9dNkWCI8fzyPSocGLZeUhPFCKgpcJYynUVAEUEV0eXsfwC7Bi/qloUIiprURGOlMS2wTQKY+yCy61qeo6NBR8uqVQSXAWc6F7gB7zDCgcEI6EYYAbEnIAkH1piulmeDiQCIwMNEMJoD2WCTAGcNsIM1keMcJCc0zUfT3/baJgGEbvp2Zyk9O7armqN/+PVhuu+1TfTmhLOodUqcv/U45XMICvHEu0X0wbrDdAUzKRZakODkIfTWWW03VwqIYAQw6MsriWDHgqnB88t3URW7ub97VB61YBdmwUq23vwDz8WX8uCcdm0n74Af2jND/Ya5rt2kjIDeKeJwYEfoerbEgoNH7MNCMxACsIuu5A8GAlgYBBFc1gdEQPQc+7p8dIGHiaCAUhLtawTvfXWIeuU1Z6/Z8X43zxYJHOQ1AAzGnAzfF/bKb+F3BU714C4OB4YtwE17jyuXToPZIagyApJwYLaxlQKCiwBk+AT7uhzxo0yK5e3DZ9lD0dT5W+me0fkq7L3VtGFXKV03Zz1N5LUGRFf2N1l/I78B+53G2WrzCzFYMe+J43m7lbQR4cA4OmzxDxUcDiyP+ndhIyDxBGQFSskToghgrRBEcEmvVsrW5dmPdioiwOBFEBwrCdaJN7DJ8vqdP9CRsiq/CcUWCUDth1/2Nd8fUfN1I/3qmW+VCe95FuK8ry06Ro8iHBgXNmlkQa0RkFgBWZELyRPCCGhEUKNCnMGO4GloBPzxu4eJIIPX1xqTjh6vonU8drA2N2X+FsJW+rUX+Of70BYJYLsD56YfnLdZmevms9HO3FV7CWsFsOBrbFoBI6DF21UsuDvYBqCQbQHECKixKMrz4YQApgZwgz+kZ7paLPzLBzt5EG+le8cUqIV2f9Pyb4pZc66iogPHqXl8LM3/fH9gSAAVxEIgTkjBPvqDrw9RYeskmnJ1x0avdi764gAHd9ipewLKoSwxAvK3/+W5MEcAy1zYBRvUXSOCpzjIyR942/2+MYWUmeIfEWBBkKI89PSHO5XrMxyo+4Z37bq1T24QHVuagFE6YrjbieP+MmsP2C7p2zFVeXHFFAOgSBIE3IIApgbQCAZ2banW0+YwETz0xma6/4pC9UE8XcIa2sa9pTT3jp5e7eGeuZvoBXaAOu26Tg1C6AgJNPiWUzyA7b6/LtvJDhj2K40CnlniEQ5MCMAqpJIvjBGARlDKRHAhG9phqj2bQ549/CZrBJcXUNvTbPl9xLY5vQtSfKYPP2Ez5ftf30wlvECY1sCOQ9BIAPb+CPKIKYTZCAgBHiQJAm5FAEQAjeD8LmlqajCLg6A+/MZWRQTZ6fXv/V/ZL4uS6pxD+BFv0z93c1e/zicEhQQQDmwGPAFtPsK+2n2NgNza+dJuQcBAQGkE/JHsz/4xoimXnmCN4KE3N9MDPDVon55wElCnWkBsaBphFBRwEsBBo8fYE9BmnsfcOjSXjYDSVXgn47SViIIgIAhoCEBbPpeJYHx0njKd//28LfTwNV2pbeJx2rB+HSUkJFLhGT1twxVQEtCMgLbSYd7KGC+egGx3nhQQ2QjANAZEgFOz40fk07PLD9G4Kc/T/nVL6dtDiey9qIIu6pJIM6ZMpnY5BZbBCBgJfLn1qNIAlCcgNobokSOegCz3mksz+pwX8ePwyOkf4bva/09K3mummyc95817cgn15cdL6q3PKepgrhRsCQacmUZ7t35DVz36IsX2vpla9O5ElZ4aevXrD+jgbffS23OfocTkFEuS0SQksPqTD2nhkmV82KeGhg/uT4kdBtKMxTuoJZ//v3N4DnsCiiwjoFN1bmN6pH6BrZWQ0wm030JXj+CqvKcTdv2+H2PO1NwoLhLfMd9cmtHnqU0/1Z1T3PZe5h/mRxAM1JvNdM8oS3PyxXlMmfCb41Gra94/fKXub2zbac8a9/S/uTzlx5ZvqmfUT+0eBqz2t29e8/PqGb1OyGc8W43rvC7ONdP+NpWXmJxAi95eQHGdhlFa+zPIU35E4duq9whatnILrfrkY7p4+KjGiJz3WcdJ4ImZM2jSs6uoIvsiioqOpSfff51yW71Nl//yXuUIJCuVbQDKa3xcMJ+eIU/BtHUu133q5JgPEEw9NSTRZoHxQuWf8NYnrKrL9RvGfR+h1KXUfM8sqIYQe4VNl3qz8HmFyiSQ9Qmnj6DiWRY21E8TxFqBNH4bQm3k832uVuBxTBb7OiofyjSVhTLMAu39N87Y64Ju3Fd/oyyfwVAnvzefedDVbYOeB2XpA7J2gJoGqj7AjXZpuNfW3+g6NciNgetDGAZx1RKE6kejHFPfG2Rl7nsAgOeVdJq1Al1GNZKOpri4eFrz9X6Kz+pOVH1Cl0oPRXvYr2ZSJu07wMZCFpOjJLDui89o8nMrKO68iZSWyKuY3MDKDufSpuXT6Izqf1NGWgfaV3zchwUN0H06QWdfL5sa7GoGv46QeYVJF2xzh5sFFwJmMLX2TK2AKeFVLO8ryF4B1gVdCaiJ+b3l6+ytlVkrhKrM+vKeJKD6QFIDwBBE0xfKPEjrCG/tlwcDux6B9Qqm+YvnKzU+wml8QZWA1iZNYDWczNdrSVe7atYw8Fu7z6KuC7r6W/2uFX445MQ1GMtg2uj9zc/g2K1WDv9Wz+hl6f/WytPuRfF/EC04hk/mxvAAMl/Hfdjp41oMyjTK4xu15SK/fo+fQVmqLqpc/d18HXUwP4d/a3/wrG/ZRhnmsrTnat9rtNt83ahfQkIcvVzWk+5/az0ltO/NAlLFdYpm68AaSiorou7dLrNIAcRrCw6mVf/8lErTelHrpCRWV46pkps1S6LEgvNo5txPaHVFXyo/UaapTroQ1zKtJqWKFLwsqkmb+YuGp5RqxSaSPl/X2n/qefgCOk9vn1kL0K7CJwF6QRcuk6Aa7uF9BVYXbkP4+HntOUMY6whv3euGIOtCAmMQTcghjGbhwXVDMDQhMZ6DYHmFzEd4DcHz/VsJXN08ZqFT94wyDUH3fYfKj7rrAuuti2nQGfUy19s78Lzlmwe2PljRBjVYtHeYiaH2d20+hbbpWcUfOpHU9pXe5zrZOCjeQS9q/C3X03vLxtLqz+ZRfIf+zAPlVPn1QvZX2J569elnuX6OkkByUqJiKPO3Iop7ubKigvJzUuicwuZUXdlMdbyPMOv/RkdqA8J03xgg5jz1PqPn8d4zfRlMXwvFvLrAnfTFMT9nsL8x6I17aJ0qQxdI733T18n0xdKEWWuTRhnaf05S/yx3oWR0CwLJqa1owUuz6eEp0+jLba/zVKCKRv2iB90xbhyE0jIMjpLAkMGDKPvJ8bR/Xz9qmZWjvrZHjxRT/O4VNPVPk6l770zLFZWMgoAgQNQqK5tmzXqcHZceZz0eh4vsD2H7JZh6JjuvIz3zwA009oEnaffWbvxZb0Ytj62jGRNGMQFYV1ek8wUBQaAOArGsdTuUHCUB1GnEqCtoTY/utHLVaqrmLcJz+15FXbr1cqi6UowgIAg4jYDjJIAK5hZ2oWv5jyRBQBAIfQSahARCv9lSQ0FAEDAQEBIQWRAEXI6AkIDLBUCaLwgICYgMCAIuR0BIwOUCIM0XBIQERAYEAZcjICTgcgGQ5gsCQgIiA4KAyxEQEnC5AEjzBQEhAZEBQcDlCAgJuFwApPmCgJCAyIAg4HIEhARcLgDSfEFASEBkQBBwOQJCAi4XAGm+IPD/GSWz9gfiq4QAAAAASUVORK5CYII=)
Let us assume,
![](data:image/png;base64,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)
⇒ AB = x, AC = 2x and BC = √3x
As we see that, AC2 = AB2 + BC2
So by converse of Pythagoras Theorem, ΔABC is right angled and right angle at B.
Let ∠C = θ,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ θ = 30°
∴ Option (b) is correct.
Question 16.In ΔXYZ, m∠X : m∠Y : m∠Z = 1 : 2 : 3. If XY = 15, YZ = …
A. ![](data:image/png;base64,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)
B. 17
C. 8
D. 7.5
Answer:![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAM4AAAB5CAYAAABx0B4JAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAABeFSURBVHhe7V0JeFXVtV6ZZzKTEDJAEpBRBo2zoCKDoCgKgpW2WK3vqXVotUgYZAxYQG0dKs9Xq20tKCiRwakqjq0+9aFV0VrIwBgIQ5hC5qT/2vee5Ga+547n3Ls2X8jNuXvvs89a6z97WuvfwU1IJEkkIBLQJYFgXbkls0hAJKAkIMARQxAJOCABAY4DQpMiIgEBjtiASMABCQhwHBCaFBEJCHDEBkQCDkhAgOOA0KSISECAIzYgEnBAAgIcB4QmRUQCAhwv20DRoSpa884+CgokYh+OUHxYODWbgoMCvNwyuX1XEhDgeNk+UuNCadaoXhQeGkSPvrabQoIaFYgkGVsCAhwv6ycqLIgGZ0TTJztPUHVtAz1y60AKCJDexstq6fb2ApxuReT+DHuPVtPKzaX0yI/7U3iIdDful7jzdxDgOC9Dp2qoqWukVQDN7Ml9KLtnhFN1SWHPSUCA4zlZd3inx97YQ9/tr6Sjp+po3d8PYq4TSOOHJVIk5jySjCsBAY6XdXNR/1hKiAqh/ceqqaGxiSLqggi/JBlcAgIcLyto1IB44h9J5pKAAMdM+mqspQ+3vUvVNTV06ajRFBEjgPOW+gQ43pK8zvseLT9As+5bRm+V9iAKCqWhT26mv666lwYMGaazJsnuCgkIcFwhRQ/UsfrJP9LWvTmUeO5YCmhqpO3FO2jOymep8LlHKCAoxAMtkFvYSkCAYwp7aKKvS49TaOb5FFhzXPnmxKTl0Pf7PqPq0xUUEdvTFE/hS40U4JhCmwE0MKMHvf55MQUkXgDgNFBl2V46KzkAoIkzxRP4WiMFOCbQaNmJWjqZeDElHFlDh/95hJqCwyns4KcUM/V6OloZRIlRJngIH2uiAMfgCv132Rla/HIRZWWcRe+uXU2ffvQu1dbW0Yi8+fRqUQLN/ssPVDAjl9hZVJLnJCDA8Zysdd9pe8kpWlZYTGdnRtOdY5MpMiKMBgz4GbELaGMDUVY2+7hVUv66nbT8plzqFRem+x5SwDEJCHAck5vbS73/XQU9snU3XT44nm4Z3Zua8O90VS0FVFluzc4FsZHBNOe6vspBNH/dLlqOnictXsDjduXgBgIcT0hZ5z02//9hWvP2Prr+vJ5044WpVFffSPXww7ENNuDPVXAQjQkPogfhILpySynNWQvwoOdJTxDw6BS57uwCHN0ic2+BP39Yppw9b7ksjSaOSFLgaOzEeU0DT5QVPKsAHjVsm55LGUnh7m2on9cuwDGIAXDY9JNv7aG3/nmU7rkqk0YNjKPKmgYVTt1VYvBUA1zsTT37mj60GsO7OWrYlkNZyRKm4C71CnDcJVkd9dZiKLZy827aXnJSzVlG9ImhymqAxs46FHhQRwSiSX8N8PDciMGzAsO2PgIeO6WoL5sAR5+8XJ77ZFU9FWwsoT2IAl1wQzb17xVJpwEavUnreTiC9P5rsugx8BfkY85TAPBIgJxeaXafX4DTvYzclqMcG5sLsUdTW9dEi6bmUBom9Y6ARmug1vOEBQM8kxg8e9RqW8H0HMpNjXTbc/hjxQIcL2m9pLxKgSYuMoTmTM6i+OhgOoM5jbOJwVODYRuD55eTMul3iDCdy+DBUnU/9GaSXCMBAY5r5Kirlm/2nKYlrxSrYdndEzJVuPSZ2sZWy826KmyT2RY8903MpMff2Ev5L1p6nrPSxD/HGdlqZQU4rpCijjr+/sNx+g02LC8+K45+fkVvRQXFq2KuJoTSwBMKYsN7AZ4n3kTPA/Asw1L1wN4CHh0q6zCrAMdZCeoo/8ZXR5UBX3NOMv3oklSqb2giXlFzNWhs5zy1uAeDh3u2p7DcPQ/gWQrwDE4X8OhQXbusAhxnpKej7Iv/OER//vAA3QzAXHtuT9XLMDmHu0DTOXj20YKXdtHiG3NoKIgQJTkmAQGOY3LTVep/wA29+YsjdMfYdLpiSIJaBPAkkw2Dsw49Twh6nrvGp9PTcOd56KUiBR52IJWkXwICHP0ys7sEGyvzQf8D85oHsLdyXm4PXRubdt/Izow8bGPw3DkuXfnCLViPngfL4MOx4SpJnwQEOPrkZXdudpdZXlhCOxFPM//6bBqEOYUzezR237iLjNzz8LyKT0L4b/R+gViYeGh9ES2alkMj+wp49MhYgKNHWnbmPQJWTl5uPnGmThllJhwuvQ0a26Zr4PmvKwEeUFUv2lBEC3C0SF42GHQk2SUBAY5dYrI/ExOoL9xQrMjT2RsgMSZEOWsaLTF4CCy7t4+x9DxLXi6Gy09fOi8n1mhNNWR7BDguVMv34IBmA2SXft545CM8XLmx6cKmqqoUeJBuw36SAs8rJTR/Sl+6oJ+ApztZC3C6k5Cd339edIIKCkvp3OwYzB8y1OFQHEvj7uVmO5vXaTYGD4cu/OyKNDVsW7qxmObCQ5s3aCV1LgEBjgus451vjtFvX99N44Yl0U9xuhrvz9TUu3+PxgVNV1Vwe6meVPBcUGCAWtTg8IZLBwh4OpOxAMdJ69v4WTn9Ydt+uvGCFLrhgp5UA09nT2xsOtnsdsW5zbUAz09GoefBsG3FqyXUeG0fGj1Q+Kk7krUAxwkL/OP7B+iV/ztEt13eG71NIlXBUbOxu5BNJ+7n7qIW8DTSzEtT4UNH9JtNpQS2XboMhCGSWktAgOOARbCBscfxezuO0X0Ic74I8wF7wpwduJXHi2g9z0y4BgVi2MYkIA14GYyBx4OkFgkIcHRaQzV6lYfh3fzNnlOUjxUodlkx0h6NzsfpMLtljkb0o4tSiE+NXw3wcE86dmiiK6r3iToEODrUWFFZR0uxZHuIIzexR8Mhyb4GGk0cCjyYr824CD0PwMM8Bg0Ytk3AkFSS8KrZbQMHKmpoEfZoeO128bRsSgHl7GlsbBp9udnuB+wgI2/zVNU1KG43XjDglUOewl01XMAjPY4dlrXr4BmEORdTSo9QFY7cIyIYcxrj79HY8WjdZmlEL1NV20BTsWrIcx4Oxebe6OqRSd2W9eUMApxutPtlKfibwUIzJCOK7hqXQaGI5XdHxKaRjYy3eXjFkJlFedjG/G9N6Ho4IM9fkwCnC81/+H0Frdqym0YPiqdbL09TwxTmL/Pl4Vln4uDFAe55rmPwAD1P/W2fmvNcl+ef4BHgdGIpW8Df/HsYB79lp2OCzPzNdR6I2DTyG5x7Hg7Cu/bcZDXnWfP2XrXaxjLytyTA6UDjL3xURmvB3zxrdC+ahOFIlTVi0x97mrbi4V6XwXPNOUlq2PYMolsZUFPP9y/wCHBsLIN9hZ96ay+9CVKNX4zPUEM0X9nYdGWPwEBhuUzCAgEP29jliBcMpl+Y4srbGLouAY5VPexqwvOZL4pO0mz4aJ2DiEgBTee2yz0Py4eXplXP8+5+darCTRenGtrgXdU4AQ4keQpczcsQsbn7SDU9BP7mfmmO8Te7SilmqUcDz3hsijJ41rwD8ABQzOTj68nvgcP8zbyxya40ixHmzPzNfFKAJPskoIFn7NkAD9DDCyo8bPsJwit8Ofk1cEoPg78ZYc68obkQMffM38zDD1kE0GfyCjx42bAjKK+2MfEhr7bNGp2mryIT5fZb4Hy718LfnJsSQXdflUURLuZvNpENuKSpvLDC4OEzS3nY9sSbvFSNyFIEx/li8kvgfLrzhArU4th6RVaBkGF/8wZwhzFr4LkMq5Hc8zwOul9eMGBOA19LfgccPirwcfhbTRyZTBxzwofSminM2egGyOBhj/FLcRQjB8Oxbxv3PLeP8S3w+BVwXvrkED2PqE0mPJ+S5zn+ZqMbu6vbp4HnEnAW8IIBe1U3wFv0DpCY+EryG+DwPsOmz8sVg+WVQz3P3+wrBqPnObjnuah/LIZtlqMV2dP6Lmws+0LyeeAw/dFjeON9/K/j6mzM83Njvcrf7AtGo+cZGDw8l7zfeqgvD9vunmB+8Pg0cNinihcB/nXgDM2bkk2DERrgqxGbeozZ03lZ5ufl9KAHrs5SkaQcknAPzusx87q/zwLn2Ok6Wozl5gr8XoSIzawk3w1z9jQQHLkfgycP4OHj5JnDgEMSmO2UFxDMmHwSOMzfvAgbm6HBAYr0PAmRm0bkbzajwTjTZgbPOWA6fRC+gHycI7Pn/AqnY/O+j9mSzwHnBwzLFuE05/QEC39zdAT4m8UbwDB2yeDh83jyAZ6HwdvG+zx8dhDv+5gp+RRwPodn83LMaUZCMczfzOfAcMivuVRiJvNxrK0MnrOzAB7Q7DJ4Vm7ejSFclqLfNUvyGeBs+7ZCnX525dkJ4G/mMGfe2BTQGNUQGTxDwUnHBO8rNpUo1tDZk/uol50Zkk8ApxD7M8/ApX0aAqmmmZi/2QwG48o2MngGYaWTwfPwq6Wq9+H5Dx+3aPRkeuA8/8EBWg+PAPaHGo+gqirQNpmZv9noBuPq9rFj6EAc8zh3Sh8Ms0vVSQlzwZBqdPCYFji8kfYE/KDe+Rb8zVgE4PNcJGLT1WbtmfoYPGelRalDrQowR12GM3rm4TNTcRk1mRI47Mm8EsuZX4HzLP+6PjQME03Z2DSqidnXLn7p9esViYOG+6peZym47Bg8fCSkEZPpgHPiTL06NaysgvmbsyknVcKcjWhYjrSJwZObAvDAy6OgsBg83cXqxG6OlTJaMhVwysDfvBhhzhwKwKDpFS9hzkYzKGfbw+DJRnAhA6aAex6Ah3kgwg0GHtMAp+gQ+JvhDZCMU5x/BZ+nHpES5uyskRq1PIOnD06CWIBhG895FmJDexFelBGhOCbbIMkUwPlq9yn15hmUHq34zsIwaZSNTYNYkJuawd4emcnoeXjBAD3PQ+sBHrhP8UneRkiGB85HCAdYhYUAjii8FUcGskutv/I3G8FgPNkG7nkyEluGbQo8OJcoBm5U3k6GBs7W7Rb+5usQrTkDp4PV4STnOkRDGX97zNtq9Y37s54rQfSenhiOYVs2LcOCwUMbitT5RMxM5M3k3bt38eRrPz5IL3xcRj++tJc6TqIKS9DsECig8aa5eP7eCjzoedLiQxV4eNi24KUiWjI9h2K9CB7DAYfj1Z9+ex+9jt6Gz6PhE495vMsbnpL8UwKWnqeRUnEKnrbPM//FXbT0xlyKi/KOCXvnrp3ovw5hzqsQ5PTZrhPK4e+c7B4S5uyfWGn31AyeMwBPSiyDJ9vimgPwLEPPkxAd4nEpGQY4p6rqVUhAcXk1uuQcGtBbNjY9bg0Gv6EGnuQeITTP6mEwD8O2Ai+AxxDAOXwSYc5Yq+exLK/X82RQXGgMbsVeap4GniTs5zF4Vlh7noLpuZSIa55KdgOHjyqPj2ppGLPHsKHHYiPSmcT8zUx6Ho31efYG4G6X5zSSRAKdSUADTyLscS7cc5iQJR/DNgYP90aeSHZb/fMflCFCjyzsJEgPrt1Jk0Yk0RUg2nY07dhXSUvQ0/SFf9K9qDciLFCNY2XlzFGJ+k85BR6stMZjcYDDEB5m8KzbSQUzctU8yN3JbuD8FMc23LJmh2LAZMJyZqh3BjTt+ZuxsQlBCGjcrXLfqZ9thbcp4jDqyQd4OIqUwbMc4EmNC9P1oDzKibTxSuBVXD73NawT72y7gcNDKI57eeCFnZQAlC/BUqCj6W9fH1WcwhOHJ9FM7NPweSoS5uyoNP27nAYe3tNhAhBmz8lft4uW35RLvXSA51Os5L7x5VFaNbMf8el8XMcdYH3Nhfd9R8lu4HDhywcl0HPvlVEe2DAdHUtu+PQQPQf+5hk4yXkKTiuuwRuDgSM9jX8DwJmn18ATA/DMUeDZTflrsVSNnicdB4XZk0YNjKcNiCTmaOIDx2vgJxfeKWi4Pl3A4QJ9UGEKeMocSXzIauFn5Yq5nk/wko1NR6QoZTqSgAaeqPBgxVuwGj3PXPQaBTNy4O8W3q3QgsGws/LmfjRhxZeKRGTNbQO7LKMbOEzho5cCi3uU376+hz74vkKFBDCXsIQ5d6tLyaBTAgwenidHIfzg19hAX43DkLVhW6Yd4OE4Lz4riYdq3SXdwPnlpExdseCKvxmTtu/3VyoX8cEZ0bJH051W5HuHJaDAA8PniT57n6zeWkpzsAK84qZ+oEHuvOfhlzvnY68E9lx59LU9YBm1rCB3lHQDR49XKvM383GBR5m/Ge7gPMyTjU2HbUIK2ikBredhvgLmqmaidwYFLxj0RYxPR+nZ9w5QUkwojccU4uL+cTTr6R3EJ8uN7BvjGuDY2Xbad6wG/M1FimCOQdMTG1PC32yv9CSfsxLQeh4Oerwf0wMmq+RhG/u2Ma9B28RnJmUkWhYSosOD6OlbB6hFK5f1OPY8EPM3swtNGjgB7gOpdgwawt6tsnJmj/Qkj6skwPbG2xwKPLBDnmfPA3iWwsOgPxh1bFM2QrVtU3I3C2C6h2rdPdT2kpPgxSqh4aBsumNcOojlEOYsG5vdiU2+d5MEbMHD8/Pfvb6Xlm85SDMHVtAXn2yjquo6mjZ5LA0ZnqerBS4Fzns7jqnx5JghiTQLx3Qzf7OEOevSh1czdzki6OLLjr+yXsWvjr5vvmbzZbt8zWVbf2NXWZakzb150BUTGkwLMFSb81ghjf35X4myx+NqED22+Ql6edkMGjdhot3ydxg4ddWnaf2GjVS6v5wuu3A4nYzPoyff2EvXY1PzRnA419Y1KRonTw3PWt3HESV3omCr/C0C7UrJrRTlgKI7vD+W/tvc127NdpURVtSEf1pSn2yG880fVT5L4pdgc7aOrre6xuUsS7pcTP1Y6+Bpg+Vv/NM+44M2nWD6Yu16u8/W+rTrlros+VuuaX9brll+kA/N4RZFhgXQv7e/TyGDp1N8r2z1xeHoVFrx7Fa6cswYCgyxb8PUIeBUnqygm++cT5uKk4h6gHXmlY00IGMbPbw4n8YPjVEbm3yoU6ijqLFqy0ZV7cygQ2VrCmI1tVFkW+VrylT5IDyLYjUFt1ds8/esKBsF8h+aMXAttkrs7HOzMq1KtTUEDg+3NQI+cJbrb3WNy3V0XV1raYMqp+prc92az3Jd+04zsLb5LcbZ2khtrjWDo3UelpICgw1o+DWqDN1aph0gba5b4arkafvGspqGqtjyWuErFkNT+4tcAB/4M/8ZwJ+t1hOITZqgplravruKovolU1PdGQWcyJg4OloeQg31Ve4FziuFm2hTURIl5d1AgbhZddYwKvv2GTq+7zt6kwZSdVV1s0KaFdlKWe2V1KxkPKStstlANMW1NaiW61oe69uK6+jgbWQx5BYjslVis5HbGIL1JdvqrasBtkWBLZhuBnMA37yLAUzbnstmSNKicFa81QD4e5Wn5W8+SkYZhfW7QOt3fEATX2v5bcmn5efffIS6+s3XseHHu+acJ0gra/1e/Y3vVb625axltTq0PLxBrt2Lvelbl+2gHps8WjtbtY+fz6a92rNp7bf83f4ZLXLR2tIij2A0Kr2iPy3a8jHFj7xagaryhw9o1HmxFBIR2+4F3dkFh3qckn2HieL7AjRnqKm+msLDY+lEUDK98PZ31HtoT2qorbIq1EZh1jdC6wdq+b6tAFop2sYYWNEh+C8QLW9rAN0qyar8FoW2UaRSEAyoI2Nh48Iz8HcWhVh/VN4WJWkG1mwEzUasGWsHhmxjHF2Bpfm7Zlla2qTJVAOd3dr304yz77mNivcsocLtf4BCQ2hyVjUtfmB+q56tO9E4BJxL8oYQbXxN9TRhYWF05PBh6h96iP73gV9QXFKq6iFs34btFd7y9uyugfK9SMDVEoiIjqM/rXmU5u/4Ens1DTRg6HDcQh8U9OW2PsGYceNpyRdf06rC39PxiFTKbCyhpxbOpF5paa5+RqlPJOA2CfQbPMLhuh0CDgYntGDugzR98pd0sLycBg++nRJTMhxuhBQUCZhNAg4Cx/KY/YeMoP5me2Jpr0jABRJwCjguuL9UIRIwpQQEOKZUmzTa2xIQ4HhbA3J/U0pAgGNKtUmjvS0BAY63NSD3N6UEBDimVJs02tsSEOB4WwNyf1NKQIBjSrVJo70tAQGOtzUg9zelBAQ4plSbNNrbEhDgeFsDcn9TSuA/mXCF+ZzSVOgAAAAASUVORK5CYII=)
Let ∠X = A,
⇒ ∠Y = 2A and ∠Z = 3A
As sum of all angles in a triangle is equal to 180°
∠X + ∠Y + ∠Z = 180
⇒ 6A = 180
⇒ A = 30
⇒ ∠Z = 90°
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ YZ = 7.5
∴ Option (d) is correct.
Question 17.In Δ ABC, ∠B is a right angle and
is an altitude. If AD = BD = 5, then DC =
A. 1
B. √5
C. 5
D. 2.5
Answer:![](data:image/png;base64,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)
As we know that,
If an altitude is drawn to hypotenuse of a right angled triangle, then the length of altitude is the geometric mean of lengths of segments of hypotenuse formed by the altitude.
BD2 = AD.CD
⇒ 52 = 5.CD
⇒ CD = 5
∴ Option (c) is correct.
Question 18.In Δ ABC,
is median. If AB2 + AC2 = 130 and AD = 7, then BD =
A. 4
B. 8
C. 16
D. 32
Answer:![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAToAAACsCAYAAADrCmC2AAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAADLlSURBVHhe7V0JfFTV9T7ZJvu+TfaZhFUWWVWQRQQRRRQVq7a12LrVpf4VFXFBurdqXequrVoXQFu1VgEtKoi4L2ALWoXsZCMJEBKyTbb/d+5kkkkySWYyM8nMm3P7a0uS9+6797v3nXfv/c75TmAHCkkRBAQBQUDDCARquG/SNUFAEBAEFAJi6GQiCAKCgOYREEOn+SGWDgoCgoAYOpkDgoAgoHkExNBpfogH72Bzazu1tnZQeEjA4BfLFYKAFyIghs4LB83VTX5gczF9kVdLr66a7OqqpT5BwCMQEEPnEcMwco2oa2yl78vqqa29gz7eV0Ozx8SMXGPkyYKAmxAQQ+cmYL2l2nf2HKYJGRE0IzuS/vlFpRg6bxk4aadDCIihcwgu7V28eVc1TcwIpwB/f/ps/1GqPGqipGid9joqPfJpBMTQ+fDwlxxupv0VDRSi8yf+d1s70ZavD9Gl81N8GBXpuhYREEOnxVG1s08vfFBGCyfG0boV2eqOj/cdpd+8lk8XzU6mkCB/O2uRywQBz0dADJ3nj5FbWsgRzg2mNrrgJH1X/SeOiqLj0sKpoLKRxuP/pQgCWkFADJ1WRtLRfvgR/eqCHPL3wz86S4C/H913yRhqBQMrRRDQEgJi6LQ0mg70hc2bn5WRs741EAZPiiCgJQTE0GlpNAfpy7atW+jt9z+j8GAdrVi+hCYcP92Hei9d9WUExND5yOjfc8+9dNsLe6g97WSiliZ68LXf0/rfXkJnLlvuIwhIN30ZATF0PjD63/73K/rV+q8ofNYqCg3iPWsAHaoaTbfe/zzNnz+PwqPifAAF6aIvIyCGzgdGf++evdQYNY4SdQHUYTqmehwdr6fSklgqKcylsZNP8AEUpIu+jIAYOh8Y/czMDAozfUZtfkFgWf3JD1EQJlMT6YMaKSM93QcQkC76OgJi6HxgBpw0Zx5NS36Bdn72d4o47lQKCgygY19soHBDHH17LIFmyM7VB2aBb3dRDJ0PjP8739ZTzImX0Tn7X6Oi0mcof///6OaLl1LEzMvpro3f0eUL9HTeCUk+gIR00VcREEOn8ZH/T1EdPbgpn06eZKCb77ifvv3vHlp1/bV0yoIFdMqiTHp8czs9tvUAlSLW9cqFaRQsoV8anxG+2T0xdBoe97IjzXTPG4VkTAqjS+cmUlNzByWmjaJ4fQbt3v01LTxtAV2xKJ30McH0ws5yKsf1q87KooRIpmalCALaQUAMnXbGskdPGprblJFrhyLJdYsz4CQcAEPXSmFhIZSWlkFFBXlUV99GHLq/fGYSJcPYPY6V3a0b9tOtZxtoTEqYRpGRbvkiAmLoNDjqHKn68L8P0P7yBrr9XCOlxwdTPQwflyCdHxlzRtE7b2+m2poaiomNo4bmFpo1JpoSsZJ76O0DtAbGjld2c8bGaBAd6ZIvIiCGToOjvuHDcnrnv4foGqzkjjdEUn2T2chx6cAKz2jMprq6Oqo8WEEJifFIjEPqmpzkUFp7vpGeeKeEfvNqPl1+ahrUTZI1iJB0ydcQEEOnsRHf8e0Rem5HOZ0LFnXRpHhq7FzJWbrZBpuXlpFJAQH+VFxUSJOmTOhCoN7UTrHhQXQTVnPrcWb35LslxOd8V+EcT/TpNDZRfKw7Yug0NOCc5Ob+zUV04qhoumiWnkxIY9hbcakNMsKJiUkUExNHhfl5xLp0iAnjtZ7636aWdgoK8KOfYTXH53bPw2iWHzHB+GVSYpRIrGtouvhUV8TQaWS4q+tM9Id/Fap8D1fATYS15Thfa+/ShiVdZFQkVnUZlJ+fS6bmdsg1wcx1StCxsWtt6wCJ0UFnT0+kZNT32NYSkBS5ICmyaGyqCHJqZMr4VDfE0GlguNmg/enNIuLUhetW5FAMtp+sHmxLVa4DFi0oCISEMYe2b3uHamtrKSoqCud0OKizKrwSZAKDV4cJkTqQFMVdJMXccbEaQE264EsIiKHTwGjzWdrXhXW0+hyDIhSOgVgYTDrTkJNDtf98RREScXExipCwVdjYGZNC6K7zsjtJigKQFCb6wSwhKTQwdXymC2LovHyoORfrG19W0c9OSaMTcqJ7MKz9da0VhEQ6CAlWGC45UEwTJo4bEIVGkBTRYYG0apmZpHjqvRIqPdJEV5+WISSFl88fX2m+GDovHunPco+CGS2lM6Yk0NJpCdSI7ao92R5aQTgkJ+lBSMRSft5+GLzFg6LA2+NAJikWpFJKjI6e7SQpbgZDK3lgB4VPLhhhBMTQjfAADPXxhVWNdC/O5SZmRNCP56aohDb25rTpgDMdExKpaWmIkMiHZBMYV6zu+PxuoGIhKZaCpEiMDlYxsqs7Iykka9hQR1LuGw4ExNANB8oufkZtQyv94fVCFdZ19WnpKhCf3UIGO5ezNKMdcWGh2Ioas0fRhzvfByFRRxERkTinaxm0pYqkwBngCTlRlHB+Nj0MkuK2jbl0w5mZdMpxQlIMCqBcMCIIiKEbEdiH/tA2WJoHthRTRU0zohiyKQm+bf0xrAM+BQaLCYktm16nqsqDFB3NzKv97WKSwpAYQmtBUjAZ8vt/FihRgItP7s4Ta39tcqUg4F4ExNC5F1+X1/7s+2X04fc1tGpppkoyzQzrUAoTEllZRoSEtdOB4iIaN340NTc7VhOTFFGhIClwTsdhZ09vK1WRFBx6FqpjuQApgoBnICCGzjPGwa5WvPV1Nb388UH64Rw9zYEvm3UMq10VWF3U1grJJkRIREXHqAgJokWOVqGut5AUK+enqkiKv20vo3KsNm8+ywD5J4mkGBKocpPLERBD53JI3VMhC2g+AkWSUybEKlmlZpzJ2cOw9teatrZWio6NppTUNDCviJDA8Zw9hISt+iwkxZlTEygZW+lHWe5p/T7l1zchPcI9gEitgoADCIihcwCskbqUt4N3I7wrBwKaPz0lVYVr8VmdM4UZVh1IDAMiJL78/BOqh5pJaFg4sQEcSrGQFNOzo+gunB0+ArknC0lx6gRJSjEUTOUe1yEghs51WLqlJhbQvBsCmmzWrj09HUxrIDW2DB75YFdjQNNmQ5tu69ubqKqqEv/OgaGz685+L2KSIjMhhO44z6h8/JgdZlGAH2G7LUUQGCkExNCNFPJ2PFcJaGJllAsBTTYcaXEhKv7UXjeSwR7Bi7f0zCwVwF9SUkyjx+QMdotdf28ASREREkg3gjDZ+FEFPfs+kxRNdN3pTFIE2FWHXCQIuBIBMXSuRNPFdW34sILe2XMIKzkIaGZFDplh7a9ZrYh2SElJpcjISCrMyyO/hQtc0gM2xCwRFQgFlZXzUhBJEUxPby9VJMXqZUYhKVyCslTiCAJi6BxBaxivfV8JaJapNIQsoGmRQndlE9hxOCoqmvQgJArAvLZghTdUQsJWu1S0RksHLZkSr7TsHt1aTLe8uI9uBUnBER1SBIHhQkAM3XAh7cBzvoOA5oNwCj5xdDRdCAFNxbA6xz3YfDoburBwHRkM2fT17i9BSByjkNBQnNM5eVBn9TQmKdjXb5oxEiRFDkiK7kiKhROFpHBgWsilTiAghs4J8NxxaxUENP9oEdCEyq8/tn+8DXRXYdFNYza06d7bStXVVZRlNLrU0FnazSvSDCTpuQORFH95r1T1kdlkjtN11ZmjuzCSer0fATF0HjSG7Hx7HwL1eQW07vwsCGgGIrzL/hjWoXSFCYksg1EJb5aWHKCcUUYyDaUiO+7hvnB87vVnZCjlYpZpZ2PHJAX/Xoog4C4ExNC5C9kh1MvZt1hAk8+wjBDQ5MgHd6922iCbnqxPofDwcHVON//UeUNouX23WEgKlnm/ZJ6eUmJ19BeEjXGM7C3IJZsWG2xfRXKVIOAgAmLoHATMXZe/9rlZQPNy5HuYCflyZ8K7HGkjn8fFxMaSXp+qDB0LmLiSkLDVFnZ2boI01GKQLExS8LkdR1Lcssyg2GUpgoCrERBD52pEh1AfC2iyai+LZ54JEc1GJKwZrmKWbNKps7lvv9lD9fUNpAsOpg4XEhK2+qJICpzbsWG7C3kuHoW/4J0v59EvlmTQ4snxw9V9eY6PICCGboQHuqBTQHNSRqSVgKYbKNYB+omdJBlzRtPOHdvpUHU1ZWRmksnNhs7SHI78SI8LptvPNdJfsY1lMVGOpOCtrT8zJVIEARcgIIbOBSAOtYoaCGjebRHQXJxOukDHBDSH+tze92FRB8kmA7W0mKisrAQsLAyduxgJG41mkiIs2F+t5ljx5AXkpeBIil8syUSEhZAUrhpnX65HDN0IjT6fU7GvXEWNie4836jOqlwZ3uVIt1oh2aRPTaWwsDAlrT5n7mxHbnf6WjNJ0aFy0f5wTgoY2WCQFCUKm9VMUmDFJ0UQcAYBMXTOoOfEvc9AQPNjJaCZReMgoDkcDGt/zVWEREwcJSWnUF7ufiijuJ+QsNUWNv6c4Oe0yWgL3E8e+TciKVjuCcZuipAUTsw2uVUM3QjMgS1KQLOCfozVy5xxMcTnVCNZOKg/NDRI+dPt3/c9NdQ3Icl1kFschwfrJ0eAsB/h5MwI+BIikgLG7s6Xcuk6bGOXHC8kxWD4yd9tIyCGbphnBvvJMcPIGm3nQECTk9q4I7zLsW51kD+Owliy6ZOPdtLhQ9XIEJY+IobO0m7exvN53W3LjUoQ4L5NRerc7qfz0+D+4ljv5GpBQAzdMM6B0sMQ0IS2XE6yRUCzw2kBTVc1vx2Lykys6ExgIcrKSikjKx0HZ66qfWj1NOIjwLknrj09E0YvGJJPB6n8sImuPzODIiEDJUUQsBcBmS32IuXkdewzdg+MHBeWXQpDyBO/yJ6yOGE5dJZVD4YPHeeQmD3nRCd77Pzt3SQF0UWz9aTHud1TiJOt2Ai5J5zbZcSHOP8QqcEnEBBDNwzDzFvTR5HvIe8gBDTPzVYsIhs+TzFyDEE7GIjYWCYkks0REi6WbHIGZgtJcSrUTpidZjHS1S/uV2FjrIoiRRAYDAExdIMh5IK/r0cqwHf+e0gFr0/OilCH7Z5k5LiLzLyGhQWrFIiFhQXU1NiilFM4t4QnFAtJMQE6dutWICcFSIq1L+eq1TEn5ZEiCAyEgBg6N8+P7d8cpuc/KKfzT0wm1l8baYZ1oO76IRWrEYTEl198pggJDvZvbx/hg7peDWb8WPnktuXZ9CxIivs3M0nRrJIGsR+eFEHAFgJi6Nw4L74rNQtongQBzR+clIwcqFDc9YwFks1e86rJYMyGe0k9lYOQSM9IQbSEGwEaYtV8thkc6EdXI5qESYqX4KrDYWMs/xQdJlN6iLBq+jaZFW4a3qraFvoDxCX5RbxiGAQ0XdEN1qZjQoJVhouK8umEk2a4olqX18HrthaQJ7yC+8GsZDgXB9FTyDh2+0uck8JAWYlCUrgcdC+vUAydGwaQpc//tKlQbVNvWZZF0RDQbEQ8p6eX1tY2io9PpITERMrPzXUqQfZw9JVJigZEUpwCn8QkkBQPMUmxYT+MXRZxflkpgoAFATF0bpgLT0Jy6T9wDGZnV0OSWUDTG0pHB0s2gZCAP11xYSE11ptASPjjnM5zjTRvtxnf8ekcSZGNBDwHaO3f8+jaxRlK9kqKIMAIiKFz8Tx41SKgie0qryoavMTIMQzMsAYGmQmJ3V99QUeOHKHEpGSPIyRsDRmvnjk+ds05RuSRLaMHthRRKUiKyxYISeHiKe6V1Ymhc+GwfbL/KBK/QEBzaqJZQBPbKg/mHmz2nCMkDIYcuJc0UXl5Kc7skj2SkLDVeCYpdF0khY42MkmBXLI3nJEpJIUL57k3ViWGzkWjll/ZqBLbTM6MpB/N1avDck9mWPvrNjsKp6WnU2BgILavBTTzhGkuQsj91VhICn98XS4Ay82RFE+8W0K3bcxVkRQGISncPwge+gQxdC4YmJp6CGiCYY0MDaCfn2YW0GRCwhsLOw7HxiVQQmeExEhJNjmDHauxMEkx97hYSlCRFMW0ulPuaYaQFM5A67X3iqFzcug4RpTPgw4eNdFaFtCMDFIpCr21sKGLjAyljIxMKoQIZzNHcUAuxFMiJOzF1UJSsNYf56R4DCF4TFJcgw/RsumJ9lYj12kEATF0Tg4kC2h+su8o3XQWBDRTw1V4l7cXJdmUPZq+3buXao4cpriEBI9mXgfCm0kK/vhwCsm/7SijP79VTCUgKa5YkEaBARJJ4e1z1d72i6GzFykb123ZbRbQ/Mm8VDp5bIzXuJEM2mUsSA3Z2XTsWB1VlJchFCxRpUH01sKaf0EwalctQiQFZNrXf1SucsneeGYWxcLHUYr2EZBRHuIY72YBza0ltGgSC2gmIk+p9zGs/XW9BYvStPQM0ul0VFxUSFOnHz9ElDznNiaH/LGfPf/EJCXo+TiSha+BczGv9LLh6yhF2wiIoRvC+LKA5r1vFNFofagKJufDb7xHmiltiJBISEwCKREHySZESGikb+z3zCQFr74TsJ1luadbYexuQt4OjkeWol0ExNA5OLZ8BscqwX5+HXT1aRkUCgHNJi8mH2x1nyMhwsLDQEhkgZAooGacc3l6hIS9w6hICvRnTEo4rYXc0+NYlf/qlXxsa9NoOaTtpWgTATF0DowrvyS8CshnAc3zsikVApqeLLvkQNd6XMoMK3Lj4Jwuh7a+tYlqao5gdRfvtYSELRx4ZRcfEaTiYp+DjNYjYGVZ7umKhWk4z4NelRRNISCGzoHhZAHN9/YeUomVOUuVFhjW/rqPsFckss5RhERlRQUlJiZ4NSFhq59MUuhAUlwJ45YSi8TZMHhlkHtatTST4mAEpWgHATF0do4lC2g+B/cE9rhnSW/e/mi5wJ1OERK8ZWVCYvLUiZrrriWSgp2il89IUomzH4MowJoNiKQASTEqWUgKrQy6GDo7RvJ/nQKas8fG0gXQP+OoB60c0PfX/Va8/YqQiI3vIiS80XHYjuFFqB4iKfDhmjUmmuJBUjwCX7s160FSLMukWaNj7KlCrvFwBMTQDTJAlbUmFd6VAgHNy09NVbkeWAdN66VdRUhEQGU4w6sjJOwdJx5RlnviVdxayD09AfcTJimuhO/deUJS2Aujx14nhm6AoeGVGwfq89f+ZhbQhEy3NwhoumK2mQkJP0VIbHt3K9XW1lJ0dLSmCAlbOPH4xoYHYTWXRS/uLFehY2VwJ2JnY3Y6luKdCIihG2DcWPniv8XHlFOpIRECmho/l+sNBa9ymJCog5GrPFgB9eFYlQZR66Wp1RxJcRnCxDiS4rkPyhQjezPC/ISk8M7RF0PXz7i9+lklvflVFdwN0mlGjncJaLpqKjIhkQ5fOj6bO1BcRBMnjXdV1R5dD6/bWKyBHYzPRtQLC3oySbEa53b80RutD/Po9kvj+iIghs7GrPh0fw0ywpfQWdMSaQkLaGIlp/1Tub5AtGFlwwrDMbGxVJDn+TkkXP2CK5IC53YcNcGJsx8CSWGJpODoCineg4AYul5jxQKa9+JcbqoBAppz9Piyt3ulgKYrpqBZsimS0tLSwbzmUYupwyslm5zBQpEU+NAZk0KUDBcfZ/zm1Xy6HL53K5CrV4p3ICCGzmqcLAKaTDpchfCuIAhoslOprx5BMyGh05kJiZ07tlPt0aMUGRWFczofOKjr9f6yxmAMkxSIi13/YQU9CVaWz+2uxNFGSJBEUni6uRND1zlCfCZzPwQ0KyGgeRfcCxLgGc9hQr5q5KwnLhMSm/71GlVVVVJMXIxPEBK9X1yeB8zCs4bdz5BwhxVQnoMWIcs9rYLx462tFM9FQAxd59g8s72MPu0U0ByTEkbHfIxh7W+K8uItI9PQRUiMP24MNXvufHZ7yywkBZ/fJnEkxb/53C6Xbj07i8ZCeFWKZyIghg7jwgKaf/+0gi6Zl2IW0MRKTooZgTacUSaBkIiMjKZCnNP5+Z3m89AwScHndieMioLcU47KSbEGCXh4ZTd3XIzP4+OJAPi8ofuaBTThFHoaC2gi3rERqpNaD+9yZCKyZBOfy3FmMCYkmpt9j5DoDy+OpODMYmuhZMMkxW9fy8e2No0uRJigFM9CwKcNHSc4Zm053qr+ZL5ZQNODk9KPyMxhQxcaGqgIiU8/2kkN9fUUEhqGlZ7vERK2BoBJikjgw6u5jVC3+cs2M0lxNZLwCEkxIlPW5kN91tDx1uPu1wvJH86wVy/OoDBdAFZzvsuwDjQl2cUiO2cU/XvLm3QQERI5o0bB0HnOJB7JljBJYYK/IZMUK6E2nYyY6GdBUlQgcTZHUghJMZKj0/1snzR0vDVl50/2mVt7nhFaZGYBTWFYbU/Kdhi1jMwssK1tVFpygMaOG4UtrGdMYE9phZmk6KAzpyZA7kmnhDxvQSTFmnOMyA4nkRQjPU4+aehexBZj294jENDMoIkaF9B0xQRrbe2gpGQ9gvpj1Dkd0QJXVKu5OljUhncK05Ekex1k2h+CGjUn4GEhz3njYzXXX2/qkM8Zum0Q0HweSrIrfERA0xWTkc/j2MjpU1IoP2+/T/rROYIjG7vMBCYpjPSkIikKQFI000Wz9Y5UI9e6EAGfMnQsoPnnLcVwIYlWSsEc9SAM6+CzSUVIBPsrQmL3l19QXd0xEBRCSAyEHJMUESGBKnfsRuT+/es2swKKSqikk0iKwWeda6/wGUNXBQHNP0JAMzUuhC6HCwAhi1eblnIUunZe9KnNHweY2dmj6L2tb9Gh6moyGA1CSAyAeRdJAeB+Av9MPUiKZ7aXIpLCpLTu9DjHkzJ8CPiEoWsGK3YPAvVZVHE1PNijEMvKX1whH+yfaBwhkWUwKgfikgNFYF4N9t/sw1e2sstSSwdUcOIpKQoy7SApbu2UezouTSIphmtq+ISh46zseyGguWa5gbISzAKaYuQcm2JMSCTrUyg8PBzS6nm0YOF8xyrw4asVSQHn4mlGkBSIo2ZjdxtIihuwrV0wQUiK4Zgamjd0LKC5CQKarDIxA2yYllMUunPCtMPHJDommlJS06BNB8mmFuz+4YPI53dS7EOAP7AZICnuAEnx1Lul9IfXC5QowA8hBybFvQho2tB9uv+oEtBcNj2RzoCAphaTTbt3enTXzhES4eE6tX3du+c/VF/fQMHBwdjKiuewI2PARybhwYFYzWXSSyAp2LmYSYprTk9XTutS3IOAZg2dWUCzEAKaUfhippDJhwU0XTl1skeNpg/e30bVVVWUmZUlhs5BcK1JChaR0MeE0NPbS6gckRS3LDMo+ScprkdAk4aOBTT/+K8CJZR4FWIOg8B8ccITOZdzbgJxHHBmlgFKwyYqKz2AsLAswj+lDAEBJinaoNi8eHIc5J5AUsC5+JYX96mcFBMzIoZQo9wyEAKaM3Q8ge7fXExVtS1KQDMeApqNIqDpkreACQl9SiqFhUeoXK9z589xSb2+Wgkfb/K53ZSsSBVJwcbujpdy6fozMmnhxDhfhcUt/dacoXt6Wyl9lltDNyGgmrM1+VqKQrfMks5K+TwuJiYO7KseoWC5nX50vE4WQsIZ3HmOpsYF0+3nGuFYXKoSpvNW9sc4cpHiGgQ0ZehYQPMfnx6klfNSafaYGCEfXDNHumphhjUkNAjOwtn0/f++BSHRiJwSOjmncwHO7OMZiuiT68/I6JRpL6dSJM7meGwhKZwHWDOGbjcENNk/afHkeDp7RqLarso6w/kJYl0DGzp/RC8Zc3Loo53vI0KiCnlfM8XQuQBmXhe34GggAOfJ7G7Cck9/hccAyz2tPttAKfhZytAR0IShK8GXj5f7LKC5EgKabeyNLlZu6LNigDtZsoldTExgIcrLSinLmAlBNrc8yicr5bnbCJKCFa+TkHDnEci03/IiEmfD2E2C0o6UoSHg9YaOk9jc/UaBEj68FgKarOoqAppDmwz23MWhYHp9qgrqZ0Ji9txZ9twm1ziAAJMU7NjOhu2uFTmKpLi9k6RgAyjFcQS82tDxhHgYApqFlU10J7zNOXBaUhQ6PgkcuYMlm2LjELcJfTrWpmNFdYmQcARB+69lB/eUWB3ddq6BmGS7B7L/5Uea6JK5qcDc/nrkSiKvNnQv7ISA5jdH6PolmTQhPUIY1mGY0eYcEhwhYYA2XR41NDRTYGCAhIK5CXtFUkDW6RenZ6hzuhc+qKAyHNVchzkfESKRFPbC7rWGjgU0X4CA5gXIuHQKAqN5JSdleBBQhAQkmz7/9BM6cuSQEuRsxwspxfUIWJMUF5+sV4wsi3mW15iUEk8aZMekDI6AVxo6FtB8AE7Bc5BDc8WJySqDusSWDz7YrrqCiZ4sQzY1NTVReWkJmNdUREu4qnapxxYCTFLwx3zBhDjkktXRw0icvRpyT8zIHg+HYykDI+B1hq6yU0AzIz6ELjs1TYV18SSQMnwIMCHBKiYc1F9UWEAnzj5h+B7uw0+yRFJwnpNfMkkBd6o7Xs6j67GtXXx8vA8jM3jXvcrQsfT5vUpAs03FBHI+TQnvGnyQXX1FG7KBxcUzIZGMc7pclQtXCAlXo9x/fUxScKYx1ld8BhLtf9pURGXwt2MlY07fKaUvAl5l6J5gAc0Dx+g2pJDLxIpOBDRHZkp3dDAhEYwUiAYqLiqgpsYWMXTDPBTsQhUS6E/XQt6JmdkNH4KkgNwTE3NCUnixoVMCmruq6apFaTQtO1IptkoZGQQ4QiIQn0gjklrv/uoLOnzoECXpk4WQGMbhUCQFcp5wJMWFsxBJER2sSIqKGnYuNoKkkEgK6+HwihXdx/tq6C8IhzlnRgKdfnwCNTYLwzeM75TNR/F21YAIiYaGBqqoKKPUND1BdFjKMCNgISnY84BzUnAu2ZtZ7gkkxRSDkBSW4fB4Q5d3sJH+hHM51tu/+OQUfMXaEd4l5MMwv099HseERGpaupmQQITEzBOmjXSTfPb5iqTADmc8fElZ7ulRRVLk0nWIFDpjaoLP4uI1K7oj9S1KQDMWmnJXYsvKy3TO6CXHrSM/d1myKS4+keITEpRkkxDfIz8mfGbN8bFrcIb97I5Suh85jMvgb/fTU4Sk8NgVXSvOH1hA81CdWUAzDsZOUhSO/MtkaQFHSESEh0Bx2EiF+QXU2NAKZRN/MLByrDBSo8QLAPZM0AX50c8XIZIC53acPLsMYWP/BzHPKHgp+Grx2J4/jWS/n+cepZuhoz8KqiS8NJeVnOdMUyXZxIREdg7t+c9uOnr0CFZ4CWLoPGCIzHJPRCtOSqYkjqSAt8JtG3PVuV0mspD5YvFIQ7cZ7CoLaF4KyaVZo6OFYfXQmQkvE2Xo6uvrlWRTcnIitQoj4RGjZSEp5o0HSYFIij+z3JOKpMii6Tjv9rXicYZuV0EtPbrVLKDJaQrZX0iKZyIAv2FFSAQFBVFxYSFNmzHFMxvqo62ykBRj08JUJAWTFGsRScFyZkun+RZJ4VGG7sChJiVFMy41vFtAU065PfY15QiJhMQkSkhIRITEfiEkPHSkGuCOlRAZRKsRTfTcjjJ6kEkKOBf/9JRUpePoC8VjDB0LDd4NIxeEw4WrRUDTK+YeEw9h4aFKTr2ooIBMGEMhJDxz6HhnpINRu2phOulBUmz4qFyFjd0AkiI6zGPMgNvA84ge8qLtIZwhFFU10VoloKmT8C63DbnrKmZCIqgzQmLLm69TzREmJOKFkHAdxC6ryRJJwRJb55+YpN6xxxVJwed2RjIkapuk8AhD9yIENLdDQJMp8AlI3surO99YULtsHo9YRXwOxFnBmJA4eLCCEpMTqEUIiREbj8Ee3N4p93Ty2Bi1neUFxq0b9invhpnZ2iUpRtzQvbfXLKB54WwIaB4XKyrBg81UD/s7G7WMzCwKCAhQAf5Tpk70sBZKc3ojYJF74mRS684HSQHybx1IimsgEHDWtERNAjaihu6bkmPqYHTu+Bi1nGZnR4nu8q55xhES8SAj4uLjVA4J4Y68Z/zqoQodh1UdS549t6NcvYucS/Zy6DxyFJKWyogZusqjJpWikB0YWUCTDZwIaHrf1FIREpHhlJ6epQxdsxASXjOIlkiKIJAUVy5MU+d2fIxUdsREq5Zqi6QYEUPXhC8JC2iaELd6HdRRI4IDJEWh17wePRvaLdmUQ+9ufZtqa49SdHSsEBJeMp5s7DjcsgNL8XNnJilBTyYpbt2wHzGzBpAUoV7Sk4GbOSKGjoHkbettAJIl0UVA07vnEkdIZEObrg5G7mBFBcVjG8vqJlK8BwHYOmLl4tljmKRATgqQFJyT4uazDHTCKO8nKYbd0HFo1+bd1Qg6TqOpCEVhIyfFuxHAMR0iJDLhQxdAB4oLadLk47y7Qz7aehY/4/dxtD5MCWk8xiTFK3l4V9OhBendJMWwGrqPvmcBzVJaPjNRJfOQFIXaeKNacQSRlJREMbGxVIBcr0Ioefe48nvJ0mi3QASAPSIeweqOc8lejnM8Ps/zxjJsho4FNO9DEo8ZOSygCTVaKCzIC+GNU6Zvm9vb20BIRFBaegYVFOSRCWewnKRFBFK9d3zZA4KNGhs3Tpz9/AdlKpLixjMzlWSat5VhMXRHjrUqAU0GiJfB/BI0QynYO78N3jbE7m+vipCABprRmEM7tr8LQqKWoqKiqF0O6twPvpueYCEp2MH4bGxbWe7pMYgCrAFJwe4oOclhbnqye6p1u6HjBB73by6iwzB2d51vpJhwTlEoRs49wzmytbJk05v/eo2qKg9SbFyMEBIjOxwueTr7RTJJcdKoaErAQoVJijUbcmnVWVlKQs1bitsN3TMsoJl3VO33+Ssg5IO3TA3H2smLt3RESHA5UFxE4yeMdawCudpjEbCQFNl4f9eCpOC0o79+JV+lN2CXFG8objV0LKD5yqeVtBJyMPxF4C+DFG0iwL5YiUnJFB0TA0IiF8cSi7XZUR/uFSeLjwkPopuWZdGLH1SorSzLPV0Jh/8g5Jj15OI2Q7eroE4J/Z0OdnUZRP6aAJLk7vLkqeBc29rhYxIVFUmpqWmdhASk1oWQcA5UD7zbQlJcdmqqSpz9N4SOcdjYzdjKejJJ4RZDVwIBzbvfKKDxaeF0ybwUFdrFDolStItAB7yGdbpAJa3+8Uc7qa6ujsLDIxDXJ57DWhp1a5JiGQQAOJLiESxo2Ll4zXLkd/FQksLlhq6usQ0MayEFYyl7DcK7QoL8VbC+FO0jwN8yIyIk3tr0hiIkokdHETsTS9EeAkxSHMNR1MycaOSS1dFDb0HuCcbuJkRSzB7jeSSFSw0d+00xK1NcDQFNHFomwdrLuZz2Jnl/PWKjxpJNPA8OHCiiceNHU3Oz7/TfF3vK5CKLdvL7ztnGfv1qPl2BMztWI/Kk4lJD9+LOCnr/WwhowqnwuPRwJaApxXcQaFMREnqKhA9dUX6+zY73cBLHPkh8Kb1/fnC+5WjkjL0R53QbPyynJ941kxRXgZXVeQhJ4TJDpwQ0IfFy4axkmo8Ua+JG4v0T2NEetMMJPCo6GnGvaZSfn6v86PxASLBDMRs4TsQSFRqAatm8dVAzomMs80QMnqNoe871PHbN+MgFQsPuUnhYJCOS4m/vI5ICibN5K8tKxiNdXGLovi2pV6J9nENSBDRHekhH7vmQTaUQnT+k1XPoy88/BSFxjEJDQ3FO16bCifgMbztW/EfrW2EAicYgeHxiVgQdrmsBYcVGceTaLk92HoFWHNy1t3TQ0qkJiqRgr4vV6/dB7slIrGbsbPkKqVC3YUHF5/4XIDm3HgbV3mK3odv5XQ09va2UIrBEbcWsXDgxTj2sAvFvf0B4VxYENH+2IFU9VwQ07YVfY9dh2eYHdyqWbHrn7c10qKqSDNnZWNkhFjYkgHYV1il2jt2NeKX3IrY5Z05JgOhjOh2EEKsU70eASQpepXNM+10rsulhkBQcNrZqaRbNGRcz5A6ux1x56eODKtfzkfoWuuzJb+nBlWOV0oo9xW5D95+iOuQF8KOrT0un2oZWevCtIkqPC6HPc4/CR66dbl8uApr2AK71a9ibJDPToIQ3S0oOUM7obNVlXs2Z8LWfgQQsv71wlPodfyRXPvYNzUeukNTYYKjZSGigVuZHPc7nefFzJ0iKp94ppd++lk+XLUijC3C05Wipqm1R/npPXTGejElmIdBUxN7mVTS63tCxhjyHcE0AycDlzV1hVI6JWocOXYVAfRHQdHT4tHl9G87dkvR6ioiIRFLrXFqwaH5XR3lryqt9Vq45dKxFbT04WfnXhcdoFL7MnGhZ2AntzAuOaY8MCaQbIMu+EXlkn8KOsBQ2g4U9ePtpb9lTXEdJUbouI8f3XTLPvHu0t9i9ogsL9qe/bjtI5ThgrKgxITC/jX6xJENNVnYMFvLBXsi1fV0bJJuio2NIzxESyCHBWcJ4m9q7WPzH+QPaitWfHM9pc15wugQmKVbO50iKYHpmexlsiDmSIhHGy57CZ39MZDlT7DZ0vD1lNpWN22Hskd/8skrtmVmfig2fHCQ7MwzauZe3rOHh+Poasmn37q+o/lgDIibMh8bMvLJhCwr0Q7Z4nfo5r7JB6RMyAytFmwiYSQqiJTiP5ZXZI2+bIylY7olX9IOVsSAyqmpNKvDAshJklXIOPeOkPvYUuw0dyy0xbcx75ITGILAqwcQsiBRBwBYCHCGx7b2tdKi6Sqma+Plh1YaP8ndl9bRpV5VKi8iiD3yYPN0YSdXYysrHUrtziZ3I+dxuGtInrLuASQrWtoPcE7a17K0xUMlCgp6phki69pnv6JrF6TB6LSr66s7zjHYDZrehi4WO3D8+LQU71kxs9EoRz7oWD+J9uBRBwBoBLOoo02DEtrWFSg4UIywsSxFWOcmhtOT4BPoQDL4/VnZzwcKtAHNfixeA7xFDp/15xEdcfJ5/J7Qpn3q3lH73zwJsZU0qgf1A5VcX5KiYWk6sFYCJsg4kx6JJcXYDZrehO2dGEqWBZeXD5Ci4mIzCpI3AQSN/iTWW69Zu8ORC2wi0YhuarAiJCCoszKd5fnPVBzECzsL8cbQUXtWxq4AJ18sc8p3ZxOx6OFKc3oBjLz7++is0K9m5+OrF5th4W4XP6Pj6oRa7Dd0rnx2kL/JqlXvJWOyr6xpbFXMmE3So0Gv3vlaERCQkJpLBYKDivO8pHMEQHRE6nNeZwKxy8Gv3wTKv4mQOaXcu2OqZOZICBAMG/sdz9SpxNvvoctjYmvPGUDxc40oLvkecdBNlj5uIqzmaxrnSr6GrLi+mosICSgF7tqcuhdZ/WEk/Otm8qjsMA8dFthrOga/Fu5lgYLmmjpZGqobD8Ns7iqBycQdd/IPlNGXmTKqDsXOOP9Miar7XJ54DvDtshm/l4snxiqR4/L2DdMffPifTnpdo694aamoPopmprfTAL6+ncROOdwokm4bu2af/Quue3kZVbfGUEFhDsQmpdNV1N9L5OE9pbkH4jrJyTj3XpTc71xSru23/0+m29ts+Bxvu4OV9291ZQf/1DP0Jljv9A7D1aG+ia1atoVcPGChs0kX06J4SennHH2nDvdfRKacuoHo4nA9Uht4KO4aqV+VufZYdvXTo+Q5dbAcWw3yJdfOZY+efLVw7R01kJIbR0ot+Tt8cM1D05ItwjutPbxfupcob/kjbX/4zRcUNXRGlj6HbtvUtuvyB7RQ88xqKgHDiUWw1Sj55kkJL38GZ3E/g7OlPyFNs1cSeaNnlJNB5kV3XWp40QG7EfuvBH6z/1rsK899wCt77Oqvu9ambA9Stumy7TvMF5r+Zr7b6Z/fPnX/oUV/3jQNe1/Oy7jb1eU4frPummbS+p6sttjCxAsrWddyKiIhQeveNf9LG3cgNOvtiCmg5RmHxGVRdmkS/+fN6SjROplYKRvyrFYllhal1vWZ4utGxxtrybwu61j/36Y/qS3eH+O89rrHqa9fzLL+zutfSEnW/1aB1P9t8U8/rugeg67nq/p6/t9zXG1dLe/o+z/yXrt/3em5XfVY3WuajdTvUdT3mSPf86I0jX2fBu8c70/n7HjhYtc1SD7e4x7Os6gsICqb6in10qNGf4qacSQFtDVjywfl89FTa9cke+uSTj+n0pct7GhsHfupj6F7ZsoPas06nyFD4OZmOUmigjtonnUMPbHiTSiJPJlMbgrNBkfUeEPOL1wWD1YvePam6Bq23IbFMJpvgWIxGd91+FmAtHbUCrMeE6Px77wEzz0bz96V7vvaapNYT2QrQrglqqbu/62xYt6524NF97HaPyWbVMQsmls+h1YS28YiuHnWPj/m7yT/74T89zbT5cqsu9Jg61rj1bFGvznN/On8VHBZJu974koKTxlJgWxO1t+GYA/+NS86gXZ+Z6I5nv6RQGL72VvPxR88xGGDm9sK5a3Uw2Cq8zwqu77Koz29srJxsXWP9ux5/tzrT6X2fzeMeXMRzmpc4PethPHjUbBf1+z79N493112df++vjz1/j7tUQ8ylx5Ot2mbpQ/e9fJ/lnu42Wf5u7TBu3X/rMQwM6qCasqNU19xBoVjJWSTJ/fhF8QukhsamASbH4H/qY+iaW6Ah58+/5i9u5xuJB7UgMJtzPPpz1Hbn6bGlA90d6n5gV8c7R8MyKPz/vUEx398Jaw9Au6HuWV/n+FquxR9VvZ0D31VfL/DNf7e6tqstVgPVa7Lxc9UdvQa6zzOsntU9mOZ6e7bL8nOvydHZfvO9PdtjPemsceg9kXqPQzfO3e23tMUMhRW+1thZ/94mht0vQY/xxH0BgUG0s20SrX5+L0WMmkfU2ox5qqOa6kqamhJIa384hQJ0cBKF9Lpd88Aad+tx6DUetjDqHO7O2dX5IlobIesx6+x/D+x7zCerceuc5r2NVveL23feWlVvPdv7GLeu+dL1KnXPBes6up5l432yxsLmPdbvpNX9fa/tnOXWD7P0vat93b/oeZnlDbG8mJ34W99nPbcwtq0mPR3c/S96e/8uShg7Q60GqqsrKM0vn06YeUXvJzr0cx9Dd/bCmfTMXW9TU+pYCtbpkGjaj+r3rKdb/m8WXXaBORhbiiAwEAIzL19B23Z8SG99uZmCDdPJ1FhBgbmv0rrf/YBOGud4ULeg7RsIBISE0sO/uZEuu/U++vSr/dThryODfyE99suVlJblnO3pY+jOOW8F/W5fId33+hN0KCCZItuqafXZKXTpypW+gbb00mkE/IMjacPjv6cHH3uGPtr7JiXF6+iyay+lUxctcbpuqUDbCIyCO8m2fzxKO7ZtA2nVQCfP+TnFJdkX5jUQMjZYV3+6fc1qunD5HioohBRTWgqNmzRd2+hK71yOQEx8Mv1y7W2olxlWu901Xd4OqdD7EAjQhdGpS85yacP7nYE54yYR/1eKIOAcAmLknMNP7nYFAjILXYGi1CEICAIejYAYOo8eHmmcICAIuAIBMXSuQFHqEAQEAY9GQAydRw+PNE4QEARcgYAYOlegKHUIAoKARyMghs6jh0caJwgIAq5A4P8Bm9GHMnKn8OsAAAAASUVORK5CYII=)
In Δ ABC, using Apolloneous Theorem, we know that,
For AD to be the median, we have,
AB2 + AC2 = 2(AD2 + BD2)
⇒ 130 = 2(72 + BD2)
⇒ 65 = 49 + BD2
⇒ 16 = BD2
⇒ BD = 4
∴ Option (a) is correct.
Question 19.The diagonal of a square is 5√2. The length of the side of the square is
A. 10
B. 5
C. 3√2
D. 2√2
Answer:![](data:image/png;base64,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)
Let the side of square be a.
Diagonal = √2a
⇒ 5√2 = √2a
⇒ a = 5
∴ Option (b) is correct.
Question 20.The length of a diagonal of a rectangle is 13. If one of the side of the rectangle is 5, the perimeter of the rectangle is …
A. 36
B. 34
C. 48
D. 52
Answer:![](data:image/png;base64,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)
Let the other side of rectangle be x
By Pythagoras Theorem,
AD2 = AC2 + CD2
132 = AC2 + 52
⇒ 169 – 25 = AC2
⇒ 144 = AC2
⇒ AC2 = 122
⇒ AC = 12
Perimeter = 2(AC + CD)
⇒ Perimeter = 2(12 + 5)
⇒ Perimeter = 34
∴ Option (b) is correct.
Question 21.The length of a median of an equilateral triangle is √3. Length of the side of the triangle is
A. 1
B. 2√3
C. 2
D. √3
Answer:![](data:image/png;base64,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)
Median of an equilateral triangle is also a perpendicular to the base.
Let the side be a
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ a = 2
∴ Option (c) is correct.
Question 22.The perimeter of an equilateral triangle is 6. The length of the altitude of the triangle is …
A. 4
B. 2√3
C. 2
D. √3
Answer:![](data:image/png;base64,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)
Let the length of altitude be x and side be a.
⇒ 3a = 6
⇒ a = 2
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ x = √3
∴ Option (d) is correct.
Question 23.In ΔABC, m∠A = 90.
is a median. If AD = 6, AB = 10, then AC =
A. 8
B. 7.5
C. 16
D. 2√11
Answer:![](data:image/png;base64,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)
In Δ ABC, using Apolloneous Theorem, we know that,
For AD to be the median, we have,
AB2 + AC2 = 2(AD2 + BD2)
⇒ BC2 = 2 (36 + BD2)
⇒ 4BD2 = 72 + 2BD2
⇒ 2BD2 = 72
⇒ BD2 = 36
⇒ BD = 6
So, BC = 12 as BC = 2BD
By Pythagoras Theorem in ΔABC,
AC2 = BC2 – AB2
⇒ AC2 = 144 – 100
⇒ AC = 2√11
∴ Option (d) is correct.
Question 24.In Δ PQR, m∠Q = 90 and PQ = QR.
,
. If QM = 2, PQ =
A. 4
B. 2√2
C. 8
D. 2
Answer:![](data:image/png;base64,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)
![](data:image/png;base64,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)
Since ∠Q = 90 and PQ = QR , therefore ∠QPR = ∠QRP = θ
As sum of all of triangle = 180°
⇒ 2θ + 90 = 180
⇒ θ = 45°
![](data:image/png;base64,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)
⇒ PQ = 2√2
∴ Option (b) is correct.
Question 25.In Δ ABC, m∠A = 90,
is an altitude. So AB2 = ……
A. BD.BC
B. BD.DC
C. ![](data:image/png;base64,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)
D. BC.DC
Answer:![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAMIAAACwCAYAAABdC7H7AAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsQBlSsOGwAAABl0RVh0U29mdHdhcmUATWljcm9zb2Z0IE9mZmljZX/tNXEAAB/nSURBVHhe7Z0HYFRV1sf/k2TSEwKEAIEQmvQu0hEEAQHBgru6rp/LimVRse1SU2ihyYqLgKiri4qiiAXRFaRJcRHsIIKA1NCrlPRM5jvnvnmTSQxkWiZTztVhJu/de997557fO+fc+969IWZKkCQSCHAJhAT49cvliwSUBLwKBFORGbn5RTCRkTIYDIgJD5ZmEgl4RAJeBcILq47gg22nUTMuFCaTGY1rRSLtzoaoEulVp+mRhpGDeFYCXqVhh87kYmjHBDzUpw7YOkxcth+vrj+Gv9+a7FmpyNECTgJeBYIx2ID4GCOqRmmnVb9GBAoJCEkigYqWgFeBEG4MwvxVmdjw83lcyjHhyNlcrBjdtqJlIPWLBLwrWM4vNGNQh3jc170WcguK8On3ZzCXwJh1b2MEUfAsSSRQURLwKotQWFSEDkmxaFYnSl1vGFmI4Qt/JijMiAwVECpKCaReb+s+NQHvbTmJ7DwT8tki/HAGA9pWR0RokLSVSKBCJeBVFuEPXWpi7c7zyKTYgNMDvevgjhtqQGxBheqAVE4S8CoQejSLA38kiQQ8LQGvAsHTFy/HEwnoEqg0ENZ8tgLvf7oOV3IK0LNjMzwwYgRCw7UgWZJIwNMSqBQQ5j7/HJ5auBmGxrciODQSS175H9ZsHonFr85DZHQVT8tAjicS8HyMsG/3dkxatAlRPUYjKiIUKDKhqF5rfLjhRfRfvBiPjHxcmkUk4HEJeNwibN/xE36LaIyEyEiY8y+rCw4yZMGQ3AVbf/wOj3hcBHJAkUAl9BpVjYtDcGE2ihBU3C1qIB7zriChWrS0iUigUiTgcYvQpWt3dKz2Orbt/Ro1mnYiGMzIvnQe5v3/xU3/eKZShCAHFQl4HISo2Kp45dmxGPFMBnZs+Roh4TGIzdqL5jd0wvoLTdH+Yj5qVqHYQZJIwIMS8DgIfG1t2t+ATZ+8idT5y7H94AXMHPkgrkQ0xD8/3ouMD7ORPqwBasQKDB7Ug4A/VKWAwFKPoG7S5t2G4GLCJSQ2TEa1CDMe618X8+gttSkfHqQ30xogQWAIeAX1lAAqDQS+wLycLBTkZiEvNxfZQSHoSY9X8KQa81dnkmXQYBDL4ClVCOzjVCoItqLn99Cy8024sXlVtXnB55mYKjAEtnZ68Oq9BgS+Zp5hyRaG+QRDxkdkGe5oiPhYowfFIocKNAl4FQilYWAwlJtEMKTeWZ/eZ5YAOtAU1FPX63Ug2MLQq4XmJjEM0wiGlDsaCAye0owAO45XglAaBo4fOGZgGFJpnqPq0eImBZieVvjlei0ItjD0JsvAvUkvrj6KjA8OEAwNUF3cpApXjkA6gFeDYAvDTS2rqXZZyDCwZaAAujrNgSRJJOAOCXg9CGXB8KIeM7BlEDfJHXoQ8HX4BAgl3CSyDNyb9OKao5huCaCrCQwBr8iuCsBnQNBhyKFBtz6tyE2iqS2UZVh+EBNuF8vgqiIEenmfAsHWMjAMbBkWkmWYQTBw12rVKIkZAl2hnb1+nwPBCgNNAtaXYaANL1EAzW7SBIHBWT0I+HI+CYKtZWAY2DS8tPaoddBNLEPA67XDAvBZEGxjhr6tq6sLf4kD6OWHkHJ7fcSJm+SwMgRyAZ8GoSQMWszAlmHGxwxDA8TKSjuBrNsOXbvPg1AMQxH6tbFYBoJhOgXQ425rQJbBLy7RoUaVzI5LwG+0hK0Bd632o9mzVQBNbtLMjw9iPFkGWYPNccUItBJ+AwI3HK8yxVPK9yfLwGC8otwkbZwhNsKvLjXQ9LTCr9fvtEO3DP3JMnB6ZR2PMxwiy1BfYKhwdfLdA/gdCLplYDfJFobpH2kwiJvku8pakWfulyDoAXSugkEbdGM3adaKQyqAjo2QhcwrUql8sW6/BUG3DLn5RWr5Kd1NmkVdq+NuS0aMxAy+qK8Vds5+DUIxDCbcwr1JFEC8su4YZlLMMFZihgpTKl+s2O9BsI0ZbmkXr3qWXl1/THOThjYgyyBuki8qrrvPOSBAsLUMg9rH07NJwGtfMAzaoFt0uMDgbsXytfoCBoQSMNCi5pxeJRh4nGHc0PoSM/ia5rr5fAMKBNsAelB7HoE24zXdTRLL4GbV8q3qAg4EDQYzcguAwewmUXrti+N49hMKoMkyRIWJm+RbKuyesw1IEIphMGNwhxqam8SWgbpWxwxNppghYMXiHq3ywVoCusW5B4kH3Qa3r6F6kxZRzPDsChAM9SWA9kFlduWUAxoEa8xQUIQh12tu0iJyk2aTm8QwiJvkimr5VtmAB8HqJuWbMcTiJi3awJZBYPAtVXbtbAUEi/yUm1Rowq3UtcpPsC7aeAyzVxzGaIoZxDK4pmS+UFpAsGmloiJaxafQjKHKTTJj0YYTMHwCjB6SjEjpTfIFfXb6HAWEUqIrItPAXatDO9ZQT60u2nBcTSY2Zkh9RIQGOS1oKejdEhAQymgffZxh6PUJ6nGMRRuPq8XR/3GrWAbvVmfnz05AuIrs2DLkFZiUZeD0OsHA6R/sJoXKoJvzKuedJQWEa7QLB9B5BWbcxm4S/X5jk+YmjSbLECEweKdGO3lWAkI5gmM3iQNoBQPlfZNgMNB/7CZJzOCk1nlhMQHBjkYxUW9SPmFw+w0aDIsVDMDfBQY7pOcbWQQEO9vJRH5SfiFwB8HAftKbm0/CQDQ8M7ieuEl2ytCbswkIDrSOFYZOCRY36YQqzZYh3Chdqw6I0uuyCggONgnDkEeW4U6CgdPiTRYYBhMMMs7goDS9J7uA4ERbqK5VCwzcm/TWZhqBpnrYTQqX3iQnJFr5RQQEJ9uAYeCY4c7OmmV460uLZaBxhrAQcZOcFGulFRMQXBA9u0mwwMC9SW9/qQfQBIPEDC5I1vNFBQQXZa7DMIwsA8+btETBYMBTg8hNEhhclK7nigsIbpC1DsNdnWuq2pRloO+nKWYIFTfJDRKu+CoEBDfJWMFA6a4uDIOBLIMWMzxFMEjM4CYhV2A1AoIbhVtsGfipVTPe/t9JDQZykyRmcKOgK6AqAcHNQtUtwzCyDGwj3mEYyE96mmAQN8nNwnZjdQKCG4WpV2XrJvE4w7tbToI7VJ8clEwwcPQgydskICBUUIvoMPyhK8cMZrIMp8gyGPDkwCSxDBUkc1eqFRBckV45ZXUY/ti1FqFgwLvkJrE9YBiMTvYmXcgqUJORnb5YoB76qxZtxN3daqJBjYgKvBL/r1pAqOA21mG4mywDu0lLyU1iBX5yYD2EBDvuJh0/n0/vRJzAo/2T1Jpw+05k4f4FO/HSg83Rsm50BV+N/1YvIHigba0w0J2b09KvtN6kUbc47ibxi0LJ8REY2a+u9cx5ho15KzPx0kPNPXA1/nkIAcFD7VoSBjPBQDEDpSduqUdukv2Wga0Jw3Al12SdlpIXWk9591cU0Jt0jtTloUv3icMICB5spmI3iWMGYBnBwIr9BLtJQfbDUPqUg6ksP97B9RtVFCLJUQkICI5KzMX8rKysqvd0IxiIhmVb2TIYyDIk2R0zMDy2YxJ7T2QjNtIo70O40DYCggvCc7ZooeVxjHu611JVKMtA348TDMZyAmiG53KOCYfP5CjX6CB9z1+VSWMUSc6ejpQjCQgIlaQGDAPPjvQngoGV+32yDAYadRs1IAns6lwtxUWGIJoWQHx68R6VzxgchIf6JtK71Np7EZKck4CA4Jzc3FJKjxnu7cGWwYxl205rluEaMCTFh+PTMe0pMC5ScYY86u2WphCL4B4xOl+LDsOfutcmy2DAB19rAfSj/a4eM7DBkIf4nJd5WSXFIrhXnk7VVmwZaJzBYMYH204hKDgEj/VLxJljB3DyxEkkJddH1RqJTtUvhcqXgIBQvow8kkPBQKbgXooZDIZgrPrhFNa/8zx27DuK03mRqG28iDHD+2PEQw975HwC7SACghe1uMlEXasU/N7XKxHrls7Fh1t/Q0y3x2A0huHIlYt4cM4rqBIbjbvuvteLzto/TkVA8LJ2DAoOxskjB7Bz72HEdHkM4TzqbKJxgpgYFLS4BwuWLMcddw5DMMEhyX0SEBDcJ0u31BRM4wjnz53BhYIIhISG0/MU2apesykPYdFxOE8TchfkZgkIbpF2cSUCgpsF6mp1BbTCZ3LDpqgfZ8LOy2cQE0cTDxfmIigsGjmZ36FNUiTCY+JcPYyULyUBAcHLVCKEpoDJNYSgVsgFfLNuBgq7PQpjVDXkHdkF80/voMGtT6OI3neTKcTc23ACgnvl6XRtPLocGRmK8+cvYObUSbicnYvxD9yIHQfW48SpPDRKMKLB6FH4IbcZXlp9VD2GzeMNktwjAQHBPXJ0qRaGIIIhOHcBM6ZOxI/ff4unRo/HsGFDkJ1dgJysK4iOjYOZRtL+s+aIelCPIfjbzQKDS4K3KSwguEuSTtajW4Jz585j2qR0/LT9ezwzJgWDbxuMi5dpPkl66MIYEYucPBM9W2TG/TfSCDRt1R/Ue/jmOggS0+Ck9CVYdllw7qigGIILyJiYip07tuPvY1MxaMgg5GQXoogXfrZJtB66elT7LzTOwDS8Z3m55xGxDC43h1gEl0XoXAU6BGfPnlMQ7Nq5A/8YTxDcOhBZBIG5FAT6UQpp0C2EHlv9S6/aFDSbNRjITVIwOHcqUookICBUkhpwYHz69Blyh9Kw++edBEE6Bg4eQDHB1SEoCYMBw8kysPIzDPz9cF+JGZxtTgHBWck5WU63BAxBBkGwZ9fPGDMhHQMG9UcWTdXCr1zakzTLoLlJXER3kwQGe6T3+zwCgnNyc7pUZFQoTp1kS5CKvb/sxpiUieh/y80EAVkCOyGwtQz8RpuCgTbyhAA8Jf3Dfes4fX6BWlBA8FTL8zgBQXDyxGlMnZiCX/ftJQjS0W8AQ2C/JSh9uvlkGUIJBnaTOKnpJclPGnFTHRlncKBtBQQHhOVsVr7RRxEEJ06cwtS0FOzfvxfjUiehb/8+yLriPAR8PhwbFBAMbBn+2ltzk5bQjHr8/VAfsgwSQdvVbAKCXWJyLRNDcPz4KUxJG4eDBw9gfNoU9Lm5N0HguDt0tTPRYXiAYeC5VrfQAocEwYMMg6RyJSAglCsi1zJw79CxYycIggk4cuigBYJeLluCss5Kd5Me6E1uEWXgKekN9N+IPvJmW3mtKCCUJyEX9jMER48ex9R0guDwIYyfOBW9+/SsEAhs3STuTforwcAR9JL/8co9ZoJBLMO1mlJAcEHRr1WUA+PMI8cIghQczTyMCekaBFdcjAnsOV3uWlUxAwXMPDbNK/ewm/QA/S2pbAkICBWgGWwJjhxmCCaQW5SJ8QRBr949PAKBfjm6mzTiJs0tesuy2icH1JJ+LwEBwc1awYHx4UNHMTltPHWVHkdKegZ69urmUhepM6fIMUIxDHVUL9Kbm0+o8QYOqCWVlICA4EaNYAgOHTiCSQTBmVMnkUoxQXeGwAPuUFmXYdu1yr1HDMFbBANvF8sgILhR9YurYggOHDiMySnjcPbsaaROnobuPbt41B262oXpXat6V6qCgWjQB+EqRCA+VqlYBDc0GEOw/9dDyh06f+4s0ibPQNcenbwCgtIxg4KB/KTFtOoOJ4FBk5CA4CIIUdGh+HXvQUxJH48L584RBNPQpbsGgXLMvSTZuknclcpnxjDwOAM/0h3oSUBwQQPYEuzbc4AGy8bj4sULyh1SEFzOd6HWii1awk3iAHrTceUm8ZtvgZwEBCdbny3Bvj37MTl1PC5dukiWYDo6demoAmNvTwwDP6j3EL2/wJbh9Y00WRKlQIZBQHBCa6MJgj2//IpJqeNI8a8gbQpB0Pl6zR1SquX9ibtWedCNH9nWYWDL8H89A9MyCAgO6ixbgl927yMIxiI7KwvpU2agY+cOXu0OXe0S1TgDTSn5CL+/QDS8voHcJMp8XwDCICA4AIKCYNcecocmICcnBxMzZuL6G9r7JAR82WrQjVbiZBi0l3nMVjcp0GAQEOwEgSHYtXOP6h3Ky81VEHTo2JYg8P6Y4FqXaAuDNWYgy8CU3NcjcNwkAcEOEDgm2PnTL/TsEEGQn6fcIQ0C7+0dsuOyrFl0GMJoSnp+55ndpEVfcG+SAX+2LHjoSH2+mFdAKKfVFAQ7ditLUFhYiIlTZ6J9hzZ+A4F++QxDnqkIYewm0dQwnP6z/hgv4ANtjTf/TgLCNdqXIfhpxy6KCcbRPENmpBME7Tq08jsISsBAMYOCgeZWLaIBwde+IBiIEl7905+TgHCV1mUIdjAEKWNVjnSKCdq2b+m3ENjCwAG0kWAY2Z/XbjbgVbIMnPwZBgGhDBAYgu0/7lQjxgZa/Hji1Olo3db/IbAVBcMQFhKEv5Fl4MSWwZ9hEBBKgcAQ/PD9DvVmWQgt45SeMQOtWjenwTL/CIztdW9UzEBrObOb9AjBwBMC6G7SPd38z00SEGw0Q0Hw3Q4VGBtDQ8kSzEDLVs0CDoKyYoa/3UxuEgXO/+YAmjLc7WcwCAiq1Q2Ijjbi+29/VJNvhYaFUWA8Ay1aNg1YCH4HA63kw24SrftJMGjPJvkTDAICNWh0jBHffv0DMugd4/CISBosm4FmLZrQc0SB5Q5dzW2yukkEw8j+mpv0yjruTTLgj11pkXQ/SAEOAlkChmDb92reoajoKDVi3LT5dQJBKeXWYNAC6Ef7aW4Sw8B+0h+7+D4MAQsC382iyB36eut3yhJEx8aqwbKmzRoFvDtkj2V4VHWtEgxrCQaCwtctQ4CCQBBEGbH1q28wjRbpiGUIyBJc11QgKM/LKekmWWBYd1QNuv3Bhy1DwIGgW4KvvtyG6ZPTUCUuTvUONSYIJCYoDwNtvw5DuJEG3TiAVm7SUbWW27DOCfZV4mW5AgoEfj+X3aEtm7di+pQ0VK1WnR6gm47GTRoKBA4qpv6gXnhosDWAfmltprIMd3byPRgCBgTdEny5cQtBkI74+BpqsKxRo/oCgYMQ2GbnQbcIgoFjBn7T7cU1R9VuX4MhIEBgCHicYBNDMDkVNWrUVF2kDRmCLOkidYEDy8s9RYgkGB4fwL1JmVhIMLDM77ihhitVe7Ss34OgW4KNG74kCNJRq1ZtGiybjgYNkwUCN6paLlsGowbDvM8z8eLqTKrdTDD4hpvk1yAoCKh3aOP6zbSifTpqJ9ZRgXFy/SSBwI0Q6AG0goEsw2MEwwIFg+Ym+QIMfguCbgk2rNuEmRkTCYK66rGJ5Pp1BQI3Q6BXxwE0wxDFMFDMsICswkKCgdvi9o7e7Sb5JQg6BF+s3YhZ0yYhsU4S9Q5NQz2BoIIQKK5W71qNCrdYhlVkGcg68PbbvBgGvwNBh2Dd6i/w7LTJSEpOVpNvJSXXoelXJDCucBIsB8grsFgGcpPmEwj84eStMPgVCHrv0JrPGYJJZAEaqHGCuvUSBQJPEWBzHOUmhVkCaLIMHDfw0rdDrvc+N8lvQNAtwZrP12FWxmTqFWpEM9BNQ526BAE/RSrLrHocBWvMEB6EJ25JwtyVRzBv1RHVFkM6eBcMfgGC3jv0+co1mD19Cho1bkLuUAbFBrWRnS0QeJwAmwOqmKHAjEiKGUYNrKdAmEfWgR/LGOpFlsHnQTAEBSEqMgSfr1yN2TOmahCQJUhMrKV6h3jIX1LlSyCXYoZogoHHGeZb3SQDbu0QX/knR2fg0yAEEQQRBMGqz1bjuZlT0fi6pmpq9sTEmsoSCAReoWPqJDTLUISYiBA8Tm4SWwXlJlHyBhh8FoQgiroYgpWfrsScWdPRpFlzpEyaitq1NQgkeacEFAzhIRilYDiCFyhu8AYYfBIEhiCSRow/W7ES/5yVgWbNWyJ1UgZq1k4QCLxT/0ucFT+ox5Zh1C31CATNMnBv0qD2rrlJvL70B9tOYffxbCRVDwPPtsG9VvYknwOB3aGoqBB8+vFneI4gaNGqNVImZiChZg3VRSrukD3NXvl5OGZgGJ4YSL1Jnx1RPUrsPw1q5xwMV3JNeOy1X0Dqgc6Nq+C7A5ewavs5LBzRHPH0Om55yadAYAgiyR1a8dGnmPPsdLRq247WMZ6CGjXjkSMxQXlt7VX7bWOGJwbV02CgDydnYHiNppkppGk5336slfU6R7+1Fz8cuox+rauVe+1uA+FCVgFGvb4HT5K5u6FRbLkHdjSDHhivWP4Jnn92Btq0a48JtI5xfHw1sQSOCtOL8rNliCXL8CTB8AKBwB+GZKCDlmHL3t9wT6n5WWff1wRFBIc9yW0grN95Hl/tvYi61U65HQQFAQlrxYcr8K/neA7SDphAlqA6QyCWwJ529to8atDNAgOPM3DwPJfiBvaTBrarbvd58+o/ceQtlE4cT9qT3AICD4589uM5zP7zdXhv6ylknsulYCXcnuOXm0eHYPmHy/HCc8+ibYfrFQTVqlelVWskJihXgD6QQYeBFfkJgkF3k1iHB7S1D4YqdKM88VvJ3sJfjmchO8+EDg3K91DcAsKOzCvIJ6r59byf6fcH207jKTJ1riYdgo/e/wgvzJlFi3N0xri0yQRBnMQErgrXy8ozDDmkQ1UIBnaTGIbn/3tYnaU9MPRrUx1LvjypdDCcJiLjlUOfJFed//YYCCt/OItdx7IwbflBFZxczC5UvQE8q4GzSYfgg2XvY96cf6Jjpy4Ynz4JcTTrhATGzkrVu8vpATRbBobhXwTC8xxAs2UgRb9WurtbTdVTdOdz29E2ORo/H81C41qRGN4r0a6Ldtki5OQXUWzwG63EWAsJsaGoHx+uJordsOsC+rQsP1ov6yyDaRbqMOr/XbZ0GeY/PxudunbH+LRJNPVKFYHArmb17UxsGRiGpwYnk1XQLAND0v8aMISQHzXn/ibgoHkPjSMMaBuP3i2q2i0Il0H45Pszqu/26UHJ1oOevlSAl2kGND4RR62CFYJ3l2LB3Dno3K0HxqdOQmyVWIHA7mb17Yy2McPTg8kyfEaWgYDgdC0YeH+3JnHq42hyGQQezRtxEy9NWpzuokmeLucUUqCiPWhlb9IheO+dpVg4bw66dO+JcanpaiY6CYztlaL/5OPeJOUmUQD9L0vMEBwShr4tonH2xCG6MdIIcsPryHUqf8CsPKm4DMK9ZaytxT1G6cMalnfsEvsZgiCaYHbp20uwcMFcdCUIxqYIBA4J0Q8zMwxV6XEa7k1asPYU5n/8IxbNWYav952l96ND0DyhCFNHP4xO3Xq6dPUug+DK0SOiYhAaQUPtsREICwUWv/E2Xp4/F91v7I3RKWmIjYlBbq50kboiY2fLXrObo9RO7U/6V/vfmtRvmw3Wnypf8Q7bfFc7br34CDzRryYGDhuD3TkNEdNmJIKCQ7D6+F7sGDkN696i9Sxat3f2civvMez9v/yEz99Ygt1HL2JD1UG4fPky/r1wPrr3ugljJqQhJibaq9yh3zXQVVqs7M2WraUURW812861spTlqopSluKV1kY+iGVwldc1KCupraV2Wf+0li1ZkjebaQCJV940F0F9q9+0o/hb367ntfmmEV8e9OWPVk+pvFSnibfTDv42hkbi562rcTw7AtU7342gwitUMA81G7bByYvHseit9zB7lo+B8NGydzFi8pvITuiBkPDauG/SMlQ5tRbDhw/HM2NpnYKoKLIEBTAGl2zp0kqmKZD1fmS9+5TIV6IK3sNrvpR5SyvR0lpO7R9WoOLf+jZ9f8l9Wj5qQL2sKl+kFMSyS6tPKYDW2MWKZFEGi2Lp+2y/rQrDeazKxApjqYvrpD9UPt7GiqTymdWzOHq+IhO0v2m72m/JRyGfpYxWv7W8+q0dQ99mPX91LFZYG6Wmv3UoSiq7JY/lGpUcLFAV/y4mk98+5BQaEYP9m7cjL7wm6M1POm8lYZgLsmGoWhfHzmqLHTqbPO4aHTu8H6Nmvo2s1o8jLr42XZEJpgY9cXqjGfXpPeOoKlE4dzZXKYutADXFsbmLsBAsdwtuXP3OoX6rxiq+o+iNpCmOts82v/XOY1Ec2/yq0dWdSVMQXZFKKIiuCBZlUMfRlVSdR5F2TL0ey52OFYDz6VBw0/I1llaO0gAV/22B8hqWhhWJR2hZnay/6Q/uzVPb9W9SLn1b6X38mEKw5cNl+HcoLTKotlH5YLphcRktj7a/+KP9zXVwF2dxXbSdyoToZalPRdV1lbJhtKbd4cY3YnjKy/SCTyFCg0JITiSx0CiYz+xDy86uLXDocRC+/eYbHAtqiITqtcmyXeTmQXBIOKKa9MULSzdiR8iNyMvLtyoNK0aJO41SFP1uU3x3tt65dZt+DddFWYRib0X95oY00DLzvI9vQtqHG1rfpymOakjepzc6KxI3sEVBuB4jaUMQdWRww3LXsq4EZTWypjhBV1Egm+2WukrWoSmOfmyuS1dIPq52/OI8+rnr56sUnvJpMBRfn7pOHRb6oUGkfZR3fxXwnL0b21uuQ3JvbF77KZ5f+TLC29+FYGMosvasQwvDt/jr/fPsrabMfB4HwWg0KsHauqTcEIUFJlSjiXpbJEai0BSmKWCpuwvfUahzyXoXUkpkuSPpdxPtrmNRWMtvtc+qFLbKYlF0m3y6stsqQvHvkndSVgj9rqoriwaUS20iha8hgVnTp6Ja3Gx8vOkNFJhD0KJeGKZMmIXEJMd6KUsfwuMgdO7aFdcZF2P/iYOIr9OQ3BsTPX1oQt6+VUh/7l4MGGDfkLhoS2BKwBgWgdS0dIzNuYT8vFxExblnkmGPg1A9IRHz0kbgL2Pm4eSRdkB4LIJPfY1J97bHgMG3BWbrylU7LAFjRCz4467kcRD4xPsPHIItjRthzbovUFBoQrtWo9C9dz93XZPUIxJwWAKVAgKfZYPrWuBh+kgSCXiDBCoNBG+4eDkHkYAuAQFBdEEkQBIQEEQNRAICguiASECTgFgE0QSRgIAgOiASEIsgOiASsEpAXCNRBpEASeD/AQ9dbHfupLKsAAAAAElFTkSuQmCC)
By Pythagoras Theorem in ΔABD,
AB2 = AD2 + BD2 ...... (1)
As we know that,
If an altitude is drawn to hypotenuse of a right angled triangle, then the length of altitude is the geometric mean of lengths of segments of hypotenuse formed by the altitude.
AD2 = BD.CD ...... (2)
From equation (1) and (2)
AB2 = BD.CD + BD2
⇒ AB2 = BD(CD + BD)
⇒ AB2 = BD.BC
∴ Option (a) is correct.
Question 26.In Δ ABC, m∠A = 90,
is an altitude. Therefore BD.DC = ……
A. AB2
B. BC2
C. AC2
D. AD2
Answer:![](data:image/png;base64,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)
As we know that,
If an altitude is drawn to hypotenuse of a right angled triangle, then the length of altitude is the geometric mean of lengths of segments of hypotenuse formed by the altitude.
⇒ AD2 = BD.DC
∴ Option (d) is correct.
,
,
are the medians of ΔABC. If BE = 12, CF = 9 and AB2 + BC2 + AC2 = 600, BC = 10, find AD.
Answer:
From the given info,
As AB2 + BC2 + AC2 = 600
and BC = 10,
⇒ AB2 + AC2 = 500 ...... (1)
As AD is median,
⇒ BD = 5 ...... (2)
In Δ ABC, using the theorem of Apolloneous, we know that,
For AD to be the median, we have,
AB2 + AC2 = 2(AD2 + BD2)
Substituting values for (1) and (2) in above equation,
⇒ 500 = 2 (AD2 + 52)
⇒ 250 = AD2 + 25
⇒ AD2 = 225
⇒ AD2 = 152
⇒ AD = 15
Question 2.
is the altitude of Δ ABC such that B—D—C. If AD2 = BD ⋅ DC, prove that ∠BAC is right angle.
Answer:
Since AD is the altitude to BC,
⇒ ∠BDA = ∠CDA = 90°
So, Δ ABD and Δ ACD are right angle triangles.
Using Pythagoras theorem,
AB2 = BD2 + AD2 ...... (1)
And, AC2 = AD2 + DC2 ...... (2)
It is given that AD2 = BD.DC …… (3)
From equation (1) and (3), we get,
AB2 = BD2 + BD.DC
⇒ AB2 = BD(BD + DC) = BD. BC
Now, from equation (2) and (3),
AC2 = BD.DC + DC2
⇒ AC2 = DC (BD + DC)
⇒ AC2 = DC(BC)
Also, BC = BD + DC,
Substituting the value of BD and DC in above equation, we get,
⇒ BC2 = AB2 + AC2
So, by the converse of Pythagoras Theorem, we can say that ∠BAC is a right angle.
Question 3.
In ΔABC, , B—D—C. If AB2 = BD ⋅ BC, prove that ∠ BAC is a right angle.
Answer:
Since AD is the altitude to BC,
⇒ ∠BDA = ∠CDA = 90°
So, Δ ABD and Δ ACD are right angle triangles.
Using Pythagoras theorem,
AB2 = BD2 + AD2 ...... (1)
And, AC2 = AD2 + DC2 ...... (2)
It is given that AB2 = BD.BC
From equation (1) and (2), we get,
AB2 – BD2 = AC2 – DC2
⇒ AB2 – BD2 = AC2 – (BC – BD)2
⇒ BC2 + BD2 – 2BC.BD = AC2 + BD2 – AB2
⇒ BC2 – 2BC (BD) = AC2 – AB2
Substituting the value of BD above, we get,
⇒ BC2 – 2(AB)2 = AC2 – AB2
⇒ BC2 = AB2 + AC2
So, by the converse of Pythagoras Theorem, we can say that ∠BAC is a right angle.
Question 4.
In ΔABC, , B—D—C. If AC2 = CD ⋅ BC, prove that ∠BAC is a right angle.
Answer:
Since AD is the altitude to BC,
⇒ ∠BDA = ∠CDA = 90°
So, Δ ABD and Δ ACD are right angle triangles.
Using Pythagoras theorem,
AB2 = BD2 + AD2 ...... (1)
And, AC2 = AD2 + DC2 ...... (2)
It is given that AC2 = CD.BC
From equation (1) and (2), we get,
AB2 – BD2 = AC2 – CD2
⇒ AB2 – (BC – CD)2 = AC2 – CD2
⇒ AB2 – BC2 – CD2 + 2.BC.CD = AC2 – CD2
⇒ AB2 – BC2 + 2.BC.CD = AC2
Substituting the value of CD above, we get,
⇒ AB2 – BC2 + 2AC2 = AC2
⇒ AB2 + AC2 = BC2
So, by the converse of Pythagoras Theorem, we can say that ∠BAC is a right angle.
Question 5.
is a median of ΔABC. If BD = AD, prove that ∠A is a right angle in ΔABC.
Answer:
In Δ ABC, using Apolloneous Theorem, we know that,
For AD to be the median, we have,
AB2 + AC2 = 2(AD2 + BD2)
As it is given that AD = BD,
⇒ AB2 + AC2 = 4.BD2
Since AD is the median to line BC, we have,
Substituting the value of BD in above equation, we get,
AB2 + AC2 = BC2
So, by the converse of Pythagoras Theorem, we can say that ∠BAC is a right angle.
Question 6.
In figure 7.25, AC is the length of a pole standing vertical on the ground. The pole is bent at point B, so that the top of the pole touches the ground at a point 15 meters away from the base of the pole. If the length of the pole is 25, find the length of the upper part of the pole.
Answer:
Let the length of upper part of pole be x,
The length of AB becomes 25 – x.
Applying Pythagoras theorem in ΔABC,
(AC’)2 = AB2 + (BC’)2
⇒ x2 = (25 – x)2 + 152
⇒ x2 = 625 + x2 – 50x + 225
⇒ 50x = 850
⇒ x = 17 m
∴ The length of the upper part of the pole is 17 meters.
Question 7.
In ΔABC, AB > AC, D is the mid–point of .
such that B—M—C. Prove that AB2 — AC2 = 2BC . DM.
Answer:
From the given information, we construct the figure as:
As AM is perpendicular BC, ΔACM and ΔABM are right triangle.
By using Pythagoras Theorem, we get,
AC2 = CM2 + AM2 ...... (1)
Also, AB2 = BM2 + AM2 ...... (2)
Eliminating AM2 from equations (1) and (2), we get,
AC2 – CM2 = AB2 – BM2
⇒ AC2 – AB2 = CM2 – BM2
By using the identity a2 – b2 = (a + b)(a – b) on above equation,
⇒ AC2 – AB2 = (CM + BM)(CM – BM)
⇒ AC2 – AB2 = BC(CM – (BC – CM))
⇒ AB2 – AC2 = BC(2CM – BC)
As D is mid point of BC,
⇒ AB2 – AC2 = BC(2CM – 2BD)
⇒ AB2 – AC2 = 2BC(CM – BD)
⇒ AB2 – AC2 = 2BC(CM – CD)
⇒ AB2 – AC2 = 2BC(–DM)
⇒ AB2 — AC2 = 2BC.DM
Hence, proved.
Question 8.
In ΔABC, , D
and ∠B is right angle. If AC = 5CD, prove that BD = 2CD.
Answer:
ΔABC is a right triangle, right angle at B and BD is perpendicular from B vertex to hypotenuse AC.
So, as we know that,
If an altitude is drawn to hypotenuse of a right angled triangle, then the length of altitude is the geometric mean of lengths of segments of hypotenuse formed by the altitude.
⇒ BD2 = AD.CD
⇒ BD2 = (AC – CD).CD
As, AC = 5CD,
⇒ BD2 = (5CD – CD).CD
⇒ BD2 = 4CD2
⇒ BD2 = (2CD)2
⇒ BD = 2CD
Hence, proved.
Question 9.
In ΔPQR, if m∠P + m∠Q = m∠R. PR = 7, QR = 24, then PQ =
A. 31
B. 25
C. 17
D. 15
Answer:
In ΔABC, we know that sum of all interior angles in triangle is equal to 180°
⇒ ∠P + ∠Q + ∠R = 180
Also, given that, ∠P + ∠Q = ∠R,
⇒ 2∠R = 180
⇒ ∠R = 90°
Using Pythagoras Theorem,
PQ2 = PR2 + QR2
⇒ PQ2 = 72 + (24)2
⇒ PQ2 = 49 + 576
⇒ PQ2 = 625
⇒ PQ = 25
∴ Option (b) is correct.
Question 10.
In ΔABC, is an altitude and ∠A is right angle. If AB =
, BD = 4, then CD =
A. 5
B. 3
C. √5
D. 1
Answer:
As we know that,
If an altitude is drawn to hypotenuse of a right angled triangle, then the length of each side other than the hypotenuse is the geometric mean of length of hypotenuse and segment of hypotenuse adjacent to the side.
⇒ AB2 = BD.BC
⇒ 20 = 4. BC
⇒ BC = 5
Also, CD = BC – BD
CD = 5 – 4
CD = 1
∴ Option (d) is correct.
Question 11.
In ΔABC, AB2 + AC2 = 50. The length of the median AD = 3. So, BC =
A. 4
B. 24
C. 8
D. 16
Answer:
In Δ ABC, using the theorem of Apolloneous, we know that,
For AD to be the median, we have,
AB2 + AC2 = 2(AD2 + BD2)
⇒ 50 = 2(32 + BD2)
⇒ 50 = 2 (9 + BD2)
⇒ 25 = 9 + BD2
⇒ BD2 = 16
⇒ BD = 4
⇒ BC = 2BD = 8
∴ Option (c) is correct.
Question 12.
In ΔABC, m∠B = 90, AB = BC. Then AB: AC =
A. 1:3
B. 1:2
C. 1 : √2
D. √2:1
Answer:
In ΔABC, applying Pythagoras Theorem, we get,
AC2 = AB2 + BC2
As AB = BC,
⇒ AC2 = 2AB2
⇒ AB:AC = 1:√2
∴ Option (c) is correct.
Question 13.
In ΔABC, m∠B = 90 and AC = 10. The length of the median BM =
A. 5
B. 5√2
C. 6
D. 8
Answer:
In Δ ABC, using the theorem of Apolloneous, we know that,
For BM to be the median, we have,
AB2 + AC2 = 2(BM2 + CM2)
Also, as Δ ABC is right angled at B,
⇒ AB2 + AC2 = AC2 = 102 = 100
⇒ 100 = 2 (BM2 + 52)
⇒ 50 = BM2 + 25
⇒ BM = 5
∴ Option (a) is correct.
Question 14.
In ΔABC, AB = BC = . m∠B
A. Is acute
B. Is obtuse
C. Is right angle
D. Cannot be obtained
Answer:
Let us assume,
⇒ AB = BC = x and AC = √2x
As we see that, AC2 = AB2 + BC2
So by converse of Pythagoras Theorem, ΔABC is right angled and right angle at B.
∴ Option (c) is correct.
Question 15.
In ΔABC, if then m ∠C = ……
A. 90
B. 30
C. 60
D. 45
Answer:
Let us assume,
⇒ AB = x, AC = 2x and BC = √3x
As we see that, AC2 = AB2 + BC2
So by converse of Pythagoras Theorem, ΔABC is right angled and right angle at B.
Let ∠C = θ,
⇒ θ = 30°
∴ Option (b) is correct.
Question 16.
In ΔXYZ, m∠X : m∠Y : m∠Z = 1 : 2 : 3. If XY = 15, YZ = …
A.
B. 17
C. 8
D. 7.5
Answer:
Let ∠X = A,
⇒ ∠Y = 2A and ∠Z = 3A
As sum of all angles in a triangle is equal to 180°
∠X + ∠Y + ∠Z = 180
⇒ 6A = 180
⇒ A = 30
⇒ ∠Z = 90°
⇒ YZ = 7.5
∴ Option (d) is correct.
Question 17.
In Δ ABC, ∠B is a right angle and is an altitude. If AD = BD = 5, then DC =
A. 1
B. √5
C. 5
D. 2.5
Answer:
As we know that,
If an altitude is drawn to hypotenuse of a right angled triangle, then the length of altitude is the geometric mean of lengths of segments of hypotenuse formed by the altitude.
BD2 = AD.CD
⇒ 52 = 5.CD
⇒ CD = 5
∴ Option (c) is correct.
Question 18.
In Δ ABC, is median. If AB2 + AC2 = 130 and AD = 7, then BD =
A. 4
B. 8
C. 16
D. 32
Answer:
In Δ ABC, using Apolloneous Theorem, we know that,
For AD to be the median, we have,
AB2 + AC2 = 2(AD2 + BD2)
⇒ 130 = 2(72 + BD2)
⇒ 65 = 49 + BD2
⇒ 16 = BD2
⇒ BD = 4
∴ Option (a) is correct.
Question 19.
The diagonal of a square is 5√2. The length of the side of the square is
A. 10
B. 5
C. 3√2
D. 2√2
Answer:
Let the side of square be a.
Diagonal = √2a
⇒ 5√2 = √2a
⇒ a = 5
∴ Option (b) is correct.
Question 20.
The length of a diagonal of a rectangle is 13. If one of the side of the rectangle is 5, the perimeter of the rectangle is …
A. 36
B. 34
C. 48
D. 52
Answer:
Let the other side of rectangle be x
By Pythagoras Theorem,
AD2 = AC2 + CD2
132 = AC2 + 52
⇒ 169 – 25 = AC2
⇒ 144 = AC2
⇒ AC2 = 122
⇒ AC = 12
Perimeter = 2(AC + CD)
⇒ Perimeter = 2(12 + 5)
⇒ Perimeter = 34
∴ Option (b) is correct.
Question 21.
The length of a median of an equilateral triangle is √3. Length of the side of the triangle is
A. 1
B. 2√3
C. 2
D. √3
Answer:
Median of an equilateral triangle is also a perpendicular to the base.
Let the side be a
⇒ a = 2
∴ Option (c) is correct.
Question 22.
The perimeter of an equilateral triangle is 6. The length of the altitude of the triangle is …
A. 4
B. 2√3
C. 2
D. √3
Answer:
Let the length of altitude be x and side be a.
⇒ 3a = 6
⇒ a = 2
⇒ x = √3
∴ Option (d) is correct.
Question 23.
In ΔABC, m∠A = 90. is a median. If AD = 6, AB = 10, then AC =
A. 8
B. 7.5
C. 16
D. 2√11
Answer:
In Δ ABC, using Apolloneous Theorem, we know that,
For AD to be the median, we have,
AB2 + AC2 = 2(AD2 + BD2)
⇒ BC2 = 2 (36 + BD2)
⇒ 4BD2 = 72 + 2BD2
⇒ 2BD2 = 72
⇒ BD2 = 36
⇒ BD = 6
So, BC = 12 as BC = 2BD
By Pythagoras Theorem in ΔABC,
AC2 = BC2 – AB2
⇒ AC2 = 144 – 100
⇒ AC = 2√11
∴ Option (d) is correct.
Question 24.
In Δ PQR, m∠Q = 90 and PQ = QR. ,
. If QM = 2, PQ =
A. 4
B. 2√2
C. 8
D. 2
Answer:
Since ∠Q = 90 and PQ = QR , therefore ∠QPR = ∠QRP = θ
As sum of all of triangle = 180°
⇒ 2θ + 90 = 180
⇒ θ = 45°
⇒ PQ = 2√2
∴ Option (b) is correct.
Question 25.
In Δ ABC, m∠A = 90, is an altitude. So AB2 = ……
A. BD.BC
B. BD.DC
C.
D. BC.DC
Answer:
By Pythagoras Theorem in ΔABD,
AB2 = AD2 + BD2 ...... (1)
As we know that,
If an altitude is drawn to hypotenuse of a right angled triangle, then the length of altitude is the geometric mean of lengths of segments of hypotenuse formed by the altitude.
AD2 = BD.CD ...... (2)
From equation (1) and (2)
AB2 = BD.CD + BD2
⇒ AB2 = BD(CD + BD)
⇒ AB2 = BD.BC
∴ Option (a) is correct.
Question 26.
In Δ ABC, m∠A = 90, is an altitude. Therefore BD.DC = ……
A. AB2
B. BC2
C. AC2
D. AD2
Answer:
As we know that,
If an altitude is drawn to hypotenuse of a right angled triangle, then the length of altitude is the geometric mean of lengths of segments of hypotenuse formed by the altitude.
⇒ AD2 = BD.DC
∴ Option (d) is correct.