Construction of Quadrilaterals — Check Your Progress

Construct a quadrilateral ABCD such that AB = 4.5 cm, BC = 4 cm, CD = 3.9 cm, AD = 3.2 cm and ∠B = 60°

Answer

Given, in quadrilateral ABCD, AB = 4.5 cm, BC = 4 cm, CD = 3.9 cm, AD = 3.2 cm and ∠B = 60°.

Steps of construction :

1. Draw a line segment AB = 4.5 cm.

2. At B, construct ∠ABX = 60°.

3. From BX, cut off BC = 4 cm.

4. With A as centre and radius 3.2 cm, draw an arc.

5. With C as centre and radius 3.9 cm, draw another arc cutting the previous arc at D.

6. Join AD and CD.

Hence, ABCD is the required quadrilateral.

Construct a quadrilateral ABCD such that AB = 5 cm, BC = 4.2 cm, AD = 3.5 cm, ∠A = 90° and ∠B = 60°

Answer

Given, in quadrilateral ABCD, AB = 5 cm, BC = 4.2 cm, AD = 3.5 cm, ∠A = 90° and ∠B = 60°.

Steps of construction :

1. Draw a line segment AB = 5 cm.

2. At A, construct ∠BAX = 90°.

3. From AX, cut off AD = 3.5 cm.

4. At B, construct ∠ABY = 60° on the same side of AB as that of D.

5. From BY, cut off BC = 4.2 cm.

6. Join CD.

Hence, ABCD is the required quadrilateral.

Construct a quadrilateral ABCD in which AB = 3.5 cm, BC = 5 cm, CD = 5.6 cm, DA = 4 cm and BD = 5.4 cm

Answer

Given, in quadrilateral ABCD, AB = 3.5 cm, BC = 5 cm, CD = 5.6 cm, DA = 4 cm and BD = 5.4 cm.

Steps of construction :

1. Draw a line segment AB = 3.5 cm.

2. With A as centre and radius 4 cm, draw an arc.

3. With B as centre and radius 5.4 cm, draw another arc cutting the previous arc at D.

4. Join AD and BD.

5. With B as centre and radius 5 cm, draw an arc on the side of BD opposite to that of A.

6. With D as centre and radius 5.6 cm, draw another arc cutting the previous arc at C.

7. Join BC and CD.

Hence, ABCD is the required quadrilateral.

Construct a quadrilateral PQRS in which PQ = 3 cm, QR = 2.5 cm, PS = 3.5 cm, PR = 4 cm and QS = 5 cm

Answer

Given, in quadrilateral PQRS, PQ = 3 cm, QR = 2.5 cm, PS = 3.5 cm, PR = 4 cm and QS = 5 cm.

Steps of construction :

1. Draw a line segment PQ = 3 cm.

2. With P as centre and radius 4 cm, draw an arc.

3. With Q as centre and radius 2.5 cm, draw another arc cutting the previous arc at R.

4. Join PR and QR.

5. With P as centre and radius 3.5 cm, draw an arc on the side of PR opposite to that of Q.

6. With Q as centre and radius 5 cm, draw another arc cutting the previous arc at S.

7. Join PS, RS and QS.

Hence, PQRS is the required quadrilateral.

Construct a quadrilateral ABCD given that BC = 6 cm, CD = 4 cm, ∠B = 45°, ∠C = 90° and ∠D = 120°

Answer

Given, in quadrilateral ABCD, BC = 6 cm, CD = 4 cm, ∠B = 45°, ∠C = 90° and ∠D = 120°.

Steps of construction :

1. Draw a line segment BC = 6 cm.

2. At C, construct ∠BCX = 90°.

3. From CX, cut off CD = 4 cm.

4. At D, construct ∠CDY = 120°.

5. At B, construct ∠CBZ = 45° such that BZ meets DY at A.

Hence, ABCD is the required quadrilateral.

Construct a parallelogram ABCD in which AB = 5.2 cm, AC = 6.5 cm and BD = 7.8 cm

Answer

Given, in parallelogram ABCD, AB = 5.2 cm, AC = 6.5 cm and BD = 7.8 cm.

Let the diagonals AC and BD meet at O.

As the diagonals of a parallelogram bisect each other,

⇒ OA = OC = $\dfrac{AC}{2} = \dfrac{6.5}{2}$ = 3.25 cm

⇒ OB = OD = $\dfrac{BD}{2} = \dfrac{7.8}{2}$ = 3.9 cm

So, in ΔAOB, all the three sides AB, OA and OB are known.

Steps of construction :

1. Draw a line segment AB = 5.2 cm.

2. With A as centre and radius 3.25 cm, draw an arc.

3. With B as centre and radius 3.9 cm, draw another arc cutting the previous arc at O.

4. Join AO and BO.

5. Produce AO to C such that OC = OA = 3.25 cm.

6. Produce BO to D such that OD = OB = 3.9 cm.

7. Join BC, CD and AD.

Hence, ABCD is the required parallelogram.

Construct a parallelogram ABCD in which diagonal AC = 4 cm, diagonal BD = 6 cm and angle between diagonals is 75°

Answer

Given, in parallelogram ABCD, diagonal AC = 4 cm, diagonal BD = 6 cm and the angle between the diagonals is 75°.

Let the diagonals AC and BD meet at O, so that ∠COD = 75°.

As the diagonals of a parallelogram bisect each other,

⇒ OA = OC = $\dfrac{AC}{2} = \dfrac{4}{2}$ = 2 cm

⇒ OB = OD = $\dfrac{BD}{2} = \dfrac{6}{2}$ = 3 cm

Steps of construction :

1. Draw a line segment AO = 2 cm and produce AO to C such that OC = OA = 2 cm.

2. At O, construct ∠COX = 75°.

3. From OX, cut off OD = 3 cm.

4. Produce DO to B such that OB = OD = 3 cm.

5. Join AB, BC, CD and AD.

Hence, ABCD is the required parallelogram.

Construct a rectangle PQRS such that PQ = 3.5 cm and ∠RPS = 75°

Answer

Given, in rectangle PQRS, PQ = 3.5 cm and ∠RPS = 75°.

Steps of construction :

1. Draw a line segment PQ = 3.5 cm.

2. At P, construct ∠QPZ = 90°.

3. At Q, construct ∠PQX = 90°.

4. At P, construct ∠ZPY = 75° such that PY meets QX at R.

5. With R as centre and radius 3.5 cm, draw an arc cutting PZ at S.

6. Join RS.

Hence, PQRS is the required rectangle.

Construct a rectangle whose one diagonal is 6.8 cm and an angle between two diagonals is 105°

Answer

Given, one diagonal of the rectangle is 6.8 cm and the angle between the two diagonals is 105°.

Let the rectangle be ABCD, in which the diagonals AC and BD meet at O and ∠COD = 105°.

As the diagonals of a rectangle are equal and bisect each other,

⇒ BD = AC = 6.8 cm

⇒ OA = OB = OC = OD = $\dfrac{6.8}{2}$ = 3.4 cm

Steps of construction :

1. Draw a line segment AO = 3.4 cm and produce AO to C such that OC = OA = 3.4 cm.

2. At O, construct ∠COX = 105°.

3. From OX, cut off OD = 3.4 cm.

4. Produce DO to B such that OB = OD = 3.4 cm.

5. Join AB, BC, CD and AD.

Hence, ABCD is the required rectangle.

Construct a rhombus BEST such that BE = 4.5 cm, ET = 6 cm

Answer

Given, in rhombus BEST, BE = 4.5 cm and ET = 6 cm.

Here, BE is a side and ET is a diagonal of the rhombus.

As all the sides of a rhombus are equal,

⇒ BE = ES = ST = TB = 4.5 cm

Steps of construction :

1. Draw a line segment BE = 4.5 cm.

2. With B as centre and radius 4.5 cm, draw an arc.

3. With E as centre and radius 6 cm, draw another arc cutting the previous arc at T.

4. With E as centre and radius 4.5 cm, draw an arc on the side of ET opposite to that of B.

5. With T as centre and radius 4.5 cm, draw another arc cutting the previous arc at S.

6. Join BT, ES and ST.

Hence, BEST is the required rhombus.

Construct a rhombus whose diagonals are 7 cm and 5.3 cm

Answer

Given, the diagonals of the rhombus are 7 cm and 5.3 cm.

Let the rhombus be ABCD, in which the diagonals AC = 7 cm and BD = 5.3 cm meet at O.

As the diagonals of a rhombus bisect each other at right angles,

⇒ OA = OC = $\dfrac{7}{2}$ = 3.5 cm

⇒ OB = OD = $\dfrac{5.3}{2}$ = 2.65 cm

⇒ ∠AOB = 90°

Steps of construction :

1. Draw a line segment AC = 7 cm.

2. Draw the perpendicular bisector XY of AC, cutting AC at its mid-point O.

3. From OX, cut off OB = 2.65 cm and from OY, cut off OD = 2.65 cm.

4. Join AB, BC, CD and AD.

Hence, ABCD is the required rhombus.

Construct a square whose one diagonal is 5.8 cm

Answer

Given, one diagonal of the square is 5.8 cm.

Let the square be ABCD, in which the diagonals AC = 5.8 cm and BD meet at O.

As the diagonals of a square are equal and bisect each other at right angles,

⇒ BD = AC = 5.8 cm

⇒ OA = OB = OC = OD = $\dfrac{5.8}{2}$ = 2.9 cm

⇒ ∠AOB = 90°

Steps of construction :

1. Draw a line segment AC = 5.8 cm.

2. Draw the perpendicular bisector XY of AC, cutting AC at its mid-point O.

3. From OX, cut off OB = 2.9 cm and from OY, cut off OD = 2.9 cm.

4. Join AB, BC, CD and AD.

Hence, ABCD is the required square.

Construct a trapezium ABCD in which AB = 7 cm, BC = 4 cm, CD = 3 cm, ∠B = 60° and AB ∥ DC

Answer

Given, in trapezium ABCD, AB = 7 cm, BC = 4 cm, CD = 3 cm, ∠B = 60° and AB ∥ DC.

As AB ∥ DC and BC is a transversal, the co-interior angles ∠B and ∠C are supplementary.

⇒ ∠C = 180° - 60° = 120°

Steps of construction :

1. Draw a line segment AB = 7 cm.

2. At B, construct ∠ABX = 60°.

3. From BX, cut off BC = 4 cm.

4. At C, construct ∠BCY = 120° such that CY is on the same side of BC as that of A.

5. From CY, cut off CD = 3 cm.

6. Join AD.

Hence, ABCD is the required trapezium.

Construct a parallelogram PQRS in which PQ = 6 cm, ∠Q = 120° and ∠RPQ = 30°.

Answer

Given, in parallelogram PQRS, PQ = 6 cm, ∠Q = 120° and ∠RPQ = 30°.

As the opposite sides of a parallelogram are equal,

⇒ RS = PQ = 6 cm and PS = QR

Steps of construction :

1. Draw a line segment PQ = 6 cm.

2. At Q, construct ∠PQX = 120°.

3. At P, construct ∠QPY = 30° such that PY meets QX at R.

4. With R as centre and radius 6 cm, draw an arc.

5. With P as centre and radius = QR, draw another arc cutting the previous arc at S.

6. Join PS and RS.

Hence, PQRS is the required parallelogram.

Construct a rectangle ABCD in which diagonal AC = 7 cm and the diagonals BD and AC intersect at O such that ∠BOC = 45°

Answer

Given, in rectangle ABCD, diagonal AC = 7 cm and the diagonals BD and AC intersect at O such that ∠BOC = 45°.

As the diagonals of a rectangle are equal and bisect each other,

⇒ BD = AC = 7 cm

⇒ OA = OB = OC = OD = $\dfrac{7}{2}$ = 3.5 cm

Steps of construction :

1. Draw a line segment AO = 3.5 cm and produce AO to C such that OC = OA = 3.5 cm.

2. At O, construct ∠COX = 45°.

3. From OX, cut off OB = 3.5 cm.

4. Produce BO to D such that OD = OB = 3.5 cm.

5. Join AB, BC, CD and AD.

Hence, ABCD is the required rectangle.