##### Class 10^{th} Mathematics Gujarat Board Solution

**Exercise 12.1**- Draw bar ab of length 7.4 cm and divide it in the ratio 5 : 7. Construct the…
- Divide a line-segment into three parts in the ratio 2 : 3 : 4 in the same…
- Construct a triangle with sides 4 cm, 5 cm, 7 cm and then construct a triangle…
- Draw ΔPQR with m∠P = 60, m∠Q = 45 and PQ = 6 cm. Then construct ΔPBC whose…
- Draw ΔABC having m angle ABC = 90, BC = 4 cm and AC = 5 cm. Then construct…
- Draw bar pq of length 6.5 cm and divide it in the ratio 4 : 7. Measure the two…

**Exercise 12**- Draw a circle of radius 5 cm. From a point 8 cm away from the centre, construct…
- Draw ⨀ (O, 4). Construct a pair of tangents from A where OA = 10 units.…
- Draw a circle with the help of a circular bangle. Construct two tangents to…
- Draw ⨀(O, r). bar pq is a diameter of 0(0, r) . Points A and B are on the bar…
- Draw AB such that AB = 10 cm. Draw ⨀(A, 3) and ⨀(B, 4). Construct tangents to…
- ⨀ (P, 4) is given. Draw a pair of tangents through A which is in the exterior…

**Exercise 12.1**

- Draw bar ab of length 7.4 cm and divide it in the ratio 5 : 7. Construct the…
- Divide a line-segment into three parts in the ratio 2 : 3 : 4 in the same…
- Construct a triangle with sides 4 cm, 5 cm, 7 cm and then construct a triangle…
- Draw ΔPQR with m∠P = 60, m∠Q = 45 and PQ = 6 cm. Then construct ΔPBC whose…
- Draw ΔABC having m angle ABC = 90, BC = 4 cm and AC = 5 cm. Then construct…
- Draw bar pq of length 6.5 cm and divide it in the ratio 4 : 7. Measure the two…

**Exercise 12**

- Draw a circle of radius 5 cm. From a point 8 cm away from the centre, construct…
- Draw ⨀ (O, 4). Construct a pair of tangents from A where OA = 10 units.…
- Draw a circle with the help of a circular bangle. Construct two tangents to…
- Draw ⨀(O, r). bar pq is a diameter of 0(0, r) . Points A and B are on the bar…
- Draw AB such that AB = 10 cm. Draw ⨀(A, 3) and ⨀(B, 4). Construct tangents to…
- ⨀ (P, 4) is given. Draw a pair of tangents through A which is in the exterior…

###### Exercise 12.1

**Question 1.**Construct the following with the help of straight-edge and compass only:

Draw of length 7.4 cm and divide it in the ratio 5 : 7.

**Answer:**Steps of Construction:

1. Draw a line AB of length 7.4cm

2.Draw a ray AX making an acute angle with AB.

3. We have to divide line in ratio of 5:7. So we make 5+7 =12 equal arcs on ray AX.

4. Taking A as center, cut an arc on ray AX. Mark the point as A1.

5. Now, with the same radius cut an arc on AX with A1 as center.

6. Mark till A12.

7. Join A12 with B.

8. As we have to divide in 5:7, we draw a line from A5 parallel to A11B.

9. The point where it cuts AB is P.

**Question 2.**Construct the following with the help of straight-edge and compass only:

Divide a line-segment into three parts in the ratio 2 : 3 : 4 in the same order.

**Answer:**Steps of Construction:

1. Draw a line AB of any length.

2.Draw a ray AX making an acute angle with AB.

3. We have to divide line in ratio of 2:3:4. So we make 2+3+4 =9 equal arcs on ray AX.

4. Taking A as center, cut an arc on ray AX. Mark the point as A1.

5. Now, with the same radius cut an arc on AX with A1 as center.

6. Mark till A9.

7. Join A9 with B.

8. As we have to divide in 2:3:4, we draw a line from A2 and A5 parallel to A9B.

9. The point where it cuts AB is P and Q

Here, AP : PQ : QB = 2 : 3 : 5

**Question 3.**Construct the following with the help of straight-edge and compass only:

Construct a triangle with sides 4 cm, 5 cm, 7 cm and then construct a triangle similar to it whose sides have lengths in the ratio 2 : 3 to the lengths of the corresponding sides of the first triangle.

**Answer:**Steps of Construction:

1.Draw triangle ABC of given dimensions.

2. Draw a ray AX making an acute angle with AB.

3. We have to construct triangle having ratio of 2:3. So we make 3 equal arcs on ray AX.

4. Taking A as center, cut an arc on ray AX. Mark the point as A1.

5. Now, with the same radius cut an arc on AX with A1 as center.

6. Mark till A3.

7. Join A3 with B.

8. As we have to divide in 2:3, we draw a line from A2 parallel to A3B.

9. The point where it cuts AB is P.

10. From P, draw a line parallel to BC.

11. APQ is the required triangle.

**Question 4.**Construct the following with the help of straight-edge and compass only:

Draw ΔPQR with m∠P = 60, m∠Q = 45 and PQ = 6 cm. Then construct ΔPBC whose sides have lengths times the lengths of the corresponding sides of ΔPQR.

**Answer:**Steps of Construction:

1. Draw triangle PQR of given dimensions.

3. Draw a ray PX making an acute angle with PQ.

3. We have to construct triangle having ratio of 5:3. So we make 5 equal arcs on ray PX.

4. Taking P as center, cut an arc on ray PX. Mark the point as P1.

5. Now, with the same radius cut an arc on PX with P1 as center.

6. Mark till P5.

7. Join P3 with Q.Extend PQ.

8. As we have to divide in 5:3, we draw a line from P5 parallel to P3Q.

9. The point where it cuts extended PQ is S.

10. From S, draw a line parallel to QR.

11. PST is the required triangle.

**Question 5.**Construct the following with the help of straight-edge and compass only:

Draw ΔABC having mABC = 90, BC = 4 cm and AC = 5 cm. Then construct ΔBXY, where scale factor is .

**Answer:**Steps of Construction:

1. Draw triangle ABC of given dimensions.

2. Draw a ray BX making an acute angle with BC .

3. We have to construct triangle having ratio of 4:3. So we make 4 equal arcs on ray BX.

4. Taking B as center, cut an arc on ray BX. Mark the point as B1.

5. Now, with the same radius cut an arc on BX with B1 as center.

6. Mark till B4.

7. Join B3 with C. Extend BC.

8. As we have to divide in 4:3, we draw a line from B4 parallel to B3C.

9. The point where it cuts extended BC is P.

10. From P, draw a line parallel to AC.

11. BPQ is the required triangle.

**Question 6.**Construct the following with the help of straight-edge and compass only:

Draw of length 6.5 cm and divide it in the ratio 4 : 7. Measure the two parts.

**Answer:**Steps of Construction:

1. Draw a line PQ of length 6.5cm

2.Draw a ray PX making an acute angle with PQ.

3. We have to divide line in ratio of 4:7. So we make 4+7 =11 equal arcs on ray PX.

4. Taking P as center, cut an arc on ray PX. Mark the point as P1.

5. Now, with the same radius cut an arc on PX with P1 as center.

6. Mark till P11.

7. Join P11 with Q.

8. As we have to divide in 4:7, we draw a line from P4 parallel to P11Q.

9. The point where it cuts PQ is A.

**Question 1.**

Construct the following with the help of straight-edge and compass only:

Draw of length 7.4 cm and divide it in the ratio 5 : 7.

**Answer:**

Steps of Construction:

1. Draw a line AB of length 7.4cm

2.Draw a ray AX making an acute angle with AB.

3. We have to divide line in ratio of 5:7. So we make 5+7 =12 equal arcs on ray AX.

4. Taking A as center, cut an arc on ray AX. Mark the point as A1.

5. Now, with the same radius cut an arc on AX with A1 as center.

6. Mark till A12.

7. Join A12 with B.

8. As we have to divide in 5:7, we draw a line from A5 parallel to A11B.

9. The point where it cuts AB is P.

**Question 2.**

Construct the following with the help of straight-edge and compass only:

Divide a line-segment into three parts in the ratio 2 : 3 : 4 in the same order.

**Answer:**

Steps of Construction:

1. Draw a line AB of any length.

2.Draw a ray AX making an acute angle with AB.

3. We have to divide line in ratio of 2:3:4. So we make 2+3+4 =9 equal arcs on ray AX.

4. Taking A as center, cut an arc on ray AX. Mark the point as A1.

5. Now, with the same radius cut an arc on AX with A1 as center.

6. Mark till A9.

7. Join A9 with B.

8. As we have to divide in 2:3:4, we draw a line from A2 and A5 parallel to A9B.

9. The point where it cuts AB is P and Q

Here, AP : PQ : QB = 2 : 3 : 5

**Question 3.**

Construct the following with the help of straight-edge and compass only:

Construct a triangle with sides 4 cm, 5 cm, 7 cm and then construct a triangle similar to it whose sides have lengths in the ratio 2 : 3 to the lengths of the corresponding sides of the first triangle.

**Answer:**

Steps of Construction:

1.Draw triangle ABC of given dimensions.

2. Draw a ray AX making an acute angle with AB.

3. We have to construct triangle having ratio of 2:3. So we make 3 equal arcs on ray AX.

4. Taking A as center, cut an arc on ray AX. Mark the point as A1.

5. Now, with the same radius cut an arc on AX with A1 as center.

6. Mark till A3.

7. Join A3 with B.

8. As we have to divide in 2:3, we draw a line from A2 parallel to A3B.

9. The point where it cuts AB is P.

10. From P, draw a line parallel to BC.

11. APQ is the required triangle.

**Question 4.**

Construct the following with the help of straight-edge and compass only:

Draw ΔPQR with m∠P = 60, m∠Q = 45 and PQ = 6 cm. Then construct ΔPBC whose sides have lengths times the lengths of the corresponding sides of ΔPQR.

**Answer:**

Steps of Construction:

1. Draw triangle PQR of given dimensions.

3. Draw a ray PX making an acute angle with PQ.

3. We have to construct triangle having ratio of 5:3. So we make 5 equal arcs on ray PX.

4. Taking P as center, cut an arc on ray PX. Mark the point as P1.

5. Now, with the same radius cut an arc on PX with P1 as center.

6. Mark till P5.

7. Join P3 with Q.Extend PQ.

8. As we have to divide in 5:3, we draw a line from P5 parallel to P3Q.

9. The point where it cuts extended PQ is S.

10. From S, draw a line parallel to QR.

11. PST is the required triangle.

**Question 5.**

Construct the following with the help of straight-edge and compass only:

Draw ΔABC having mABC = 90, BC = 4 cm and AC = 5 cm. Then construct ΔBXY, where scale factor is .

**Answer:**

Steps of Construction:

1. Draw triangle ABC of given dimensions.

2. Draw a ray BX making an acute angle with BC .

3. We have to construct triangle having ratio of 4:3. So we make 4 equal arcs on ray BX.

4. Taking B as center, cut an arc on ray BX. Mark the point as B1.

5. Now, with the same radius cut an arc on BX with B1 as center.

6. Mark till B4.

7. Join B3 with C. Extend BC.

8. As we have to divide in 4:3, we draw a line from B4 parallel to B3C.

9. The point where it cuts extended BC is P.

10. From P, draw a line parallel to AC.

11. BPQ is the required triangle.

**Question 6.**

Construct the following with the help of straight-edge and compass only:

Draw of length 6.5 cm and divide it in the ratio 4 : 7. Measure the two parts.

**Answer:**

Steps of Construction:

1. Draw a line PQ of length 6.5cm

2.Draw a ray PX making an acute angle with PQ.

3. We have to divide line in ratio of 4:7. So we make 4+7 =11 equal arcs on ray PX.

4. Taking P as center, cut an arc on ray PX. Mark the point as P1.

5. Now, with the same radius cut an arc on PX with P1 as center.

6. Mark till P11.

7. Join P11 with Q.

8. As we have to divide in 4:7, we draw a line from P4 parallel to P11Q.

9. The point where it cuts PQ is A.

###### Exercise 12

**Question 1.**Draw a circle of radius 5 cm. From a point 8 cm away from the centre, construct two tangents to the circle from this point. Measure them.

**Answer:**Steps of Construction:

1. Draw a circle of radius 5cm and center O.

2. Draw a line OP such that OP= 8cm

3. Draw a perpendicular bisector of OP which cuts OP at M.

4. Taking M as center and radius OM, draw a circle.

5. The two circles intersect at A and B.

6. Join AP and BP. AB and BP are the required tangents.

**Question 2.**Draw ⨀ (O, 4). Construct a pair of tangents from A where OA = 10 units.

**Answer:**Steps of Construction:

1. Draw a circle of radius 4cm and center O.

2. Draw a line OA such that OA= 10cm

3. Draw a perpendicular bisector of OA which cuts OA at M.

4. Taking M as center and radius OM, draw a circle.

5. The two circles intersect at P and Q.

6. Join AP and AQ. AP and AQ are the required tangents.

**Question 3.**Draw a circle with the help of a circular bangle. Construct two tangents to this circle through a point in the exterior of the circle.

**Answer:****Steps of construction:**

**1.** Draw a circle(Say C_{1}) with the help of a bangle, and take any three points A, B and C on its circumference

**2.** Join AB and BC and draw the perpendicular bisector of AB and BC which intersect each other at O.

**3.** O is the center of circle, now take any point P outside the circle and join OP.

**4.** Draw the perpendicular bisector of OP, which intersect OP at O'.

**5.** Taking O' as center and O'O = O'P as radius, draw a circle C_{2}, which intersects the previous circle at Q and R.

**6.** Join PQ and PR, which are required tangents.

**Question 4.**Draw ⨀(O, r). is a diameter of 0(0, r) . Points A and B are on the such that A—P—Q and P—Q—B. Construct tangents through A and B to ⨀(O, r) .

**Answer:**Steps of Construction:

1. Draw a circle of any radius(r ) and center O. Draw a diameter PQ of this circle.

2. Extend PQ both the sides as APQB is a straight line.

3. Draw a perpendicular bisector of OA and OB which cuts OA and OB.

4. Taking OA and OB as diameters, draw two circles.

5. The three circles intersect at K,L and M,N.

6. Join AK and AL. Join BM and BN.

7. AK,AL,BM and BN are the required tangents.

**Question 5.**Draw AB such that AB = 10 cm. Draw ⨀(A, 3) and ⨀(B, 4). Construct tangents to each circle through the centre of the other circle.

**Answer:**Steps of Construction:

1. Draw a circle of any radius 3 and center A.

2. With A as center, draw a circle of radius 3 cm. With B as center, draw a circle of radius 4cm.

3. Draw a perpendicular bisector of AB which cuts AB at M.

4. With M as center and AM as radius, draw a circle.

5. The three circles intersect at K,L and M,N.

6. Join AM and AN. Join BK and BL.

7. AM,AN,BK and BL are the required tangents.

**Question 6.**⨀ (P, 4) is given. Draw a pair of tangents through A which is in the exterior of ⨀(P, 4) such that measure of an angle between the tangents is 60° .

**Answer:**Angle between the tangents =60°

We first find the angle between the radius.

Angle between radius + Angle between tangents =180°

⇒ Angle between radius = 180-60=120°

Steps of Construction:

1. Draw a circle of radius 4cm and center P.

2. Draw a radius on the circle. Draw another radius making 120° with this radius.

3. From the points on the circumference on the circle, draw perpendiculars to these radius.

4. They intersect each other at one point.

**Question 1.**

Draw a circle of radius 5 cm. From a point 8 cm away from the centre, construct two tangents to the circle from this point. Measure them.

**Answer:**

Steps of Construction:

1. Draw a circle of radius 5cm and center O.

2. Draw a line OP such that OP= 8cm

3. Draw a perpendicular bisector of OP which cuts OP at M.

4. Taking M as center and radius OM, draw a circle.

5. The two circles intersect at A and B.

6. Join AP and BP. AB and BP are the required tangents.

**Question 2.**

Draw ⨀ (O, 4). Construct a pair of tangents from A where OA = 10 units.

**Answer:**

Steps of Construction:

1. Draw a circle of radius 4cm and center O.

2. Draw a line OA such that OA= 10cm

3. Draw a perpendicular bisector of OA which cuts OA at M.

4. Taking M as center and radius OM, draw a circle.

5. The two circles intersect at P and Q.

6. Join AP and AQ. AP and AQ are the required tangents.

**Question 3.**

Draw a circle with the help of a circular bangle. Construct two tangents to this circle through a point in the exterior of the circle.

**Answer:**

**Steps of construction:**

**1.** Draw a circle(Say C_{1}) with the help of a bangle, and take any three points A, B and C on its circumference

**2.** Join AB and BC and draw the perpendicular bisector of AB and BC which intersect each other at O.

**3.** O is the center of circle, now take any point P outside the circle and join OP.

**4.** Draw the perpendicular bisector of OP, which intersect OP at O'.

**5.** Taking O' as center and O'O = O'P as radius, draw a circle C_{2}, which intersects the previous circle at Q and R.

**6.** Join PQ and PR, which are required tangents.

**Question 4.**

Draw ⨀(O, r). is a diameter of 0(0, r) . Points A and B are on the such that A—P—Q and P—Q—B. Construct tangents through A and B to ⨀(O, r) .

**Answer:**

Steps of Construction:

1. Draw a circle of any radius(r ) and center O. Draw a diameter PQ of this circle.

2. Extend PQ both the sides as APQB is a straight line.

3. Draw a perpendicular bisector of OA and OB which cuts OA and OB.

4. Taking OA and OB as diameters, draw two circles.

5. The three circles intersect at K,L and M,N.

6. Join AK and AL. Join BM and BN.

7. AK,AL,BM and BN are the required tangents.

**Question 5.**

Draw AB such that AB = 10 cm. Draw ⨀(A, 3) and ⨀(B, 4). Construct tangents to each circle through the centre of the other circle.

**Answer:**

Steps of Construction:

1. Draw a circle of any radius 3 and center A.

2. With A as center, draw a circle of radius 3 cm. With B as center, draw a circle of radius 4cm.

3. Draw a perpendicular bisector of AB which cuts AB at M.

4. With M as center and AM as radius, draw a circle.

5. The three circles intersect at K,L and M,N.

6. Join AM and AN. Join BK and BL.

7. AM,AN,BK and BL are the required tangents.

**Question 6.**

⨀ (P, 4) is given. Draw a pair of tangents through A which is in the exterior of ⨀(P, 4) such that measure of an angle between the tangents is 60° .

**Answer:**

Angle between the tangents =60°

We first find the angle between the radius.

Angle between radius + Angle between tangents =180°

⇒ Angle between radius = 180-60=120°

Steps of Construction:

1. Draw a circle of radius 4cm and center P.

2. Draw a radius on the circle. Draw another radius making 120° with this radius.

3. From the points on the circumference on the circle, draw perpendiculars to these radius.

4. They intersect each other at one point.