# ICSE Solutions for Selina Concise Chapter 15 Construction of Polygons Class 9 Maths

### Exercise 15

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.

Steps for construction:

1. Draw. AB = 3.2cm
1. 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.
3. 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.

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 = 75o.

Steps of construction:

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

3. Construct a quadrilateral ABCD, when:

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

Steps of construction:

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

4. Construct a quadrilateral ABCD, when:

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

Steps of construction:

2. Draw AB = 3.8cm and ∠A = 105°.
3. Draw BP such that ∠B = 60°.
4. With A as a centre and radii 4.8 cm draw an arc cutting BP at C.
5. Join AC.

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 = 120o.

Steps of construction:

2. Draw AP such that ∠A = 120° .
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.
6. Now join CB.

6. Construct a quadrilateral ABCD, when:

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

Steps of construction:
1. Draw AD = 4 cm
2. Draw AP such that ∠A =45°
1. With A as a centre with radii 4 cm draw an arc at B on AP.
2. Now taking B and D as a centre and radii 2.8 cm and 2.8 cm draw arcs cutting each other at C.
3. Now join BC and CD.

7. Construct a quadrilateral ABCD, when:

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

Steps of construction:
1. Draw AB = 6.3 cm.
2. Draw BP such that ∠ABP = 90° .
3. With B as a centre and radii 4.2cm draw an arc AP at c.
4. With C as a centre draw a line CD with radii 4.2 cm draw a line such that ∠BCD = 90°

8. Construct a parallelogram ABCD, when:

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

Steps of construction:
1. Draw AD = 6.2 cm.
2. Draw triangle ACD.
3. Then draw triangle ABC.
ABCD is the required parallelogram

9. Construct a parallelogram ABCD, when:

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

Steps of construction:
1. Draw AC = 6.4cm.
2. Draw line BOD such that ∠DOC = 60° and OB = OD = (1/2)BD = (1/2)×8.2 = 4.1cm.
3. Join AB,BC,CD and DA.
ABCD is the required parallelogram.

10. Construct a parallelogram ABCD, when:

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

Steps of construction:
1. Since diagonal of a parallelogram bisect each other, construct OAB such that:
OA = 1/2(AC) = (1/2) 8.2cm = 4.1 cm
OB = (1/2)BD = 1/2×6.2 cm = 3.1 cm  and AB = 5.8cm.

ABCD is the required parallelogram.

11. Construct a parallelogram ABCD, when:

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

Steps of construction:
1. Draw AB = 6cm.
2. Draw AD with radii 5 cm with an angle of 45°
3. With D and B as a centre and radii 6 cm and 5cm draw arcs cutting each other at C.
4. Now join DC and BC.

ABCD is the required parallelogram.

12. Construct a parallelogram ABCD, when:

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

Steps of construction:
1. Draw AB = 6.5cm.
2. At B, draw BP ⊥ AB.
3. From BP cut BE = 3.1cm.
4. Through E draw perpendicular to BP to get QR parallel to AB.
5. With B as a centre and radius = AC = 4cm, draw an arc which cuts QR at c.
6. With A as a centre and radius = AD = 4cm, draw an arc which cuts QR at D.
ABCD is the required parallelogram.

13. Construct a parallelogram ABCD, when:

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

Steps of construction:
1. Draw AB = 4.5cm.
2. At B, draw BP ⊥ AB.
3. From BP cut BE = 3cm.
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.
7. Now join Ad and BC.

ABCD is the required parallelogram.

14. Construct a parallelogram ABCD, when:

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

Steps of construction:

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

ABCD is the required parallelogram.

15. Construct a rectangle ABCD, when:

Its sides are 6.0 cm and 7.2 cm.

Since each angle of a rectangle is 90° and opposite sides are equal. Therefore,
1. with B as a centre draw a line BX taking as a
2. Now taking radius 6 cm draw an arc at A.
3. From point A draw a line AY parallel to BC.
4. With A as a centre taking radius 7.2 cm draw an arc at D.
5. Now join CD.

ABCD is the required rectangle.

16 . Construct a rectangle ABCD, when:

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

Steps of construction:

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

ABCD is the required rectangle.

17. Construct a rectangle ABCD, when:

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

Steps of construction:

1. Draw AC=6cm
2. Draw right triangle ACB.
4. Join DC.

ABCD is the required rectangle.

18. Construct a rectangle ABCD, when:

Area = 24 cm2 and base = 4.8 cm2.

Answer Given that the base = 4.8 cm2 and Area = 24 cm2
we know that area of rectangle = base × Height.
Therefore,
24 = 4.8×height
⇒ Height = 5
With base = 4.8 cm
2 and height 5cm2, the rectangle is shown below :

Steps of construction:

1. Draw base AB=4.8 cm2
2. With A and B as a centre draw an arcs taking radius at D and C.
3. Now join AD,BC and DC.

ABCD is the required rectangle.

19. Construct a rectangle ABCD, when:

Area = 36 cm2 and height = 4.5 cm.

Given that the height = 4.5 cm and Area = 36 cm2
we know tat area of rectangle = base × Height.
Therefore,
36 = base × 4.5
⇒ Base = 8 cm
With height = 4.5 and base 8 cm, the rectangle is shown below :

Steps of construction:
1. Draw base AB=8cm
2. With A and B as a centre draw an arcs taking radius 4.5 cm. at D and C.
3. Now join AD,BC and DC.

ABCD is the required rectangle.

20. Construct a trapezium ABCD, when:

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

Steps of construction:

1. Draw BC = 6.8 cm
2. With B as a centre and radii 4.8cm draw an arc at A such that ∠B = 60° .
3. From point A draw a line AP such that AP॥ BC.
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.

21. Construct a trapezium ABCD, when:

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

Steps of construction:

1. Draw BC = 6 cm.
2. From BC cut BE = AD = 4.1cm.
3. Draw triangle DEC such that:
DE = AB = 3.2cm and CD =3.2cm.
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.
ABCD is the required trapezium.

22. Construct a rhombus ABCD, when:

Its one side = 6 cm and ∠A = 60o.

Steps of construction:

1. Draw a line AB = 6 cm.
2. At A, we construct ∠BAP = 60° .
3. From AP, we cut at D taking AD = 6cm.
4. Through B, we draw BQ॥ AD.
5. Through D, we draw DC॥ AB to cut BQ at c.
ABCD is the required rhombus.

23. Construct a rhombus ABCD, when:

One side = 5.4 cm and one diagonals is 7.0 cm

Steps of construction:

1. We construct the segment. diagonals is 7.0 cm
2. With A as a centre and radius 5.4 cm , we draw an arc extending on both sides of AC.
3. 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.
4. Join AB,BC,CD and DA.

ABCD is the required rhombus.

24. Construct a rhombus ABCD, when:

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

Steps of construction:

1. Draw AC = 6.3cm .
2. Draw perpendicular bisector to AC which cuts AC to O.
3. From this perpendicular cut OD and OB such that:
OD = OB = (1/2)BD = (1/2)×5.8 = 2.9cm.
4. Join AB,BC,CD and DA.

ABCD is the required rhombus.

25. Construct a rhombus ABCD, when:

One side = 5.0 cm and height = 2.6 cm.

Steps of construction:

1. Draw AB = 5cm.
2. At B, draw BP ⊥ AB.
3. From BP, cut BE = 2.6cm = height.
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.

26. Construct a rhombus ABCD, when:

∠A = 60o and height = 3.0 cm.

Steps of construction:

1. Draw a line AP.
2. Now draw a line AF such that ∠A = 60o
3. At S draw a perpendicular SE of length 3 cm such that it cut at AF at D.
4. Through D draw a line QR parallel to AP.
5. Now taking the radius same as AD draw an arc at B on AP.
6. Now through and B taking radius same as AD and AB draw arcs cutting each other at C.
7. Now join BC.

ABCD is the required rhombus.

27. Construct a rhombus ABCD, when:

Diagonal AC = 6.0 cm and height = 3.5 cm

Steps of construction:

1. Draw a line AP.
2. Now draw AC = 6cm and CP = 3.5cm.
3. Now draw a line BC such that AB = BC.
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.
ABCD is the required rhombus.

28. Construct a square ABCD, when:

One side = 4.5 cm.

Steps of construction:

1. Draw a line segment AB = 4.5cm
2. Draw AP ⊥AB.
3. From AP cut off AD = 4.5cm .
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.

29.Construct a square ABCD, when:

One diagonal = 5.4 cm.

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

Steps of construction:

1. Draw AC = 5.4cm.
2. Draw the right bisector XY of AC, meeting AC at O.
3. From O, cut off OB = (1/2)(5.4) = 2.7cm along OY and OD = 3.1cm along OX.
4. Join AB, BC, CD and DA.

ABCD is the required square.

30. Construct a square ABCD, when:

Perimeter = 24 cm.

The perimeter of a square

P = 4a
Where a is the length of each side.
We have Perimeter = 24cm.
Therefore,
24 = 4a
⇒ a = 6
Therefore the sides of the squares are of length 6cm.

Steps of construction:
1. Draw a line segment AB = 6cm.
2. Draw AP ⊥ AB.
3. From AP cut off AD = 6cm.
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.

31. Construct a rhombus, having given one side = 4.8 cm and one angle = 75o. Answer
Steps of construction:
1. Draw line AB = 4.8cm.
2. At A Draw AX such that ∠BAX = 75
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.

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

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

Steps of construction:

1. Draw a circle of radius 2.5 cm
2. 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.
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.

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

ABCDEF is the required regular hexagon.

(ii) Construct a regular hexagon of side 3.2 cm

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. 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.

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

Steps of construction:
1. Draw AB = 5cm.
2. Now draw ∠XAB such that it is 90°.
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.

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 = 120o BC = 3.6 cm and angle B is obtuse.
1. Draw AB = 4.5cm.
2. Now draw ∠BAS = 120° and draw EA ⊥ AB such that AX = 3.3cm.
3. Through 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.

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

Steps of construction:

1. Draw AB=5cm.
2. From A draw a line AY such that ∠BAD = 90o
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.

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

Steps of construction:
1. Draw AB = 6 cm.
2. With A as a centre draw a line AX such that ∠BAX = 45°.
3. With A as a centre and radii  3 cm draw an arc on AD.
4. Now with D and B as a centre and radii 6cm and 3cm draw arcs cutting each other at C
5. Join DC and BC.
ABCD is the required parallelogram.
Here,
∠PAB = ∠APD [Alternate angles]
∠CPB = ∠PBA  [Alternate angles ]
Now,
∠DPA + ∠APB + ∠CPB = 180° ...(i)
Also, considering △APB,
∠PAB + ∠PBA + ∠APB = 180° ...(ii)
Therefore, from (i) and (ii)
∠APB = 90°

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

Steps of construction:

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.

38. Draw parallelogram ABCD with the following data:

AB = 6 cm, AD = 5 cm and ∠DAB = 45o.

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.

To draw the parallelogram follows the steps:

First draw a line AB of measure 6cm. Then draw an angle of measure 45° at point A such that ∠DAB = 45° and AD = 5cm.
Now, draw a line CD parallel to the line AB of measure 6cm. Then join BC to construct the parallelogram as shown below.

Since, O is the midpoint of AC and E is the midpoint of BC, therefore the line is parallel to AB and OE = (1/2)AB.

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