ML Aggarwal Solution Class 10 Chapter 16 Costructions Test

 Test

Question 1

Draw a circle of radius 3 cm. Mark its centre as C and mark a point P such that CP = 7 cm. Using ruler and compasses only, Construct two tangents from P to the circle.

Sol :

Steps of Construction :

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

2.Mark a point P such that CP = 7 cm.

3.With CP as diameter, draw a circle intersecting the given circle at T and S.

4.Join PT and PS.

5.Draw a tangent at Q to the circle given. Which intersects PT at D.

Draw the angle bisector of ∠PDQ intersecting CP at E.

6.With centre E and radius EQ, draw a circle.

7.It will touch the tangent T and PS and the given circle at Q.

This is the required circle.








Question 2

Draw a line AQ = 7 cm. Mark a point P on AQ such that AP = 4 cm. Using ruler and compasses only, construct :

(i) a circle with AP as diameter.

(ii) two tangents to the above circle from the point Q.

Sol :

Steps of construction :

1.Draw a line segment AQ = 7 cm.

2.From AQ,cut off AP = 4cm

3.With AP as diameter draw a circle with centre O.

4.Draw bisector of OQ which intersect OQ at M.

5.With centre M and draw a circle with radius MQ which intersects the first circle at T and S.

6.Join QT and QS.

QT and QS are the tangents to the first circle.









Question 3

Using ruler and compasses only, construct a triangle ABC having given c = 6 cm, b = 1 cm and ∠A = 30°. Measure side a. Draw carefully the circumcircle of the triangle.

Sol :

Steps of Construction :

1.Draw a line segment AC = 7 cm.

2.At C, draw a ray CX making an angle of 30°

3.With centre A and radius 6 cm draw an arc which intersects the ray CX at B.

4.Join BA.

5.Draw perpendicular bisectors of AB and AC intersecting each other at O.

6.With centre O and radius OA or OB or OC, draw a circle which will pass through A, B and C.

This is the required circumcircle of ∆ABC








Question 4

Using ruler and compasses only, construct an equilateral triangle of height 4 cm and draw its circumcircle.

Sol :

Steps of Construction :

1.Draw a line XY and take a point D on it.

2.At D, draw perpendicular and cut off DA = 4 cm.

3.From A, draw rays making an angle of 30° on each side of AD meeting the line XY at B and C.

4.Now draw perpendicular bisector of AC intersecting AD at O.

5.With centre O and radius OA or OB or OC draw a circle which will pass through A, B and C.

This is the required circumcircle of ∆ABC.








Question 5

Using ruler and compasses only :

(i) Construct a triangle ABC with the following data: BC = 7 cm, AB = 5 cm and ∠ABC = 45°.

(ii) Draw the inscribed circle to ∆ABC drawn in part (i).

Sol :

Steps of construction :

1.Draw a line segment BC = 7 cm.

2.At B, draw a ray BX making an angle of 45° and cut off BA = 5 cm.

3.Join AC.

4.Draw the angle bisectors of ∠B and ∠C intersecting each other at I.

5.From I, draw a perpendicular ID on BC.

6.With centre, I and radius ID, draw a circle which touches the sides of ∆ABC at D, E and F respectively.

7.This is the required inscribed circle.








Question 6

Draw a triangle ABC, given that BC = 4cm, ∠C = 75° and that radius of the circumcircle of ∆ABC is 3 cm.

Sol :

Steps of Construction:

1.Draw a line segment BC = 4 cm

2.Draw the perpendicular bisector of BC.

3.From B draw an arc of 3 cm radius which intersects the perpendicular bisector at O.

4.Draw a ray CX making art angle of 75°

5.With centre O and radius 3 cm draw a circle which intersects the ray CX at A.

6.Join AB.

∆ABC is the required triangle










Question 7

Draw a regular hexagon of side 3.5 cm construct its circumcircle and measure its radius.

Sol :

Steps of construction:

1.Draw a regular hexagon ABCDEF whose each side is 3.5 cm.

2.Draw the perpendicular bisector of AB and BC which intersect each other at O.

3.Join OA and OB.

4.With centre O and radius OA or OB, draw a circle which passes through A, B, C, D, E and P.

Then this is the required circumcircle.










Question 8

Construct a triangle ABC with the following data: AB = 5 cm, BC = 6 cm and ∠ABC = 90°.

(i) Find a point P which is equidistant from B and C and is 5 cm from A. How many such points are there ?

(ii) Construct a circle touching the sides AB and BC, and whose centre is equidistant from B and C.

Sol :

Steps of Construction :

1.Draw a line segment BC = 6 cm.

2.At B, draw a ray BX making an angle of 90° and cut off BA = 5 cm.

3.Join AC.

4.Draw the perpendicular bisector of BC.

5.From A with 5 cm radius draw arc which intersects the perpendicular bisector of BC at P and P’.

There are two points.

6.Draw the angle bisectors of ∠B and ∠C intersecting at 0.

7.From O, draw OD ⊥ BC.

8.With centre O and radius OD, draw a circle which will touch the sides AB and BC.

This is the required circle.



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