# NCERT Solutions for Class 10 Maths Chapter 11 Constructions Exercise 11.1

NCERT Solutions comprise complete and tailored answers to each question in Exercise 11.1. Download free NCERT Solutions for Maths Chapter 11, Exercise 11.1. The solutions provide students a fine preparation strategy in order to prepare for their exam more methodically. Class 10 Maths Chapter 11 Constructions Exercise 11.1 questions and answers helps students to boost second term exam preparations and clear doubts. NCERT Solutions for Class 10 Maths Chapter 11 ConstructionÂ are prepared by BYJUâ€™S subject experts in an easy and logical way. Students will find it extremely easy to understand the questions and how to go about solving the problem.

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In each of the following, give the justification of the construction also:

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

Construction Procedure:

A line segment with a measure of 7.6 cm length is divided in the ratio of 5:8 as follows.

1. Draw line segment AB with the length measure of 7.6 cm

2. Draw a ray AX that makes an acute angle with line segment AB.

3. Locate the points i.e.,13 (= 5+8) points, such as A1, A2, A3, A4 â€¦â€¦.. A13, on the ray AX such that it becomes AA1 = A1A2 = A2A3 and so on.

4. Join the line segment and the ray, BA13.

5. Through the point A5, draw a line parallel to BA13 which makes an angle equal to âˆ AA13B

6. The point A5 which intersects the line AB at point C.

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

8. Now, measure the lengths of the line AC and CB. It comes out to the measure of 2.9 cm and 4.7 cm respectively.

Justification:

The construction of the given problem can be justified by proving that

AC/CB = 5/ 8

By construction, we have A5C || A13B. From Basic proportionality theorem for the triangle AA13B, we get

AC/CB =AA5/A5A13â€¦.. (1)

From the figure constructed, it is observed that AA5 and A5A13 contain 5 and 8 equal divisions of line segments respectively.

Therefore, it becomes

AA5/A5A13=5/8â€¦ (2)

Compare the equations (1) and (2), we obtain

AC/CB = 5/ 8

Hence, Justified.

2. Construct a triangle of sides 4 cm, 5 cm and 6 cm and then a triangle similar to it whose sides are 2/3 of

the corresponding sides of the first triangle.

Construction Procedure:

1. Draw a line segment AB which measures 4 cm, i.e., AB = 4 cm.

2. Take the point A as centre, and draw an arc of radius 5 cm.

3. Similarly, take the point B as its centre, and draw an arc of radius 6 cm.

4. The arcs drawn will intersect each other at point C.

5. Now, we obtained AC = 5 cm and BC = 6 cm and therefore Î”ABC is the required triangle.

6. Draw a ray AX which makes an acute angle with the line segment AB on the opposite side of vertex C.

7. Locate 3 points such as A1, A2, A3 (as 3 is greater between 2 and 3) on line AX such that it becomes AA1= A1A2 = A2A3.

8. Join the point BA3 and draw a line through A2which is parallel to the line BA3 that intersect AB at point Bâ€™.

9. Through the point Bâ€™, draw a line parallel to the line BC that intersect the line AC at Câ€™.

10. Therefore, Î”ABâ€™Câ€™ is the required triangle.

Justification:

The construction of the given problem can be justified by proving that

ABâ€™Â = (2/3)AB

Bâ€™Câ€™ = (2/3)BC

ACâ€™= (2/3)AC

From the construction, we get Bâ€™Câ€™ || BC

âˆ´ âˆ ABâ€™Câ€™ = âˆ ABC (Corresponding angles)

In Î”ABâ€™Câ€™ and Î”ABC,

âˆ ABC = âˆ ABâ€™C (Proved above)

âˆ BAC = âˆ Bâ€™ACâ€™ (Common)

âˆ´ Î”ABâ€™Câ€™ âˆ¼ Î”ABC (From AA similarity criterion)

Therefore, ABâ€™/AB = Bâ€™Câ€™/BC= ACâ€™/AC â€¦. (1)

In Î”AABâ€™ and Î”AAB,

âˆ A2ABâ€™ =âˆ A3AB (Common)

From the corresponding angles, we get,

âˆ AA2Bâ€™ =âˆ AA3B

Therefore, from the AA similarity criterion, we obtain

Î”AA2Bâ€™ and AA3B

So, ABâ€™/AB = AA2/AA3

Therefore, ABâ€™/AB = 2/3 â€¦â€¦. (2)

From the equations (1) and (2), we get

ABâ€™/AB=Bâ€™Câ€™/BC = ACâ€™/ AC = 2/3

This can be written as

ABâ€™Â = (2/3)AB

Bâ€™Câ€™ = (2/3)BC

ACâ€™= (2/3)AC

Hence, justified.

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

Construction Procedure:

1. Draw a line segment AB =5 cm.

2. Take A and B as centre, and draw the arcs of radius 6 cm and 5 cm respectively.

3. These arcs will intersect each other at point C and therefore Î”ABC is the required triangle with the length of sides as 5 cm, 6 cm, and 7 cm respectively.

4. Draw a ray AX which makes an acute angle with the line segment AB on the opposite side of vertex C.

5. Locate the 7 points such as A1, A2, A3, A4, A5, A6, A7 (as 7 is greater between 5 and 7), on line AX such that it becomes AA1 = A1A2 = A2A3 = A3A4 = A4A5 = A5A6 = A6A7

6. Join the points BA5 and draw a line from A7 to BA5 which is parallel to the line BA5 where it intersects the extended line segment AB at point Bâ€™.

7. Now, draw a line from Bâ€™ the extended line segment AC at Câ€™ which is parallel to the line BC and it intersects to make a triangle.

8. Therefore, Î”ABâ€™Câ€™ is the required triangle.

Justification:

The construction of the given problem can be justified by proving that

ABâ€™Â = (7/5)AB

Bâ€™Câ€™ = (7/5)BC

ACâ€™= (7/5)AC

From the construction, we get Bâ€™Câ€™ || BC

âˆ´ âˆ ABâ€™Câ€™ = âˆ ABC (Corresponding angles)

In Î”ABâ€™Câ€™ and Î”ABC,

âˆ ABC = âˆ ABâ€™C (Proved above)

âˆ BAC = âˆ Bâ€™ACâ€™ (Common)

âˆ´ Î”ABâ€™Câ€™ âˆ¼ Î”ABC (From AA similarity criterion)

Therefore, ABâ€™/AB = Bâ€™Câ€™/BC= ACâ€™/AC â€¦. (1)

In Î”AA7Bâ€™ and Î”AA5B,

âˆ A7ABâ€™=âˆ A5AB (Common)

From the corresponding angles, we get,

âˆ A A7Bâ€™=âˆ A A5B

Therefore, from the AA similarity criterion, we obtain

Î”A A2Bâ€™ and A A3B

So, ABâ€™/AB = AA5/AA7

Therefore, AB /ABâ€™ = 5/7 â€¦â€¦. (2)

From the equations (1) and (2), we get

ABâ€™/AB = Bâ€™Câ€™/BC = ACâ€™/ AC = 7/5

This can be written as

ABâ€™ = (7/5)AB

Bâ€™Câ€™ = (7/5)BC

ACâ€™= (7/5)AC

Hence, justified.

Construction Procedure:

1. Draw a line segment BC with the measure of 8 cm.

2. Now draw the perpendicular bisector of the line segment BC and intersect at the point D

3. Take the point D as centre and draw an arc with the radius of 4 cm which intersect the perpendicular bisector at the point A

4. Now join the lines AB and AC and the triangle is the required triangle.

5. Draw a ray BX which makes an acute angle with the line BC on the side opposite to the vertex A.

6. Locate the 3 points B1, B2 and B3 on the ray BX such that BB1 = B1B2 = B2B3

7. Join the points B2C and draw a line from B3 which is parallel to the line B2C where it intersects the extended line segment BC at point Câ€™.

8. Now, draw a line from Câ€™ the extended line segment AC at Aâ€™ which is parallel to the line AC and it intersects to make a triangle.

9. Therefore, Î”Aâ€™BCâ€™ is the required triangle.

Justification:

The construction of the given problem can be justified by proving that

Aâ€™BÂ = (3/2)AB

BCâ€™ = (3/2)BC

Aâ€™Câ€™= (3/2)AC

From the construction, we get Aâ€™Câ€™ || AC

âˆ´ âˆ  Aâ€™Câ€™B = âˆ ACB (Corresponding angles)

In Î”Aâ€™BCâ€™ and Î”ABC,

âˆ B = âˆ B (common)

âˆ Aâ€™BCâ€™ = âˆ ACB

âˆ´ Î”Aâ€™BCâ€™ âˆ¼ Î”ABC (From AA similarity criterion)

Therefore, Aâ€™B/AB = BCâ€™/BC= Aâ€™Câ€™/AC

Since the corresponding sides of the similar triangle are in the same ratio, it becomes

Aâ€™B/AB = BCâ€™/BC= Aâ€™Câ€™/AC = 3/2

Hence, justified.

5. Draw a triangle ABC with side BC = 6 cm, AB = 5 cm and âˆ ABC = 60Â°. Then construct a triangle whose sides are 3/4 of the corresponding sides of the triangle ABC.

Construction Procedure:

1. Draw a Î”ABC with base side BC = 6 cm, and AB = 5 cm and âˆ ABC = 60Â°.

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

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

4. Join the points B4C and also draw a line through B3, parallel to B4C intersecting the line segment BC at Câ€™.

5. Draw a line through Câ€™ parallel to the line AC which intersects the line AB at Aâ€™.

6. Therefore, Î”Aâ€™BCâ€™ is the required triangle.

Justification:

The construction of the given problem can be justified by proving that

Since the scale factor is 3/4 , we need to prove

Aâ€™BÂ = (3/4)AB

BCâ€™ = (3/4)BC

Aâ€™Câ€™= (3/4)AC

From the construction, we get Aâ€™Câ€™ || AC

In Î”Aâ€™BCâ€™ and Î”ABC,

âˆ´ âˆ  Aâ€™Câ€™B = âˆ ACB (Corresponding angles)

âˆ B = âˆ B (common)

âˆ´ Î”Aâ€™BCâ€™ âˆ¼ Î”ABC (From AA similarity criterion)

Since the corresponding sides of the similar triangle are in the same ratio, it becomes

Therefore, Aâ€™B/AB = BCâ€™/BC= Aâ€™Câ€™/AC

So, it becomes Aâ€™B/AB = BCâ€™/BC= Aâ€™Câ€™/AC = 3/4

Hence, justified.

6. Draw a triangle ABC with side BC = 7 cm, âˆ  B = 45Â°, âˆ  A = 105Â°. Then, construct a triangle whose sides are 4/3 times the corresponding sides of âˆ† ABC.

To find âˆ C:

Given:

âˆ B = 45Â°, âˆ A = 105Â°

We know that,

Sum of all interior angles in a triangle is 180Â°.

âˆ A+âˆ B +âˆ C = 180Â°

105Â°+45Â°+âˆ C = 180Â°

âˆ C = 180Â° âˆ’ 150Â°

âˆ C = 30Â°

So, from the property of triangle, we get âˆ C = 30Â°

Construction Procedure:

The required triangle can be drawn as follows.

1. Draw a Î”ABC with side measures of base BC = 7 cm, âˆ B = 45Â°, and âˆ C = 30Â°.

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

3. Locate 4 points (as 4 is greater in 4 and 3), such as B1, B2, B3, B4, on the ray BX.

4. Join the points B3C.

5. Draw a line through B4 parallel to B3C which intersects the extended line BC at Câ€™.

6. Through Câ€™, draw a line parallel to the line AC that intersects the extended line segment at Câ€™.

7. Therefore, Î”Aâ€™BCâ€™ is the required triangle.

Justification:

The construction of the given problem can be justified by proving that

Since the scale factor is 4/3, we need to prove

Aâ€™BÂ = (4/3)AB

BCâ€™ = (4/3)BC

Aâ€™Câ€™= (4/3)AC

From the construction, we get Aâ€™Câ€™ || AC

In Î”Aâ€™BCâ€™ and Î”ABC,

âˆ´ âˆ Aâ€™Câ€™B = âˆ ACB (Corresponding angles)

âˆ B = âˆ B (common)

âˆ´ Î”Aâ€™BCâ€™ âˆ¼ Î”ABC (From AA similarity criterion)

Since the corresponding sides of the similar triangle are in the same ratio, it becomes

Therefore, Aâ€™B/AB = BCâ€™/BC= Aâ€™Câ€™/AC

So, it becomes Aâ€™B/AB = BCâ€™/BC= Aâ€™Câ€™/AC = 4/3

Hence, justified.

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 5/3 times the corresponding sides of the given triangle.

Given:

The sides other than hypotenuse are of lengths 4cm and 3cm. It defines that the sides are perpendicular to each other

Construction Procedure:

The required triangle can be drawn as follows.

1. Draw a line segment BC =3 cm.

2. Now measure and draw âˆ = 90Â°

3. Take B as centre and draw an arc with the radius of 4 cm and intersects the ray at the point B.

4. Now, join the lines AC and the triangle ABC is the required triangle.

5. Draw a ray BX makes an acute angle with BC on the opposite side of vertex A.

6. Locate 5 such as B1, B2, B3, B4, on the ray BX such that such that BB1 = B1B2 = B2B3= B3B4 = B4B5

7. Join the points B3C.

8. Draw a line through B5 parallel to B3C which intersects the extended line BC at Câ€™.

9. Through Câ€™, draw a line parallel to the line AC that intersects the extended line AB at Aâ€™.

10. Therefore, Î”Aâ€™BCâ€™ is the required triangle.

Justification:

The construction of the given problem can be justified by proving that

Since the scale factor is 5/3, we need to prove

Aâ€™BÂ = (5/3)AB

BCâ€™ = (5/3)BC

Aâ€™Câ€™= (5/3)AC

From the construction, we get Aâ€™Câ€™ || AC

In Î”Aâ€™BCâ€™ and Î”ABC,

âˆ´ âˆ  Aâ€™Câ€™B = âˆ ACB (Corresponding angles)

âˆ B = âˆ B (common)

âˆ´ Î”Aâ€™BCâ€™ âˆ¼ Î”ABC (From AA similarity criterion)

Since the corresponding sides of the similar triangle are in the same ratio, it becomes

Therefore, Aâ€™B/AB = BCâ€™/BC= Aâ€™Câ€™/AC

So, it becomes Aâ€™B/AB = BCâ€™/BC= Aâ€™Câ€™/AC = 5/3

Hence, justified.

Exercise 11.2 Solutions : 7 Solved Questions

## NCERT Solutions for Class 10 Maths Chapter 11 Constructions Exercise 11.1

This exercise mainly explains the Division of a Line Segment. With the help of pictorial presentation, students will understand how to divide a line segment in a given ratio and how to construct a triangle similar to a given triangle as per the given scale factor using different ways.