**Construction** of **Polygons Class-9th Concise** Selina ICSE Mathematics Solutions Chapter-15. We provide step by step Solutions of Exercise / lesson-15 **Construction** of **Polygons **** **for **ICSE** **Class-9th Concise** Selina Mathematics by R K Bansal.

Our Solutions contain all type Questions with Exe-15 to develop skill and confidence. Visit official Website **CISCE** for detail information about ICSE Board Class-9th Mathematics .

**Construction** of **Polygons Class-9th Concise** Selina ICSE Mathematics Solutions Chapter-15

### Exercise – 15, **Construction** of **Polygons Class-9th Concise** Selina ICSE Mathematics

#### Question 1

Construct a quadrilateral ABCD, when:

AB = 3.2 cm, BC = 5.2 cm, CD = 6.2 cm, DA = 4.2 cm and BD = 5.2 cm.

#### Answer

Steps:

- Draw. AB = 3.2cm

- With A as a centre draw an arc at D and with B as a centre and radius 5.2 cm draw an arc at D.
- join AD and DB.
- With D and b as a centre taking radius 6.2 cm and 5.2 cm draw arc at C. Now join BC and DC.

ABCD is the required quadrilateral.

#### Question 2

Construct a quadrilateral ABCD, when:

AB = 7.2 cm, BC = 5.8 cm, CD = 6.3 cm, AD = 4.3 cm and angle A = 75^{o}.

#### Answer

Steps:

- Draw.AB = 7.2cm
- Through A draw AP such that.
- From AP cut
- With D and B as centre and radii 6.2 cm and 5.8 cm respectively, draw arcs cutting each other at C.
- Join DC and BC.

ABCD is the required quadrilateral.

Question 3

Construct a quadrilateral ABCD, when:

Angle A = 90^{o}, AB = 4.6 cm, BD = 6.4 cm, AC = 6.0 cm and CD = 4.2 cm.

#### Answer

Steps:

- Draw AB = 4.6 cm
- Through A, draw AP such that Angle A = 90°.
- With B as a centre and radii 6.4 cm draw an arc at D on AP.
- With D and A as a centre and radii 4.2 cm and 6 cm draw arc cutting each other at C.
- Now join BD, AC and CB.

ABCD is the required quadrilateral.

#### Question 4

Construct a quadrilateral ABCD, when:

AB = 3.8 cm, AC = 4.8 cm, AD = 2.8 cm, angle A = 105^{o} and angle B = 60^{o}.

#### Answer

4 With A as a centre and radii 4.8 cm draw an arc cutting BP at C.

5 Join AC,AD.

ABCD is the required quadrilateral

#### Question 5

Construct a quadrilateral ABCD, when:

BC = 7.5 cm AC = 5.8 cm, AD = 3.6 cm, CD = 4.2 cm and angle A = 120^{o}.

#### Answer

3 With A and D as a centre and radii 5.8 cm and 4.2 cm draw arcs cutting each other at C.

4 Now join AC and CD.

5 Now with C as centre and radii 7.5 cm draw an arc at B on AP.

Now join CB.

ABCD is the required quadrilateral.

#### Question 6 (Construction of Polygons Class-9th Concise)

Construct a quadrilateral ABCD, when:

AD = AB = 4 cm, BC = 2.8 cm, CD = 2.5 cm and angle BAD = 45^{o}.

#### Answer

Steps:

- Draw AD = 4 cm
- Draw AP such that ∠A =45 ‘

- With A as a centre with radii 4 cm draw an arc at B on AP.
- Now taking B and D as a centre and radii 2.8 cm and 2.8 cm draw arcs cutting each other at C.
- Now join BC and CD.

ABCD is the required quadrilateral.

#### Question 7

Construct a quadrilateral ABCD, when:

AB = 6.3 cm, BC = CD=4.2 cm and ABC = ∠BCD = 90^{o}.

#### Answer

5 Now join AD

ABCD is the required quadrilateral.

Question 8

Construct a parallelogram ABCD, when:

AB = 4.4 cm, AD = 6.2 cm and AC = 4.8 cm.

#### Answer

2 Draw triangle ACD.

3 Then draw triangle ABC.

ABCD is the required parallelogram

#### Question 9

Construct a parallelogram ABCD, when:

Diagonal AC = 6.4 cm, diagonals BD = 8.2 cm and angle between the diagonals = 60^{o}

#### Answer

3 Join AB,BC,CD and DA.

ABCD is the required parallelogram.

#### Question 10

Construct a parallelogram ABCD, when:

AB = 5.8 cm, diagonal AC = 8.2 cm and diagonal BD = 6.2 cm.

#### Answer

- Join AD,DC and CB.

ABCD is the required parallelogram.

#### Question 11

Construct a parallelogram ABCD, when:

AB = 6.0 cm, AD = 5.0 cm and ∠A = 45^{o}.

#### Answer

- With D and B as a centre and radii 6 cm and 5cm draw arcs cutting each other at C.
- Now join DC and BC.

ABCD is the required parallelogram.

Question 12

Construct a parallelogram ABCD, when:

Base AB = 6.5 cm, BC = 4 cm and the altitude corresponding to AB = 3.1 cm.

#### Answer

#### Question 13

Construct a parallelogram ABCD, when:

AB = 4.5 cm, ∠B = 120^{o} and the distance between AB and DC = 3.0 cm.

#### Answer

4 Through E draw perpendicular to BP to get QR parallel to AB.

5 With B as a centre draw an arc which cuts QR at C.

6 With A as a centre draw an arc which cuts QR at D.

Now join Ad and BC.

ABCD is the required parallelogram.

#### Question 14

Construct a parallelogram ABCD, when:

Base BC = 5.6 cm, diagonal BD = 6.5 cm and altitude = 3.2 cm.

#### Answer

Steps:

- Draw.BC=5.6 cm
- At C, draw CX perpendicular to BC.
- with C as a centre and taking radius 3.2 cm draw an arc to cut CX at Y.
- Through Y draw a straight line PQ parallel to BC.
- With B as a centre and radius 6.5 cm draw an arc to meet PQ at D.
- With D as a centre and radius equal to 5.6 cm , draw an arc to meet PQ at A.
- Join BA,BD and CD.

ABCD is the required parallelogram.

#### Question 15

Construct a rectangle ABCD, when:

Its sides are 6.0 cm and 7.2 cm.

#### Answer

- with B as a centre draw a line BX taking as a
- Now taking radius 6 cm draw an arc at A.
- From point A draw a line AY parallel to BC.
- With A as a centre taking radius 7.2 cm draw an arc at D.
- Now join CD.

ABCD is the required rectangle.

#### Question 16 (Construction of Polygons Class-9th Concise)

Construct a rectangle ABCD, when:

One side = 4 cm and one diagonal is 5 cm. Measure the length of other side.

#### Answer

Steps:

- Draw.BC=4
- With C as a centre and radius 5 cm draw an arc at A.
- Now join AB and AC.
- With A as a centre draw an arc at D.
- Now join AD and CD.

ABCD is the required rectangle.

#### Question 17

Construct a rectangle ABCD, when:

One diagonal = 6.0 cm and the acute angle between the diagonals = 45^{o}.

#### Answer

Steps:

- Draw.AC=6cm
- Draw right triangle ACB.
- Draw right triangle ADB.
- Join DC.

ABCD is the required rectangle.

#### Question 18

Construct a rectangle ABCD, when:

Area = 24 cm^{2} and base = 4.8 cm^{2}.

#### Answer

Steps:

- Draw base. AB=4.8 cm
^{2} - With A and B as a centre draw an arcs taking radiusat D and C.
- Now join AD,BC and DC.

ABCD is the required rectangle.

#### Question 19

Construct a rectangle ABCD, when:

Area = 36 cm^{2} and height = 4.5 cm.

#### Answer

Steps:

- Draw base. AB=8cm
- With A and B as a centre draw an arcs taking radius 4.5 cm. at D and C.
- Now join AD,BC and DC.

ABCD is the required rectangle

#### Question 20

Construct a trapezium ABCD, when:

AB = 4.8 cm, BC = 6.8 cm, CD = 5.4 cm, angle B = 60^{o} and AD // BC.

#### Answer

4 With C as a centre and radii 5.4 cm draw an arc at D on the line AP.

5 Now join AB,CD.

ABCD is the required trapezium.

#### Question 21

Construct a trapezium ABCD, when:

AB = CD = 3.2 cm, BC = 6.0 cm, AD = 4.4 cm and AD // BC.

#### Answer

4 Taking B and D as a centre and radii 3.2 cm and 4.1 cm respectively, draw arcs cutting each other at A.

5 Join AB and AD.

ABCD is the required trapezium.

#### Question 22

Construct a rhombus ABCD, when:

Its one side = 6 cm and ∠A = 60^{o}.

#### Answer

Question 23

Construct a rhombus ABCD, when:

One side = 5.4 cm and one diagonals is 7.0 cm

#### Answer

Steps:

- We construct the segment. diagonals is 7.0 cm
- With A as a centre and radius 5.4 cm , we draw an arc extending on both sides of AC.
- With C as centre and same radius as in step 2, we draw an arc extending on both sides of AC to cut the first arc at B and D.
- Join AB,BC,CD and DA.

ABCD is the required rhombus.

#### Question 24

Construct a rhombus ABCD, when:

Diagonal AC = 6.3 cm and diagonal BD = 5.8 cm.

#### Answer

4 Join AB,BC,CD and DA.

ABCD is the required rhombus.

#### Question 25

Construct a rhombus ABCD, when:

One side = 5.0 cm and height = 2.6 cm.

#### Answer

4 Through E draw perpendicular to CP to get QR parallel to AB.

5 With A and B as a centre and radii 5 cm draw arcs cutting QR at D and C.

ABCD is the required rhombus.

#### Question 26 (Construction of Polygons Class-9th Concise)

Construct a rhombus ABCD, when:

∠A = 60^{o} and height = 3.0 cm.

#### Answer

Steps:

- Draw a line AP.
- Now draw a line AF such that.∠A = 60
^{o} - At S draw a perpendicular SE of length 3 cm such that it cut at AF at D.
- Through D draw a line QR parallel to AP.
- Now taking the radius same as AD draw an arc at B on AP.
- Now through and B taking radius same as AD and AB draw arcscutting each other at C.
- Now join BC.

ABCD is the required rhombus.

Question 27

Construct a rhombus ABCD, when:

Diagonal AC = 6.0 cm and height = 3.5 cm

#### Answer

4 Now at C draw a line CY parallel to AP.

5 At point C and A, taking radius same as AB draw arcs cutting each other at D.

6 Now join AD.

ABCD is the required rhombus.

#### Question 28

Construct a square ABCD, when:

One side = 4.5 cm.

#### Answer

4 With B as a centre and radius 4.5 cm draw an arc.

5 With D as centre and radius 4.5 cm draw another arc cutting the former arc at C.

6 Join BC and CD.

ABCD is the required square.

#### Question 29

Construct a square ABCD, when:

One diagonal = 5.4 cm.

#### Answer

We know that the diagonals of a square are equal and bisect each other at right angles.

4 Join AB, BC, CD and DA.

ABCD is the required square.

Question 30

Construct a square ABCD, when:

Perimeter = 24 cm.

#### Answer

The perimeter of a square

4 With B as a centre and radius 6 cm draw an arc.

5 With D as centre and radius 6 cm draw another arc cutting the former arc at C.

6 Join BC and CD.

ABCD is the required square.

#### Question 31

Construct a rhombus, having given one side = 4.8 cm and one angle = 75^{o}.

#### Answer

3 With A as a centre and measurement equal to AB cut off an arc at D on AX.

4 Using same radius taking D and B as centers cut off arcs, which will intersect at C.

5 Join CD and CB.

ABCD is the required rhombus.

#### Question 32

(i) Construct a regular hexagon of side 2.5 cm.

#### Answer

The length of side of regular hexagon is equal to the radius of its circumcircle.

Steps of construction:

- Draw a circle of radius 2.5 cm
- Taking any point A on the circumference of the circle as centre, draw arcs of same radii (i.e. 2.5 cm) which cut the circumference at B and F.
- With B and F as centres, again draw two arcs of same radii which cut the circumference at C and E respectively.
- With C or E as centre, draw one more arc of the same radius which cuts the circumference at point D.

In this way, the circumference of the circle is divided into six equal parts.

- Join AB, BC, CD, DE, EF and FA.

ABCDEF is the required regular hexagon.

(ii)

Construct a regular hexagon of side 3.2 cm

#### Answer

The length of side of regular hexagon is equal to the radius of its circumcircle.

Steps of construction:

1 Draw a circle of radius 3.2 cm

2 Taking any point A on the circumference of the circle as centre, draw arcs of same radii (i.e. 3.2 cm) which cut the circumference at B and F.

3 With B and F as centres, again draw two arcs of same radii which cut the circumference at C and E respectively.

4 With C or E as centre, draw one more arc of the same radius which cuts the circumference at point D.

In this way, the circumference of the circle is divided into six equal parts.

5 Join AB, BC, CD, DE, EF and FA.

ABCDEF is the required regular hexagon.

#### Question 33 (Construction of Polygons Class-9th Concise)

Using ruler and compasses only, construct the quadrilateral ABCD, having given AB = 5 cm, BC = 2.5 cm, CD = 6 cm. angle BAD = 90^{o} and the diagonal AC = 5.5 cm.

#### Answer

3 Taking A and B as a centre and radius 2.5 cm and 5.5 cm draw arcs cuts off at C.

4 Now join BC and AC.

5 Taking C as a centre and radius 6 cm draw arcs at D on AX.

ABCD is the required quadrilateral.

#### Question 34

Using ruler and compasses only, construct a trapezium ABCD, in which the parallel sides AB and DC are 3.3 cm apart; AB = 4.5 cm, angle A = 120^{o} BC = 3.6 cm and angle B is obtuse.

#### Answer

3 Through *X *draw draw a line QR which is parallel to AB which cuts AS at D.

4 Through B draw an arc taking radius 3.6 cm at C on PQ.

5 Join CB.

ABCD is the required trapezium.

#### Question 35

Using ruler and compasses only, construct the quadrilateral ABCD, having given AB = 5 cm, BC = 2.5 cm CD = 6 cm, ∠BAD = 90^{o} and diagonal BD = 5.5 cm.

#### Answer

Steps:

1.Draw AB=5cm.

2 From A draw a line AY such that.∠BAD = 90^{o}

3 Taking B as a centre with radius 5.5 cm draw an arc at D on AY.

4 With D and B as centre and radii 6 cm and 2.5 cm draw arcs cutting each other at C.

5 Join DC and BC.

ABCD is the required quadrilateral.

#### Question 36

Using ruler and compasses only, construct a parallelogram ABCD using the following data: AB = 6 cm, AD = 3 cm and ∠DAB = 45^{o}. If the bisector of ∠DAB meets DC at P, prove that ∠APB is a right angle.

#### Answer

#### Question 37 (Construction of Polygons Class-9th Concise)

The perpendicular distance between the pair of opposite sides of a parallelogram are 3 cm and 4 cm, and one of its angles measures 60^{o}. Using ruler and compasses only, construct the parallelogram.

#### Answer

Steps:

Draw a base line AQ.

1 From A take some random distance in compass and draw one are below and above the line. Now without changing the distance in compass draw one are below and above the line. These arcs intersect each other above and below the line.

2 Draw the line passing through these intersecting points, you will get a perpendicular to the line AQ.

3 Take distance of 4 cm in compass and mark an arc on the perpendicular above the line. Draw a line parallel to line AQ passing through through this arc.

4 From point A measure an angle of 60 degree and draw the line which intersect above drawn line at some point label it as D.

5 Using the procedure given in step 2 again draw a perpendicular to line AD.

6 Take distance of 3 cm in compass and mark an arc on the perpendicular above the line. Draw a line parallel to line AD passing through through this arc which intersect the line AQ at some point label it as B and to other line at point C.

ABCD is the required parallelogram.

#### Question 38

Draw parallelogram ABCD with the following data:

AB = 6 cm, AD = 5 cm and ∠DAB = 45^{o}.

Let AC and DB meet in O and let E be the mid-point of BC. Join OE. Prove that:

(i) OE // AB(ii) OE = 1/2AB.

#### Answer

To draw the parallelogram follows the steps:

#### Question 39

Using ruler and compasses only, construct a rectangle each of whose diagonals measure 6 cm and the diagonals interest at an angle of 45^{o}.

#### Answer

To draw the rectangle follows the steps:

(1)Firs draw a line *AC* of measure 6cm.

(2)Then draw the perpendicular bisector of *AC* through *O*.

(3)At *O* draw an angle of measure 45 ‘. Then produce *OD *of measure 3cm and *OB *of measure 3cm each.

(4)Now join *AD*, *AB*, *BC* and *CD* to form the rectangle.

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