##### Class 10^{th} Mathematics CBSE Solution

**Exercise 11.1**- Draw a line segment of length 7.6 cm and divide it in the ratio 5: 8. Measure…
- Construct a triangle of sides 4 cm, 5 cm and 6 cm and then a triangle similar…
- Construct a triangle with sides 5 cm, 6 cm and 7 cm and then another triangle…
- Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then…
- Draw a triangle ABC with side BC = 6 cm, AB = 5 cm and ABC = 60. Then construct…
- Draw a triangle ABC with side BC = 7 cm, B = 45, A = 105. Then, construct a…
- Draw a right triangle in which the sides (other than hypotenuse) are of lengths…

**Exercise 11.2**- Draw a circle of radius 6 cm. From a point 10 cm away from its centre,…
- Construct a tangent to a circle of radius 4 cm from a point on the concentric…
- Draw a circle of radius 3 cm. Take two points P and Q on one of its extended…
- Draw a pair of tangents to a circle of radius 5 cm which are inclined to each…
- Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of…
- Draw a circle with the help of a bangle. Take a point outside the circle.…

**Exercise 11.1**

- Draw a line segment of length 7.6 cm and divide it in the ratio 5: 8. Measure…
- Construct a triangle of sides 4 cm, 5 cm and 6 cm and then a triangle similar…
- Construct a triangle with sides 5 cm, 6 cm and 7 cm and then another triangle…
- Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then…
- Draw a triangle ABC with side BC = 6 cm, AB = 5 cm and ABC = 60. Then construct…
- Draw a triangle ABC with side BC = 7 cm, B = 45, A = 105. Then, construct a…
- Draw a right triangle in which the sides (other than hypotenuse) are of lengths…

**Exercise 11.2**

- Draw a circle of radius 6 cm. From a point 10 cm away from its centre,…
- Construct a tangent to a circle of radius 4 cm from a point on the concentric…
- Draw a circle of radius 3 cm. Take two points P and Q on one of its extended…
- Draw a pair of tangents to a circle of radius 5 cm which are inclined to each…
- Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of…
- Draw a circle with the help of a bangle. Take a point outside the circle.…

###### Exercise 11.1

**Question 1.**Draw a line segment of length 7.6 cm and divide it in the ratio 5: 8. Measure the two parts.

**Answer:**__Step 1:__ Draw line segment AB = 7.6 cm

__Step 2:__ Draw a ray AC making an acute angle with line segment AB .

__Step 3:__ Along AC, divide (5+8) 13 equals points

A_{1},A_{2},A_{3},……………….,A_{13} on AX such that

AA_{1}=AA_{2}=A_{2}A_{3} and so on .

__Step 4:__ Join BA_{13}

__Step 5:__ Through the point A_{5}, draw a line parallel to BA_{13} (by making an angle equal to ∠AA_{13}B) at A_{5} intersecting AB at point D.

C is the point dividing line segment AB of 7.6 cm in the required ratio of 5:8.

The lengths of AD and DB can be measured. It comes out to 2.9 cm and 4.7 cm respectively.

Justification

The construction can be justified by proving that

By construction, we have A_{5}D∥A_{13}B. By applying Basic proportionality theorem for

From the figure, it can be observed that AA_{5} and A_{5}A_{13} contain 5 and 8 equal divisions of line segments respectively.

On comparing equations (1) and (2), we obtain

This justifies the construction.

**Question 2.**Construct a triangle of sides 4 cm, 5 cm and 6 cm and then a triangle similar to it whose sides are of the corresponding sides of the first triangle.

**Answer:**Step 1: Draw a line segment AB of 5 cm.

Step 2: Taking A and B as Center, draw arcs of 6 cm and 5 cm radius respectively. Let these arcs intersect each other at point C. △ABC is the required triangle having length of sides as 5 cm , 6 cm , 7 cm respectively.

Step 3: Draw a ray AX making acute angle with line AB on the opposite side of vertex C.

Step 4: Locate 7 points, A_{1},A_{2},A_{3},A_{4,} A_{5},A_{6},A_{7}( as 7 is greater between 5 and 7), on line AX such that AA_{1}=A1 A2=A2 A3=A3 A4=A4 A5=A5 A6=A6 A7

Step 5: Join BA_{5} and draw a line through A_{7} parallel to BA_{5} to intersect extended line segment AB at point B′.

Step 6: Draw a line through B′ parallel to BC intersecting the extended line segment AC at C′.△AB’C’ is the required triangle.

Justification

∆AB' C'~∆ABCby AAA Similarity Condition

And

**Question 3.**Construct a triangle with sides 5 cm, 6 cm and 7 cm and then another triangle whose sides are of the corresponding sides of the first triangle.

**Answer:** 1. Now in order to make a triangle, draw a line segment AB = 5 cm.From point A, draw an arc at 6 cm. Draw an arc at 7 cm from point B intersecting the previous arc.

2. Join the point of intersection from A and B.

Hence, this gives the required triangle ABC

3. Dividing the base, draw a ray AX which is at an acute angle from AB

4. Plot seven points on AX such that:

AA_{1} = A_{1}A_{2} = A_{2}A_{3} = A_{3}A_{4} = A_{4}A_{5} = A_{5}A_{6} = A_{6}A_{7}.

5. Join A_{5} to B.

6. Draw a line from point A_{7}that is parallel to A_{5}B and joins AB’ (AB extended to AB’).

7. Draw a line B’C’ || BC.

Hence, triangle AB’C’ is the required triangle.

**Question 4.**Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then another triangle whose sides are times the corresponding sides of the isosceles triangle.

**Answer:**△A′BC′ whose sides of the corresponding sides of can be drawn as follows.

Step 1 Draw a △ABC with side BC = 6 cm, AB = 5 cm and ∠ABC = .

Step 2 Draw a ray BX making an acute angle with BC on the opposite side of vertex A.

Step 3 Locate 4 points (as 4 is greater in 3 and 4), B1,B2,B3,B4 on line segment BX.

Step 4 Join B4C and draw a line through B3 , parallel to B4C intersecting BC at C′

Step 5 Draw a line through C' parallel to AC intersecting AB at A'.△A′BC′ is the required triangle.

Justification

∠A=60° (Common)

∠C=∠C'

By AA similarity condition

Also,

AB’=AB ,B’C’=BC and AC’=AC

**Question 5.**Draw a triangle ABC with side BC = 6 cm, AB = 5 cm and ∠ ABC = 60°. Then construct a triangle whose sides are of the corresponding sides of the triangle ABC.

**Answer:**Given in Î” ABC,

- Length of side BC = 6 cm.
- Length of side AB = 5 cm.
- ∠ ABC = 60°.

__Steps of Construction__:

- Draw a line segment BC of length 6 cm.

2. With B as center, draw a line which makes an angle of 60° with BC.

*Construction of 60° angle at B*:

a. With B as centre and with some convenient radius draw an arc which cuts the line BC at D.

b. With D as radius and with same radius (in step a), draw another arc which cuts the previous arc at E.

c. Join BE. The line BE makes an angle 60° with BC.

- Again with B as centre and with radius of 5 cm, draw an arc which intersects the line BE at point A.

4. Join AC. This is the required triangle.

- Now, from B, draw a ray BX which makes an acute angle on the opposite side of the vertex A.

6. With B as center, mark four points B_{1}, B_{2}, B_{3} and B_{4} on BX such that they are equidistant. i.e. BB_{1} = B_{1}B_{2} = B_{2}B_{3} = B_{3}B_{4}.

7. Join B_{4}C and then draw a line from B_{3} parallel to B_{4}C which meets the line BC at P.

8. From P, draw a line parallel to AC and meets the line AB at Q. Thus Î”BPQ is the required triangle.

**Question 6.**Draw a triangle ABC with side BC = 7 cm, ∠ B = 45°, ∠ A = 105°. Then, construct a triangle whose sides are times the corresponding sides of Î” ABC.

**Answer:**Steps of construction:

1. Draw a line segment BC = 7 cm.

2. Draw ∠ABC = 45° and ∠ACB = 30° i.e. ∠BAC = 105°.

We obtain Î”ABC.

3. Draw a ray BX making an acute angle with BC. Mark four points B_{1}, B_{2}, B_{3},B_{4} at equal distances.

4. Through B_{3} draw B_{3}C and through B_{4} draw B_{4}C_{1} parallel to B_{3}C.

Then draw A_{1}C_{1} parallel to AC.

∴ A_{1}BC_{1} is the required triangle.

**Question 7.**Draw a right triangle in which the sides (other than hypotenuse) are of lengths 4 cm and 3 cm. Then construct another triangle whose sides are times the corresponding sides of the given triangle.

**Answer:**Steps of construction:

- Now in order to make a triangle, draw a line segment AB = 3 cm.

2. Make a right angle at point A and draw AC = 4 cm from this point.

3. Join points A and B to get the right triangle ABC.

4. Now, Dividing the base, draw a ray AX such at it forms an acute angle from AB.

5. Then, plot 5 points on AX such that:

AG = GH = HI = IJ = JK.

6. Join I to point line AB and Draw a line from K which is parallel to IB such that it meets AB at point M.

7. Draw MN || CB.

This is the required construction, thus forming AMN which have all the sides 5/3 times the sides of ABC

Triangle AMN is the required triangle.

**Question 1.**

Draw a line segment of length 7.6 cm and divide it in the ratio 5: 8. Measure the two parts.

**Answer:**

__Step 1:__ Draw line segment AB = 7.6 cm

__Step 2:__ Draw a ray AC making an acute angle with line segment AB .

__Step 3:__ Along AC, divide (5+8) 13 equals points

A_{1},A_{2},A_{3},……………….,A_{13} on AX such that

AA_{1}=AA_{2}=A_{2}A_{3} and so on .

__Step 4:__ Join BA_{13}

__Step 5:__ Through the point A_{5}, draw a line parallel to BA_{13} (by making an angle equal to ∠AA_{13}B) at A_{5} intersecting AB at point D.

C is the point dividing line segment AB of 7.6 cm in the required ratio of 5:8.

The lengths of AD and DB can be measured. It comes out to 2.9 cm and 4.7 cm respectively.

Justification

The construction can be justified by proving that

By construction, we have A_{5}D∥A_{13}B. By applying Basic proportionality theorem for

From the figure, it can be observed that AA_{5} and A_{5}A_{13} contain 5 and 8 equal divisions of line segments respectively.

On comparing equations (1) and (2), we obtain

This justifies the construction.

**Question 2.**

Construct a triangle of sides 4 cm, 5 cm and 6 cm and then a triangle similar to it whose sides are of the corresponding sides of the first triangle.

**Answer:**

Step 1: Draw a line segment AB of 5 cm.

Step 2: Taking A and B as Center, draw arcs of 6 cm and 5 cm radius respectively. Let these arcs intersect each other at point C. △ABC is the required triangle having length of sides as 5 cm , 6 cm , 7 cm respectively.

Step 3: Draw a ray AX making acute angle with line AB on the opposite side of vertex C.

Step 4: Locate 7 points, A_{1},A_{2},A_{3},A_{4,} A_{5},A_{6},A_{7}( as 7 is greater between 5 and 7), on line AX such that AA_{1}=A1 A2=A2 A3=A3 A4=A4 A5=A5 A6=A6 A7

Step 5: Join BA_{5} and draw a line through A_{7} parallel to BA_{5} to intersect extended line segment AB at point B′.

Step 6: Draw a line through B′ parallel to BC intersecting the extended line segment AC at C′.△AB’C’ is the required triangle.

Justification

∆AB' C'~∆ABCby AAA Similarity Condition

And

**Question 3.**

Construct a triangle with sides 5 cm, 6 cm and 7 cm and then another triangle whose sides are of the corresponding sides of the first triangle.

**Answer:**1. Now in order to make a triangle, draw a line segment AB = 5 cm.From point A, draw an arc at 6 cm. Draw an arc at 7 cm from point B intersecting the previous arc.

2. Join the point of intersection from A and B.

Hence, this gives the required triangle ABC

3. Dividing the base, draw a ray AX which is at an acute angle from AB

4. Plot seven points on AX such that:

AA_{1} = A_{1}A_{2} = A_{2}A_{3} = A_{3}A_{4} = A_{4}A_{5} = A_{5}A_{6} = A_{6}A_{7}.

5. Join A_{5} to B.

6. Draw a line from point A_{7}that is parallel to A_{5}B and joins AB’ (AB extended to AB’).

7. Draw a line B’C’ || BC.

Hence, triangle AB’C’ is the required triangle.

**Question 4.**

Construct an isosceles triangle whose base is 8 cm and altitude 4 cm and then another triangle whose sides are times the corresponding sides of the isosceles triangle.

**Answer:**

△A′BC′ whose sides of the corresponding sides of can be drawn as follows.

Step 1 Draw a △ABC with side BC = 6 cm, AB = 5 cm and ∠ABC = .

Step 2 Draw a ray BX making an acute angle with BC on the opposite side of vertex A.

Step 3 Locate 4 points (as 4 is greater in 3 and 4), B1,B2,B3,B4 on line segment BX.

Step 4 Join B4C and draw a line through B3 , parallel to B4C intersecting BC at C′

Step 5 Draw a line through C' parallel to AC intersecting AB at A'.△A′BC′ is the required triangle.

Justification

∠A=60° (Common)

∠C=∠C'

By AA similarity condition

Also,

AB’=AB ,B’C’=BC and AC’=AC

**Question 5.**

Draw a triangle ABC with side BC = 6 cm, AB = 5 cm and ∠ ABC = 60°. Then construct a triangle whose sides are of the corresponding sides of the triangle ABC.

**Answer:**

Given in Î” ABC,

- Length of side BC = 6 cm.
- Length of side AB = 5 cm.
- ∠ ABC = 60°.

__Steps of Construction__:

- Draw a line segment BC of length 6 cm.

2. With B as center, draw a line which makes an angle of 60° with BC.

*Construction of 60° angle at B*:

a. With B as centre and with some convenient radius draw an arc which cuts the line BC at D.

b. With D as radius and with same radius (in step a), draw another arc which cuts the previous arc at E.

c. Join BE. The line BE makes an angle 60° with BC.

- Again with B as centre and with radius of 5 cm, draw an arc which intersects the line BE at point A.

4. Join AC. This is the required triangle.

- Now, from B, draw a ray BX which makes an acute angle on the opposite side of the vertex A.

6. With B as center, mark four points B

_{1}, B

_{2}, B

_{3}and B

_{4}on BX such that they are equidistant. i.e. BB

_{1}= B

_{1}B

_{2}= B

_{2}B

_{3}= B

_{3}B

_{4}.

7. Join B

_{4}C and then draw a line from B

_{3}parallel to B

_{4}C which meets the line BC at P.

8. From P, draw a line parallel to AC and meets the line AB at Q. Thus Î”BPQ is the required triangle.

**Question 6.**

Draw a triangle ABC with side BC = 7 cm, ∠ B = 45°, ∠ A = 105°. Then, construct a triangle whose sides are times the corresponding sides of Î” ABC.

**Answer:**

Steps of construction:

1. Draw a line segment BC = 7 cm.

2. Draw ∠ABC = 45° and ∠ACB = 30° i.e. ∠BAC = 105°.

We obtain Î”ABC.

3. Draw a ray BX making an acute angle with BC. Mark four points B_{1}, B_{2}, B_{3},B_{4} at equal distances.

4. Through B_{3} draw B_{3}C and through B_{4} draw B_{4}C_{1} parallel to B_{3}C.

Then draw A_{1}C_{1} parallel to AC.

∴ A_{1}BC_{1} is the required triangle.

**Question 7.**

Draw a right triangle in which the sides (other than hypotenuse) are of lengths 4 cm and 3 cm. Then construct another triangle whose sides are times the corresponding sides of the given triangle.

**Answer:**

Steps of construction:

- Now in order to make a triangle, draw a line segment AB = 3 cm.

2. Make a right angle at point A and draw AC = 4 cm from this point.

3. Join points A and B to get the right triangle ABC.

4. Now, Dividing the base, draw a ray AX such at it forms an acute angle from AB.

5. Then, plot 5 points on AX such that:

AG = GH = HI = IJ = JK.

6. Join I to point line AB and Draw a line from K which is parallel to IB such that it meets AB at point M.

7. Draw MN || CB.

This is the required construction, thus forming AMN which have all the sides 5/3 times the sides of ABC

Triangle AMN is the required triangle.

###### Exercise 11.2

**Question 1.**Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths.

**Answer:**Step1: Draw circle of radius 6cm with center A, mark point C at 10 cm from center

Step 2: find perpendicular bisector of AC

Step3: Take this point as center and draw a circle through A and C

Step4:Mark the point where this circle intersects our circle and draw tangents through C

Length of tangents = 8cm

AE is perpendicular to CE (tangent and radius relation)

In Î”ACE

AC becomes hypotenuse

AC^{2} = CE^{2} + AE^{2}

10^{2} = CE^{2} + 6^{2}

CE^{2} = 100-36

CE^{2} = 64

CE = 8cm

**Question 2.**Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by actual calculation.

**Answer:**Steps of construction:

i. Draw two concentric circles with radii 4 cm and 6 cm respectively.

ii. Now, draw the radius OP of the larger circle.

iii. Construct a perpendicular bisector of OP intersecting OP at point O’.

iv. Considering O’P as radius, draw another circle.

v. From point P, draw tangents PQ and PR(can see in the figure)

**Justification: **By applying Pythagoras theorem, we have;

PQ^{2} = OP^{2} – OQ^{2}

= 6^{2} – 4^{2}

= 36 – 16 = 20

Or, PQ = 2√5 cm

**Question 3.**Draw a circle of radius 3 cm. Take two points P and Q on one of its extended diameter each at a distance of 7 cm from its centre. Draw tangents to the circle from these two points P and Q.

**Answer:**Steps of construction:

i. At first draw a circle with radius 3 cm.

ii. Now, extend the diameter to A and B on both the sides.

iii. Then, draw the perpendicular bisectors of OA and OB.

Such that:

Perpendicular bisector of OA intersects it at point O’.

And,

Perpendicular bisector of OB intersects it at point O”.

iv. Considering O’A as radius, construct another circle.

v. considering O”B as radius, construct the third circle.

vi. From point A, draw tangents AP and AQ.

vii. From point B, draw tangents BR and BS.

**Question 4.**Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60°.

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

1) Draw a circle of radius 5 cm, and draw a radius OA anywhere in the circle.

2) Taking OA as base, draw an angle AOB such that ∠AOB = 120°.

3) At A, Draw a line AX such that AX ⊥ OA.

4) At B, Draw a line BY such that BY ⊥ OB.

5) AX and BY intersect at P; AP and BP are required tangents.

**Justification:**

1) Clearly, AP and BP are tangents since tangent at a point on the circle is perpendicular to the radius through point of contact.

2) In Quadrilateral OAPB, we have

∠OAP + ∠APB + ∠OBP + ∠AOB = 360° [By Angle Sum Property]

⇒ ∠OAP + 90° + 90° + 120° = 360°

⇒ ∠OAP = 60°

**Question 5.**Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle.

**Answer:**Step1: Draw a line PQ=8cm. taking P and Q as a center draw circle of 3cm and 4cm.

Step2: Now bisect PQ. We get midpoint of PQ be T.

Now take T as a center , draw a circle of PT radius , this will intersect the circle at point A,B,C,D. Join PB,PD,AQ,QC.

**Question 1.**

Draw a circle of radius 6 cm. From a point 10 cm away from its centre, construct the pair of tangents to the circle and measure their lengths.

**Answer:**

Step1: Draw circle of radius 6cm with center A, mark point C at 10 cm from center

Step 2: find perpendicular bisector of AC

Step3: Take this point as center and draw a circle through A and C

Step4:Mark the point where this circle intersects our circle and draw tangents through C

Length of tangents = 8cm

AE is perpendicular to CE (tangent and radius relation)

In Î”ACE

AC becomes hypotenuse

AC^{2} = CE^{2} + AE^{2}

10^{2} = CE^{2} + 6^{2}

CE^{2} = 100-36

CE^{2} = 64

CE = 8cm

**Question 2.**

Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm and measure its length. Also verify the measurement by actual calculation.

**Answer:**

Steps of construction:

i. Draw two concentric circles with radii 4 cm and 6 cm respectively.

ii. Now, draw the radius OP of the larger circle.

iii. Construct a perpendicular bisector of OP intersecting OP at point O’.

iv. Considering O’P as radius, draw another circle.

v. From point P, draw tangents PQ and PR(can see in the figure)

**Justification: **By applying Pythagoras theorem, we have;

PQ^{2} = OP^{2} – OQ^{2}

= 6^{2} – 4^{2}

= 36 – 16 = 20

Or, PQ = 2√5 cm

**Question 3.**

Draw a circle of radius 3 cm. Take two points P and Q on one of its extended diameter each at a distance of 7 cm from its centre. Draw tangents to the circle from these two points P and Q.

**Answer:**

Steps of construction:

i. At first draw a circle with radius 3 cm.

ii. Now, extend the diameter to A and B on both the sides.

iii. Then, draw the perpendicular bisectors of OA and OB.

Such that:

Perpendicular bisector of OA intersects it at point O’.

And,

Perpendicular bisector of OB intersects it at point O”.

iv. Considering O’A as radius, construct another circle.

v. considering O”B as radius, construct the third circle.

vi. From point A, draw tangents AP and AQ.

vii. From point B, draw tangents BR and BS.

**Question 4.**

Draw a pair of tangents to a circle of radius 5 cm which are inclined to each other at an angle of 60°.

**Answer:**

**Steps of construction:**

1) Draw a circle of radius 5 cm, and draw a radius OA anywhere in the circle.

2) Taking OA as base, draw an angle AOB such that ∠AOB = 120°.

3) At A, Draw a line AX such that AX ⊥ OA.

4) At B, Draw a line BY such that BY ⊥ OB.

5) AX and BY intersect at P; AP and BP are required tangents.

**Justification:**

1) Clearly, AP and BP are tangents since tangent at a point on the circle is perpendicular to the radius through point of contact.

2) In Quadrilateral OAPB, we have

∠OAP + ∠APB + ∠OBP + ∠AOB = 360° [By Angle Sum Property]

⇒ ∠OAP + 90° + 90° + 120° = 360°

⇒ ∠OAP = 60°

**Question 5.**

Draw a line segment AB of length 8 cm. Taking A as centre, draw a circle of radius 4 cm and taking B as centre, draw another circle of radius 3 cm. Construct tangents to each circle from the centre of the other circle.

**Answer:**

Step1: Draw a line PQ=8cm. taking P and Q as a center draw circle of 3cm and 4cm.

Step2: Now bisect PQ. We get midpoint of PQ be T.

Now take T as a center , draw a circle of PT radius , this will intersect the circle at point A,B,C,D. Join PB,PD,AQ,QC.