## Important Questions for Class 10 Maths Chapter 11 Constructions

### Constructions Class 10 Important Questions Short Answer-I (2 Marks)

Question 1.

Draw a line segment of length 6 cm. Using compasses and ruler, find a point P on it which divides it in the ratio 3 : 4. (2011D)

Solution:

Hence, PA : PB = 3 : 4

Question 2.

Draw a line segment AB of length 7 cm. Using ruler and compasses, find a point P on AB such that \(\frac{\mathbf{A} \mathbf{P}}{\mathbf{A B}}=\frac{3}{5}\). (2011OD)

Solution:

AB = 7 cm, AB = \(\frac{\mathbf{A} \mathbf{P}}{\mathbf{A B}}=\frac{3}{5}\) … [Given

∴ AP : PB = 3 : 2

Hence, AP : AB = 3 : 5 or \(\frac{\mathbf{A} \mathbf{P}}{\mathbf{A B}}=\frac{3}{5}\)

### Constructions Class 10 Important Questions Short Answer-II (3 Marks)

Question 3.

Draw a triangle ABC in which AB = 5 cm, BC = 6 cm and ∠ABC = 60°. Then construct a triangle whose sides are \(\frac{5}{7}\) times the corresponding sides of ∆ABC. (2011D)

Solution:

In ∆ABC

AB = 5 cm

BC = 6 cm

∠ABC = 60°

Hence, ∆A’BC’ is the required ∆.

Question 4.

Construct a triangle with sides 5 cm, 5.5 cm and 6.5 cm. Now construct another triangle, whose sides are \(\frac{3}{5}\) times the corresponding sides of the given triangle. (20140D)

Solution:

∴ ∆AB’C’ is the required ∆.

Question 5.

Construct a right triangle in which the sides, (other than the hypotenuse) are of length 6 cm and 8 cm. Then construct another triangle, whose sides are \(\frac{3}{5}\) times the corresponding sides of the given triangle. (2012D)

Solution:

Here AB = 8 cm, BC = 6 cm and

Ratio = \(\frac{3}{5}\) of corresponding sides

∴ ∆AB’C’ is the required triangle.

Question 6.

Draw a triangle PQR such that PQ = 5 cm, ∠P = 120° and PR = 6 cm. Construct another triangle whose sides are \(\frac{3}{4}\) times the corresponding sides of ∆PQR. (2011D)

Solution:

In ∆PQR,

PQ = 5 cm, PR = 6 cm, ∠P = 120°

∴ ∆POʻR’ is the required ∆.

Question 7.

Draw a triangle ABC with BC = 7 cm, ∠B = 45° and ∠C = 60°. Then construct another triangle, whose sides are \(\frac{3}{5}\) times the corresponding sides of ∆ABC. (2012OD)

Solution:

Here, BC = 7 cm, ∠B = 45°, ∠C = 60° and ratio is \(\frac{3}{5}\) times of corresponding sides

∴ ∆A’BC’ is the required triangle.

Question 8.

Construct a triangle with sides 5 cm, 4 cm and 6 cm. Then construct another triangle whose sides are \(\frac{2}{3}\) times the corresponding sides of first triangle. (2013D)

Solution:

Steps of Construction:

- Draw ∆ABC with AC = 6 cm, AB = 5 cm, BC = 4 cm.
- Draw ray AX making an acute angle with AÇ.
- Locate 3 equal points A
_{1}, A_{2}, A_{3}on AX. - Join CA
_{3}. - Join A
_{2}C’ || CA_{3}. - From point C’ draw B’C’ || BC.

∴ ∆AB’C’ is the required triangle.

Question 9.

Draw a pair of tangents to a circle of radius 3 cm, which are inclined to each other at an angle of 60°. (2011OD)

Solution:

∴ PA & PB are the required tangents.

Question 10.

Construct a tangent to a circle of radius 4 cm from a point on the concentric circle of radius 6 cm. (2013D)

Solution:

Steps of Construction:

Draw two circles with radius OA = 4 cm and OP = 6 cm with O as centre. Draw ⊥ bisector of OP at M. Taking M as centre and OM as radius draw another circle intersecting the smaller circle at A and B and touching the bigger circle at P. Join PA and PB. PA and PB are the required tangents.

Verification:

In rt. ∆OAP,

OA^{2} + AP^{2} = OP^{2} … [Pythagoras’ theorem

(4)^{2} + (AP)^{2} = (6)^{2}

AP^{2} = 36 – 16 = 20

AP = + \(\sqrt{20}=\sqrt{4 \times 5}\)

= \(2 \sqrt{5}\) = 2(2.236) = 4.472 = 4.5 cm

By measurement, ∴ PA = PB = 4.5 cm

Question 11.

Draw a pair of tangents to a circle of radius 4.5 cm, which are inclined to each other at an angle of 45°. (2013OD)

Solution:

Draw ∠AOB = 135°, ∠OAP = 90°, ∠OBP = 90°

∴ PA and PB are the required tangents.

Question 12.

Draw two tangents to a circle of radius 3.5 cm, from a point P at a distance of 6.2 cm from its cehtre. (2013OD)

Solution:

OP = OC + CP = 3.5 + 2.7 = 6.2 cm

Hence AP & PB are the required tangents.

Question 13.

Draw a right triangle ABC in which AB = 6 cm, BC = 8 cm and ∠B = 90°. Draw BD perpendicular from B on AC and draw a circle passing through the points B, C and D. Construct tangents from A to this circle. (2014D, 2015OD)

Solution:

Steps of Construction:

- Draw BC = 8 cm.
- From B draw an angle of 90°.
- Draw an arc BA = 6 cm cutting the angle at A.
- Join AC. ∴ ∆ABC is the required ∆.
- Draw ⊥ bisector of BC cutting BC at M.
- Take Mas centre and BM as radius, draw a circle.
- Take A as centre and AB as radius draw an arc cutting the circle at E. Join AE.

AB and AE are the required tangents.

Justification: ∠ABC = 90° …[Given

Since, OB is a radius of the circle.

∴ AB is a tangent to the circle.

Also, AE is a tangent to the circle.

Question 14.

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. (2014OD)

Solution:

Draw two circles on A and B as asked.

Z is the mid-point of AB.

From Z, draw a circle taking ZA = ZB as radius,

so that the circle intersects the bigger circle at M and N and smaller circle at X and Y.

Join AX and AY, BM and BN.

BM, BN are the reqd. tangents from external point B.

AX, AY are the reqd. tangents from external point A.

Justification:

∠AMB = 90° …[Angle in a semi-circle

Since, AM is a radius of the given circle.

∴ BM is a tangent to the circle

Similarly, BN, AX and AY are also tangents.

### Constructions Class 10 Important Questions Long Answer (4 Marks).

Question 15.

Draw a triangle ABC with side BC = 7 cm, ∠B = 45° and ∆A = 105°. Then construct a triangle whose sides are \(\frac{3}{5}\)times the corresponding sides of ∆ABC. (2011D)

Solution:

In ∆ABC, ∠A + ∠B + ∠C = 180° … [angle sum property of a ∆

105° + 45° + C = 180°

∠C = 180° – 105° – 45o = 30°

BC = 7 cm

∴ ∆A’BC’ is the required ∆.

Question 16.

Draw a triangle ABC with side BC = 6 cm, ∠C = 30° and ∠A = 105°. Then construct another triangle whose sides are \(\frac{2}{3}\) times the corresponding sides of ∆ABC. (2012D)

Solution:

Here, BC = 6 cm, ∠A = 105o and ∠C = 30°

In ∆ABC,

∠A + ∠B + ∠C = 180° …[Angle-sum-property of a ∆

105° + ∠B + 30o = 180°

∠B = 180° – 105° – 30o = 45°

∴ ∆A’BC’ is the required ∆.

Question 17.

Draw a triangle with sides 5 cm, 6 cm and 7 cm. Then construct another triangle whose sides are \(\frac{2}{3}\) times the corresponding sides of the first triangle. (2012OD)

Solution:

Here, AB = 5 cm, BC = 7 cm, AC = 6 cm and ratio is \(\frac{2}{3}\) times of corresponding sides.

∴ ∆A’BC’ is the required triangle.

Question 18.

Construct an isosceles triangle whose base is 6 cm and altitude 4 cm. Then construct another triangle whose sides are \(\frac{3}{4}\) times the corresponding sides of the isosceles triangle. (2015D)

Solution:

∴ ∆A’BC’ is the required triangle.

Question 19.

Draw a line segment AB of length 7 cm. Taking A as centre, draw a circle of radius 3 cm and taking B as centre, draw another circle of radius 2 cm. Construct tangents to each circle from the centre of the other circle. (2015D)

Solution:

Draw two circles on A and B as asked.

Z is the mid-point of AB.

From Z, draw a circle taking ZA = ZB as radius,

so that the circle intersects the bigger circle at M and N and smaller circle at X and Y.

Join AX and AY, BM and BN.

BM, BN are the required tangents from external point B.

AX, AY are the required tangents from external point A.

Justification:

∠AMB = 90° …[Angle in a semi-circle

Since, AM is a radius of the given circle.

∴BM is a tangent to the circle

Similarly, BN, AX and AY are also tangents.

Question 20.

Construct a ∆ABC in which AB = 6 cm, ∠A = 30° and ∆B = 60°. Construct another ∆AB’C’ similar to ∆ABC with base AB’ = 8 cm. (2015OD)

Solution:

Steps of construction:

- Draw a ∆ABC with side AB = 6 cm, ∠A = 30° and ∠B = 60°.
- Draw a ray AX making an acute angle with AB on the opposite side of point C.
- Locate points A
_{1}, A_{2}, A_{3}and A_{4}on AX. - Join A
_{3}B. Draw a line through A_{4}parallel to A_{3}B intersecting extended AB at B’. - Draw a line parallel to BC intersecting ray AY at C’.

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

Question 21.

Construct a triangle ABC in which AB = 5 cm, BC = 6 cm and ∠ABC = 60°. Now construct another triangle whose sides are \(\frac{5}{7}\) times the corresponding sides of ∆ABC. (2015OD)

Solution:

In ∆ABC, AB = 5 cm; BC = 6 cm; ∠ABC = 60°

∴ ∆A’BC’ is the required ∆.

Question 22.

Construct a triangle ABC in which BC = 6 cm, AB = 5 cm and ∠ABC = 60°. Then construct another triangle whose sides are \(\frac{3}{4}\) times the corresponding sides of ∆ABC. (2016D)

Solution:

Steps of Construction:

- Draw ∆ABC with the given data.
- Draw a ray BX downwards making an acute angle with BC.
- Locate 4 points B
_{1}, B_{2}, B_{3}, B_{4}, on BX, such that BB_{1}= B_{1}B_{2}= B_{2}B_{3}= B_{3}B_{4}. - Join CB
_{4}. - From B
_{3}draw a line C’B_{3}|| CB_{4}intersecting BC at C’. - From C’ draw A’C’ || AC intersecting AB at B’.

Then ∆AB’C’ in the required triangle.

Justification:

Question 23.

Draw a triangle ABC with BC = 7 cm, ∠B = 45° and ∠A = 105°. Then construct a triangle whose sides are \(\frac{4}{5}\) times the corresponding sides of ∆ABC. (2016D)

Solution:

In ∆ABC, ∠A + ∠B + ∠C = 180° ..[Angle-sum-property of a ∆

105° + 45° + ∠C = 180°

∠C = 180° – 105° – 45o = 30°

∴ A’BC’ is the required triangle.

Question 24.

Draw an isosceles ∆ABC in which BC = 5.5 cm and altitude AL = 3 cm. Then construct another triangle whose sides are \(\frac{3}{4}\) of the corresponding sides of ∆ABC. (2016OD)

Solution:

∴ ∆ABC’ is the required ∆.

Question 25.

Draw a triangle with sides 5 cm, 6 cm and 7 cm. Then draw another triangle whose sides are \(\frac{4}{5}\) of the corresponding sides of first triangle. (2016OD)

Solution:

∴ ∆A’BC’ is the required ∆.

Question 26.

Draw two concentric cirlces of radii 3 cm and 5 cm. Construct a tangent to smaller circle from a point on the larger circle. Also measure its length. (2016D)

Solution:

We have, OD = 3 cm and OP = 5 cm

PA and PB are the required tangents

By measurement PA = PB = 4 cm.

Question 27.

Draw a circle of radius 4 cm. Draw two tangents to the circle inclined at an angle of 60° to each other. (2013, 2016OD)

Solution:

Draw a circle with O as centre and radius 4 cm.

Draw any ∠AOB = 120°. From A and B draw ∠PAO = ∠PBO = 90° which meet at P.

∴ PA and PB are the required tangents.