# Frank Solutions for Chapter 20 Construction of Quadrilaterals Class 9 Mathematics ICSE

**Exercise 20**

**1. Construct a quadrilateral ABCD in which**

**(i) AB = 4.6 cm, BD = 5 cm, AC = 6 cm, CD = 4.2 cm and ****∠****A = 90****°**

**(ii) AB = 7.2 cm, BC = 5.8 cm, CD = 6.3 cm, AD = 4.3 cm and ****∠****A = 75****°**

**(iii) AB = 4.8 cm, AC = 5.8 cm, AD = 3.6 cm, ****∠****A = 105****°**** and ****∠****B = 60****°**

**(iv) AD = AB = 5 cm, BC = 3.8 cm, CD = 3.5 cm and ****∠****BAD = 45****°**

**Answer**

**(i)** AB = 4.6 cm, BD = 5 cm, AC = 6 cm, CD = 4.2 cm and ∠A = 90°

- Draw a line segment AB = 4.6 cm
- Taking A as centre, draw a ray making an angle of 90
^{0}with AB - With B as centre and radius equal to 5 cm cut an arc on the ray from A and mark it as point D
- With D as centre and radius 4.2 cm cut an arc on right side of AD
- Taking A as centre and radius 6 cm cut an arc which meets the arc from D at point C
- Now, join BC
- ABCD is the required quadrilateral

**(ii)** AB = 7.2 cm, BC = 5.8 cm, CD = 6.3 cm, AD = 4.3 cm and ∠A = 75°

- Draw a line segment AB = 7.2 cm
- With A as centre draw rays X and Y to make angles 90° and 60° with AB. Then bisect the angle between them to make an angle of 75° with AB
- With A as centre and radius 4.3 cm cut an arc on line segment making 75° angle with AB and mark it as point D
- With D and B as centres and radii of 6.3 and 5.8 cm respectively, draw arcs cutting each other at point C
- Now, join DC and BC
- ABCD is the required quadrilateral

**(iii)** AB = 4.8 cm, AC = 5.8 cm, AD = 3.6 cm, ∠A = 105° and ∠B = 60°

Steps of Construction:

- Draw a line segment AB = 4.8 cm
- With A as centre draw rays X and Y to make angles 60
^{0}and 90^{0}with AB produced. Then bisect the angle between them to make an angle of 105^{0}with AB - With A as centre and radius 3.6 cm cut an arc on line segment making 105
^{0}angle with AB and name it as point D - With B as centre draw a ray making an angle of 60
^{0}with AB - With A as centre and radius 5.8 cm cut an arc on the ray from B and mark the point as C
- Now, join BC and DC
- ABCD is the required quadrilateral

**(iv)** AD = AB = 5 cm, BC = 3.8 cm, CD = 3.5 cm and ∠BAD = 45°

Steps of Construction:

- Draw a line segment AB = 5 cm
- With A as centre draw an angle of 90° and bisect it to form ∠BAD = 45°
- With A as centre and radius 5 cm cut an arc on the ray making an angle of 45° with AB and name it as point D
- With D and B as centre and radii as 3.5 cm and 3.8 cm respectively draw arcs intersecting each other at point C
- Now, join DC and BC
- ABCD is the required quadrilateral

**2. Construct a trapezium ABCD in which**

**(i) AB = 3.5 cm, BC = 6 cm, CD = 3.5 cm, AD = 4.4 cm and AD || BC**

**(ii) AB = 4.6 cm, BC = 6.4 cm, CD = 5.6 cm, ****∠****B = 60****°**** and AD || BC**

**Answer**

**(i)** AB = 3.5 cm, BC = 6 cm, CD = 3.5 cm, AD = 4.4 cm and AD || BC

Steps of Construction:

- Draw BC of length 6 cm
- From BC, cut BE = 4.4 cm
- Draw a triangle DEC, such that DE = AB = 3.5 cm and CD = 3.5 cm
- With B and D as centres and radii 3.5 cm and 4.4 cm respectively, draw arcs cutting each other at point A
- Now, join AB and AD
- Thus, ABCD is the required trapezium

**(ii)** AB = 4.6 cm, BC = 6.4 cm, CD = 5.6 cm, ∠B = 60° and AD || BC

Steps of Construction:

- Draw BC of length 6.4 cm
- With B as centre, draw an angle of 60° and cut an arc with radius 4.6 cm. Name this point as A
- From point A, draw a line segment parallel to BC
- With C as centre and radius 5.6 cm cut an arc on the line segment parallel to BC. Name this point as D
- Join CD
- Thus, ABCD is the required trapezium

**3. Construct a parallelogram ABCD in which AB = 4.5 cm, ****∠****A = 105****°**^{ }and the distance between AB and CD is 3.2 cm

**Answer**

- Draw a line segment AB = 4.5 cm
- At B, draw BX perpendicular to AB
- From B, cut BR = 3.2 cm on BX which is the distance between AB and CD
- Through R, draw a line perpendicular to BX to get QR parallel to AB
- Taking A as centre, draw a ray AP making an angle of 105° with AB and meeting QR at point D
- Taking B as centre, draw an arc with radius = AD on QR and name it as point C
- Now, join BC
- ABCD is the required parallelogram

**4. Construct a rhombus whose perimeter is 16 cm and BD = 6.2 cm**

**Answer**

Therefore,

Perimeter = side×4

Side = perimeter/4

Side = 16/4

We get,

Side = 4 cm

Steps of Construction:

- Draw a line segment BD = 6.2 cm
- Taking B as centre and radius 4 cm, draw two arcs one above BD and the other below BD
- Taking D as centre and radius 4 cm draw two arcs one above BD and the other below BD intersecting the previous arcs drawn in step 2 at points A and C respectively
- Now, join AB, BC, CD and AD
- ABCD is the required rhombus

**5. Construct a rhombus whose diagonals AC = 7.4 cm and BD = 6 cm**

**Answer**

Steps of Construction:

- Draw a line segment AC = 7.4 cm
- Draw perpendicular bisector to AC which cuts AC at point O
- From this perpendicular cut OD and OB such that OD = OB = (1/2) BD = (1/2)×6 cm = 3 cm
- Now, join AB, BC, CD and AD
- ABCD is the required rhombus

**6. Construct a rhombus whose side AB = 5 cm and diagonal AC = 6 cm. Measure DB and AD.**

**Answer**

Steps of Construction:

- Draw AC of length 6 cm
- With A as centre and radius 5 cm, draw two arcs one above AC and the other below AC
- Taking C as centre and radius 5 cm draw two arcs one above AC and the other below AC intersecting the previous arcs of Step 2 at points B and D respectively
- Now, join AB, BC, CD and AD
- ABCD is the required rhombus
- On measuring we get, AD = 5 cm and DB = 8 cm

**7. Construct a square with each side 4.3 cm**

**Answer**

Steps of Construction:

- Draw PQ of length 4.3 cm
- Construct ∠PQT = 90° at Q
- From QT cut off QR = 4.3 cm
- From P and R, draw two arcs of radii 4.3 cm each to cut each other at point S
- Now, join PS and RS
- PQRS is the required square

**8. Construct a square ABCD with AC = 6.5 cm**

**Answer**

Steps of Construction:

- Draw AC of length 6.5 cm
- Draw perpendicular bisector to AC which cuts AC at point O
- From this perpendicular cut OD and OB such that OD = OB = (1/2) BD = (1/2)×6.5 = 3.25 cm
- Now, join AB, BC, CD and AD
- ABCD is the required square

**9. Construct a square with perimeter = 18 cm**

**Answer**

Perimeter = 4×side

Side = perimeter/4

⇒ Side = 18/4

We get,

Side = 4.5 cm

Steps of Construction:

- Draw PQ of length 4.5 cm
- Construct ∠PQT = 90° at Q
- From QT cut off QR = 4.5 cm
- From P and R, draw two arcs of radii 4.5 cm each to cut each other at point S
- Now, join PS and RS
- PQRS is the required square

**10. Construct a square whose area is 25 sq. cm.**

**Answer**

- Draw PQ of length 5 cm
- Construct ∠PQT = 90° at Q
- From QT cut off QR = 5 cm
- From P and R, draw two arcs of radii 5 cm each to cut each other at point S
- Now, join PS and RS
- PQRS is the required square

**11. Construct a rectangle ABCD with one diagonal AC = 5.8 cm and the acute angle between the diagonals is equal to 45****°**

**Answer**

- Draw AC of length 5.8 cm and locate its midpoint O
- Draw line BOD such that ∠DOC = 45° and OB = OD = (1/2) BD = (1/2)°5.8 cm = 2.9 cm
- Now, join AB, BC, CD and DA
- Thus ABCD is the required rectangle

**12. Construct a rectangle ABCD with perimeter 18 cm and AB = 6 cm**

**Answer**

Opposite sides of a rectangle are equal

Hence,

AB = CD and BC = DA

Perimeter of rectangle = AB + BC + CD + DA

⇒ 18 cm = AB + BC + AB + BC

⇒ 18 cm = 6 + BC + 6 + BC

⇒ (18 – 12) cm = 2BC

We get,

BC = 3 cm

Hence,

AB = CD = 6 cm and BC = DA = 3 cm

Steps of Construction:

- Draw AB = 6 cm
- On A and B draw perpendiculars AX and BY to AB
- Taking A and B as centres and radii 3 cm draw arcs on AX and BY. Mark them as points D and C respectively
- Now, join CD
- Therefore, ABCD is the required rectangle

**13. Construct a rectangle ABCD, AB = 6 cm, ****∠****CAB = 30****°**

**Answer**

- Draw AB = 6 cm
- On A and B draw perpendiculars AX and BY to AB
- Taking A as centre, draw a line making an angle of 30° with AB and intersecting BY at point C
- With A as centre and radius = BC cut an arc on AX. Name it as point D
- Join CD
- Hence, ABCD is the required rectangle

**14. Construct a regular hexagon of side 3.5 cm**

**Answer**

- Draw a circle with radius of length 3.5 cm
- Take a point A on the circle. With A as centre and radius 3.5 cm cut the circle at B and from B with radius 3.5 cm cut the circle at C and so on
- Now, join AB, BC, CD, DE, EF and AF
- Therefore, ABCDEF is the required regular hexagon

**15. Construct a regular hexagon of side 4 cm. **

**Answer**

Steps of Construction:

- Draw AB of length 4 cm
- With centres A and B and radius 4 cm draw arcs to cut each other at O
- Taking O as centre and the radius 4 cm cut the arc drawn in step 2 at C and F. Now, join AF and BC
- With centres C and F and radius 4 cm cut the arcs drawn in step 3 at D and E. Then join CD, DE and EF
- Thus ABCDEF is the required regular hexagon

**16. Construct a parallelogram ABCD. Given: **

**(i) AB = 5.5 cm, BC = 3.5 cm, ****∠****B = 75****°**

**(ii) AB = 3.5 cm, AC = 4 cm and BD = 5.2 cm **

**Answer**

**(i)** AB = 5.5 cm, BC = 3.5 cm, ∠B = 75°

**(ii)** AB = 3.5 cm, AC = 4 cm and BD = 5.2 cm

**17. Construct a parallelogram PQRS in which PQ = 6.4 cm, QR = 4 cm and the distance between PQ and SR is 3 cm. **

**Answer**

**18. Construct a rhombus ABCD, AB = 3.8 cm and ****∠****A = ****60°. Measure AC. **

**Answer**

**19. Construct a parallelogram PQRS in which PQ = 6.4 cm, QR = 4 cm and the distance between PQ and SR is 3 cm. **

**Answer**

**20. (a) Construct a rectangle PQRS, when its Area = 21 c****m ^{2}**

**and length = 4.2 cm**

**(b) Construct a rectangle PQRs, when its Area = 33.8 c****m ^{2}**

**and breadth = 6.5 cm**

**Answer**

(a)

(b)