# Constructions ML Aggarwal Solutions ICSE Class-10 Maths

Constructions ML Aggarwal Solutions ICSE Class-10 Maths chapter-16. We Provide Step by Step Answer of Exercise-16.1 , Exercise-16.2 Constructions , with  Chapter-Test Questions  / Problems related  for ICSE Class-10 APC Understanding Mathematics  .  Visit official Website CISCE  for detail information about ICSE Board Class-10.

## Constructions ML Aggarwal Solutions ICSE Class-10 Maths chapter-16

–: Select Exercise :–

Exercise 16.1,

Exe – 16.2 ,

Chapter -Test

### How to Solve Constructions Problems/Questions / Exercise of ICSE Class-10 Mathematics

Before viewing Answer of Chapter-16 Constructions of ML Aggarwal Solutions. Read the Chapter Carefully and then solve all example given in  your text book .For more practice on Constructions  related problems /Questions / Exercise try to solve Constructions  exercise of other famous publications also such as  / Concise Selina Publications ICSE  Mathematics. Get the Concept of Constructions   for ICSE Class 10 Maths  to understand the topic more clearly in effective way.

### Constructions Chapter 16 ML Aggarwal Solutions Exercise 16.1

#### Question 1.

Use a ruler and compass only in this question.
(i) Draw a circle, centre O and radius 4 cm.
(ii) Mark a point P such that OP = 7 cm.
Construct the two tangents to the circle from P. Measure and record the length of one of the tangents.

Steps of Construction:

1. Draw a circle with centre O and radius 4 cm.
2. Take a point P such that OP = 7 cm.
3. Bisect OB at M.
4. With centre M and diameter OP,
draw another circle intersecting the given circle at A and B.
5. Join PA and PB.
PA and PB is a pair of tangents to the circle.
6. On measuring PA, it is equal to 5.5 cm.

#### Question 2.

Draw a line AB = 6 cm. Construct a circle with AB as diameter. Mark a point P at a distance of 5 cm from the mid-point of AB. Construct two tangents from P to the circle with AB as diameter. Measure the length of each tangent

Steps of Construction:

1. Take a line segment AB = 6 cm.
2. Draw its perpendicular bisector bisecting it at O.
3. With centre O and radius OB, draw a circle.
4. Produce AB to P such that OP = 5 cm.
5. Draw its perpendicular bisector intersecting it at M.
6. With centre M and radius OM, draw a circle
which intersects the given circle at T and S.
7. Join OT, OS, TP and SP.
PT and PS are the required tangents to the given circle.
8. On measuring, each tangent is 4 cm long i.e. PT = PS = 4 cm.

#### Question 3.

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

Steps of construction:

1. Take a point O.
2. With centre O and radii 4 cm and 6 cm, draw two concentric circles.
3. Join OA and take its mid-point M.
4. With centre M and radius MA, draw another circle
which intersects the first circle at P and Q.
5. Join AP and AQ.
AP and AQ are the required tangents to the first circle from point A.

#### Question 4.

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.

Steps of construction:

1. Take a point O and with centre O, and radius 3 cm, draw a circle.
2. Produce its diameter both sides and cut off OP = OQ = 7 cm.
3. Take mid-points of OP and OQ as M and N respectively.
4. With centres M and N and OP and OQ as diameters,
draw circles which intersect the given circle at A, B and C and D respectively.
5. Join PA, PB, QC and QD.
PA, PB and QC and QD are the required tangents.

#### 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.

Steps of construction:

1. Draw a line segment AB = 8 cm.
2. With centre A and radius 4 cm and with centre B
and radius 3 cm, draw circles.
3. Draw a third circle AB as diameter which intersects
the given two circles at C and D and P and Q respectively.
4. Join AC and AD, BP and BQ.
5. AC and AD, BP and BQ are the required tangents.

### Exercise 16.2 Chapter  16 Constructions ML Solution for ICSE Maths

#### Question 1.

Draw an equilateral triangle of side 4 cm. Draw its circumcircle.

Steps of Construction :
(i) Draw a line segment BC = 4 cm
(ii) With centres B and C, draw two arcs of radius 4 cm
which intersect each other at A.
(iii) Join AB and AC. ∆ ABC is an equilateral triangle.
(iv) Draw the right bisector of BC and AC intersecting each other at O.
(v) Join OA, OB and OC.
(vi) With centre O, and radius equal to OB or OC or OA,
draw a circle which will pass through A, B and C.
This is the required circumcircle of ∆ ABC.

#### Question 2.

Using a ruler and a pair of compasses only, construct: (i) a triangle ABC given AB = 4cm, BC = 6 cm and ∠ABC = 90°.
(ii) a circle which passes through the points A, B and C and mark its centre as O. (2008)

Steps of Construction:
(i) Draw a line segment AB = 4cm
(ii) At B, draw a ray BX making an angle of 90°
and cut off BC = 6 cm.
(iii) Join AC.
(iv) Draw the perpendicular bisectors of sides
AB and AC intersecting each other at O.
(v) With centre O, and radius equal to OB or OA or OC,
draw a circle which passes through A, B and C.
This is the required circle.

#### Question 3. Constructions ML Aggarwal Solutions

Construct a triangle with sides 3 cm, 4 cm and 5 cm. Draw its circumcircle and measure its radius.

Steps of Construction
(i) Draw a line segment BC = 4 cm.
(ii) With centre B and radius 3 cm and with centre C
and radius 5 cm draw two arcs which intersect each other at A.
(iii) Join AB and AC.
(iv) Draw the perpendicular bisector of sides BC and AC
which intersect each other at O.
(v) Join OB.
(vi) With centre O and radius OB, draw a circle
which will pass through A, B and C.
(vii) On measuring the radius OB = 2.5cm

#### Question 4.

Using ruler and compasses only :
(i) Construe a triangle ABC with the following data: Base AB = 6 cm, AC = 5.2 cm and ∠CAB = 60°.
(ii) In the same diagram, draw a circle which passes through the points A, B and C. and mark its centre O.

Steps of Construction :
(i) Draw a line segment AB = 6 cm.
(ii) At A, draw a ray making an angle of 60°.
(iii) With centre B and radius 5-2 cm.
draw an arc which intersects the ray at C.
(iv) Join BC
(v) Draw the perpendicular bisector of AB and BC
intersecting each other at O.
(vi) With O as a centre and OA as a radius
draw a circle which touches the ∆ABC at A, B and C.

#### Question 5.

Using ruler and compasses only, draw an equilateral triangle of side 5 cm and draw its inscribed circle. Measure the radius of the circle.

Steps of Construction :
(i). Draw a line segment BC = 5 cm
(ii) With centre B and C and radius 5 cm,
draw two arcs intersecting each other at A.
(iii) Join AB and AC.
(iv) Draw the angle bisectors of ∠B and ∠C intersecting each other at I.
(v) From I, draw a perpendicular ID on BC.
(vi) With centre I and radius ID, draw a circle
which touches the sides of the triangle internally.

This is the required in circle.
Measure the radius ID which is 1.5 cm (approx)

#### Question 6. Constructions ML Aggarwal Solutions

(i) Conduct a triangle ABC with BC = 6.4 cm, CA = 5.8 cm and ∠ ABC = 60°. Draw its incircle. Measure and record the radius of the incircle.
(ii) Construct a ∆ABC with BC = 6.5 cm, AB = 5.5 cm, AC = 5 cm. Construct the incircle of the triangle. Measure and record the radius of the incircle. (2014)

Steps of Construction :
(i) Draw a line segment BC = 6.4 cm
(ii) Construct ∠ DBC = 60° at B.
(iii) With C as centre and radius CA = 5.8 cm.
Draw an arc cutting BD at A.
(iv) Join AC. Then ABC is the required triangle.
(v) Draw the angle bisectors of ∠B and ∠C which intersect each other at O.
(vi) Draw OE ⊥ BC, intersecting BC in E.
(vii) With O as centre and OE as radius draw the required incircle.
Measure the radius OE which is = 1.5cm

Steps of construction:
1. Draw a line BC = 6.5 cm.
2. With centre B and C draw arcs AB = 5.5 cm and AC = 5 cm
3. Join AB and AC, ABC is the required triangle.
4. Draw the angle bisetors of B and C. Let these bisectors meet at O.
5. Taking O as centre. Draw a incircle which touches all the sides of the ∆ABC.
6. From O draw a perpendicular to side BC which cut at N.

7. Measure ON which is required radius of the incircle. ON = 1.5 cm

#### Question 7.

The bisectors of angles A and B of a scalene triangle ABC meet at O.
(i) What is the point O called?
(ii) OR and OQ are drawn a perpendicular to AB and CA respectively. What is the relation between OR and OQ ?
(iii) What is the relation between ∠ACO and ∠BCO?

(i) The point O where the angle bisectors meet is called the incentre of the triangle.
(ii) The perpendiculars drawn from O to AB and CA are equal i.e. OR and OQ.
(iii) ∠ACO = ∠BCO
OC will bisect the ∠C

#### Question 8.

Using ruler and compasses only, construct a triangle ABC in which BC = 4 cm, ∠ACB = 45° and the perpendicular from A on BC is 2.5 cm. Draw the circumcircle of triangle ABC and measure its radius.

#### Answer 8 Constructions ML Aggarwal Solutions

Steps of Construction
(i) Draw a line segment BC = 4 cm.
(ii) At B, draw a perpendicular and cut off BE = 2.5 cm.

(iii) From E, draw a line EF parallel to BC.
(iv) From C, draw a ray making an angle of 45° which intersects EF at A.
(v) Join AB.
(vi) Draw the perpendicular bisectors of
sides BC and AC intersecting each other at O.
(vii) Join OB, OC and OA.
(viii) With centre 0 and radius OB or OC or OA draw a circle
which will pass through A, B and C.
This circle is the circumcircle of ∆ABC. On measuring its radius OB = 2 cm.

#### Question 9.

Construct a regular hexagon of side 4 cm. Construct a circle circumscribing the hexagon.

Steps of Construction
(i) Draw a line segment BC = 4 cm.
(ii) At A and B draw rays making on angle of 120° each and cut off AF = BC = 4cm.
(iii) At F and C, draw rays making angle of 120° each and cut off EF = CD = 4cm.
(iv) Join ED.
ABCDEF is the required hexagon.
(v) Draw perpendicular bisectors of sides AB and BC intersecting each other at O.
(vi) With centre O and radius equal OA or OB draw a circle
which passes through the vertices of the hexagon.
This is the required circumcircle of hexagon ABCDEF.

#### Question 10.

Draw a regular hexagon of side 4 cm and construct its incircie.

Steps of constructions :

(i) Draw a regular hexagon ABCDEF of side 4 cm.
(ii) Draw the angle bisectors of ∠A and ∠B
which intersect each other at O.
(iii) Draw OL ⊥ AB.
(iv) With centre O and radius OB, draw a circle
which touches the sides of the hexagon

### Chapter Test Constructions of Chapter 16 ML Aggarwal Solutions for ICSE Class 10 Maths

#### 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.

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.
6. Draw the angle bisector of ∠PDQ intersecting CP at E.
7. With centre E and radius EQ, draw a circle.
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.

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.

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.

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).

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.
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.

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.

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.

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.

–: End of Constructions ML Aggarwal Solutions :–

Thanks