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NCERT Exemplar · Class 10 · Maths

Chapter 10: Constructions

Practice each question, then open the answer or worked solution.

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Exercise 10.4

3 questions
Q1Written answer

Two line segments AB and AC include an angle of 60° where AB = 5 cm and AC = 7 cm. Locate points P and Q on AB and AC, respectively such that AP = ¾ AB and AQ = ¼ AC. Join P and Q and measure the length PQ.

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Worked solution

Steps of construction:

1. Draw a line segment AB = 5 cm.

NCERT Exemplar Class 10 Maths Chapter 10 Ex. 10.4 Question-1

2. Draw ∠BAZ = 60°.

3. With centre A and radius 7 cm, draw an arc cutting the line AZ at C.

4. Draw a ray AX, making an acute ∠BAX.

5. Divide AX into four equal parts, namely AA1 = A1A2 = A2A3 = A3A4. 

6. Join A4B.

7. Draw A3P || A4B meeting AB at P.

8. Hence, we obtain, P is the point on AB such that AP = ¾ AB.

9. Next, draw a ray AY, such that it makes an acute ∠CAY.

10. Divide AY into four parts, namely AB1 = B1B2 = B2B3 = B3B4.

11. Join B4C.

12. Draw B1Q || B4C meeting AC at Q. We get, Q is the point on AC such that AQ = ¼ AC.

13.   Join PQ and measure it.

14.   PQ = 3.25 cm.

Q2Written answer

Draw a parallelogram ABCD in which BC = 5 cm, AB = 3 cm and angle ABC = 60°, divide it into triangles BCD and ABD by the diagonal BD. Construct the triangle BD‘ C‘ similar to triangle BDC with scale factor 4/3. Draw the line segment D‘A‘ parallel to DA where A’ lies on extended side BA. Is A’BC’D’ a parallelogram?

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Worked solution

Steps of constructions:

1. Draw a line AB=3 cm.

2. Draw a ray BY making an acute ∠ABY=60°.

3. With centre B and radius 5 cm, draw an arc cutting the point C on BY.

4. Draw a ray AZ making an acute ∠ZAX’=60°.(BY||AZ, ∴ ∠YBX’=ZAX’=60°)

5. With centre A and radius 5 cm, draw an arc cutting the point D on AZ.

6. Join CD

7. Thus we obtain a parallelogram ABCD.

8. Join BD, the diagonal of parallelogram ABCD.

9. Draw a ray BX downwards making an acute ∠CBX.

10. Locate 4 points B1, B2, B3, B4 on BX, such that BB1=B1B2=B2B3=B3B4.

11. Join B4C and from B3C draw a line B4C’||B3C intersecting the extended line segment BC at C’.

12. Draw C’D’|| CD intersecting the extended line segment BD at D’. Then, ∆D’BC’ is the required triangle whose sides are 4/3 of the corresponding sides of ∆DBC.

13. Now draw a line segment D’A’|| DA, where A’ lies on the extended side BA.

14. Finally, we observe that A’BC’D’ is a parallelogram in which A’D’=6.5 cm A’B = 4 cm and ∠A’BD’= 60° divide it into triangles BC’D’ and A’BD’ by the diagonal BD’.

NCERT Exemplar Class 10 Maths Chapter 10 Ex. 10.4 Question-2

Q3Written answer

Draw two concentric circles of radii 3 cm and 5 cm. Taking a point on outer circle construct the pair of tangents to the other. Measure the length of a tangent and verify it by actual calculation.

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Worked solution

NCERT Exemplar Class 10 Maths Chapter 10 Ex. 10.4 Question-3

Steps of constructions:

1. Draw a circle with centre O and radius 3 cm.

2. Draw another circle with centre O and radius 5 cm.

3. Take a point P on the circumference of larger circle and join OP.

4. Draw another circle with diameter OP such that it intersects the smallest circle at A and B.

5. Join A to P and B to P.

Hence AP and BP are the required tangents.