Rectilinear Figures — Exercise 12.2

Using ruler and compasses only, construct the quadrilateral ABCD in which ∠BAD = 45°, AD = AB = 6cm, BC = 3.6cm, CD = 5cm. Measure ∠BCD.

Answer

Steps of construction:

1. Draw a line segment AB = 6 cm
2. At A, construct ∠BAX = 45°.
3. From AX, cut off AD = 6 cm.
4. With B as centre and radius 3.6 cm, draw an arc.
5. With D as centre and radius 5 cm, draw an arc to meet the previous arc at C.
6. Join BC and DC. Then, ABCD is the required quadrilateral.

On measuring ∠BCD, it is 63°.

Hence, ∠BCD = 63°.

Draw a quadrilateral ABCD with AB = 6 cm, BC = 4 cm, CD = 4 cm and ∠ABC = ∠BCD = 90°.

Answer

Steps of construction:

1. Draw a line segment BC = 4 cm.
2. At B, construct ∠CBX = 90° and cut off BA = 6 cm.
3. At C, construct ∠BCY = 90° and cut off CD = 4 cm.
4. Join AD. Then, ABCD is the required quadrilateral.

Using ruler and compasses only, construct the quadrilateral ABCD given that AB = 5 cm, BC = 2.5 cm, CD = 6 cm, ∠BAD = 90° and the diagonal AC = 5.5 cm.

Answer

Steps of construction:

1. Draw a line segment AB = 5cm.
2. With centre A and radius 5.5 cm, draw an arc.
3. With centre B and radius 2.5 cm, draw an arc to meet the previous arc at C.
4. Join AC and BC.
5. At A, construct ∠BAX = 90°.
6. With centre C and radius 6 cm, draw an arc intersecting AX at D.
7. Join CD. Then, ABCD is the required quadrilateral.

Construct a quadrilateral ABCD in which AB = 3.3 cm, BC = 4.9 cm, CD = 5.8 cm, DA = 4 cm and BD = 5.3 cm.

Answer

Steps of construction:

1. Draw a line segment AB = 3.3 cm.
2. With centre A and radius 4 cm, draw an arc.
3. With centre B and radius 5.3 cm, draw an arc to meet the previous arc at D.
4. Join AD and BD.
5. With centre B and radius 4.9 cm, draw an arc.
6. With centre D and radius 5.8 cm, draw an arc to meet the previous arc at C.
7. Join BC and DC. Then ABCD is the required quadrilateral.

Construct a trapezium ABCD in which AD || BC, AB = CD = 3 cm, BC = 5.2 cm and AD = 4 cm.

Answer

Steps of construction:

1. Draw a line segment BC = 5.2 cm.
2. From BC, cut off BE = 4 cm.
3. With centre E and radius 3 cm, draw an arc.
4. With centre C and radius 3 cm, draw an arc to meet the previous arc at D.
5. Join ED and CD.
6. With centre D and radius 4 cm, draw an arc.
7. With centre B and radius 3 cm, draw an arc to meet the previous arc at A.
8. Join BA and DA. Then ABCD is the required quadrilateral.

Construct a trapezium ABCD in which AD || BC, ∠B = 60°, AB = 5 cm, BC = 6.2 cm and CD = 4.8 cm.

Answer

Steps of construction:

1. Draw a line segment BC = 6.2 cm.
2. At B, construct ∠CBX = 60°.
3. With centre B and radius 5 cm, draw an arc intersecting BX at A.
4. From A, draw a line AY parallel to BC.
5. With centre C and radius 4.8 cm, draw an arc which intersects AY at D.
6. Join CD. Then ABCD is the required trapezium.

Using ruler and compasses only, construct a parallelogram ABCD with AB = 5.1 cm, BC = 7 cm and ∠ABC = 75°.

Answer

Steps of construction:

1. Draw a line segment BC = 7 cm.
2. At B, construct ∠CBX = 75°.
3. With B as center and radius 5.1 cm, draw an arc intersecting BX at A.
4. With A as center and radius 7 cm, draw an arc.
5. With C as center and radius 5.1 cm, draw an arc to meet the previous arc at D.
6. Join AD and CD. Then ABCD is the required parallelogram.

Using ruler and compasses only, construct a parallelogram ABCD in which AB = 4.6 cm, BC = 3.2 cm and AC = 6.1 cm.

Answer

Steps of construction:

1. Draw a line segment AB = 4.6 cm
2. With A as center and radius 6.1 cm, draw an arc.
3. With B as center and radius 3.2 cm, draw an arc to meet the previous arc at C.
4. Join AC and BC.
5. Again, with A as center and radius 3.2 cm, draw an arc.
6. With C as center and radius 4.6 cm, draw an arc to meet the previous arc at D.
7. Join AD and CD. Then ABCD is the required parallelogram.

Using ruler and compasses, construct a parallelogram ABCD given that AB = 4 cm, AC = 10 cm, BD = 6 cm. Measure BC.

Answer

Steps of construction:

1. Draw a line segment AB = 4 cm.
2. Construct triangle OAB such that
   OA = $\dfrac{1}{2}$ x AC = 5cm
   OB = $\dfrac{1}{2}$ x BD = 3cm
   With A as center and radius 5 cm and with centre B and radius 3 cm, draw arcs intersecting each other at O.
   As, diagonals of || gm bisect each other.
3. Produce AO to C such that OA = OC = 5cm.
4. Produce BO to D such that OB = OD = 3cm.
5. Join AD, BC and CD. Then ABCD is the required parallelogram.

On measuring, BC = 7.2 cm.

Hence, BC = 7.2 cm.

Using ruler and compasses only, construct a parallelogram ABCD such that BC = 4 cm, diagonal AC = 8.6 cm and diagonal BD = 4.4 cm. Measure the side AB.

Answer

Steps of construction:

1. Draw a line segment BC = 4 cm.
2. Construct triangle OBC such that
   OB = $\dfrac{1}{2}$ x BD = 2.2 cm
   OC = $\dfrac{1}{2}$ x AC = 4.3 cm
3. With B as center and radius 2.2 cm and with centre C and radius 4.3 cm, draw arcs intersecting each other at O.
   Since, diagonals of || gm bisect each other.
4. Produce BO to D such that BO = OD = 2.2 cm.
5. Produce CO to A such that CO = OA = 4.3 cm.
6. Join AB, AD and CD. Then ABCD is the required parallelogram.

On measuring AB = 5.6 cm.

Hence, AB = 5.6 cm.

Use ruler and compasses to construct a parallelogram with diagonals 6 cm and 8 cm in length having given the acute angle between them is 60°. Measure one of the longer sides.

Answer

Steps of construction:

Let AC = 6 cm and BD = 8 cm.

1. Draw AO = $\dfrac{1}{2}AC = \dfrac{1}{2} \times 6 = 3\text{ cm and}$ produce AO to C such that OC = OA.
2. At O construct ∠COP = 60°.
3. From OP cut off OD = $\dfrac{1}{2} BD = \dfrac{1}{2} \times 8 = 4$ cm, produce DO to B such that OB = OD.
4. Join AB, BC, CD and DA. Then ABCD is the required parallelogram.

On measuring the length of longer side AD = BC = 6.1 cm.

Hence, length of longer side = 6.1 cm.

Using ruler and compasses only, draw a parallelogram whose diagonals are 4 cm and 6 cm long and contain an angle of 75°. Measure and write down the length of one of the shorter sides of the parallelogram.

Answer

Steps of construction:

Let AC = 4 cm and BD = 6 cm.

1. Draw AO = $\dfrac{1}{2}AC = \dfrac{1}{2} \times 4 = 2\text{ cm and}$ produce AO to C such that OC = OA.
2. At O construct ∠COP = 75°.
3. From OP cut off OD = $\dfrac{1}{2} BD = \dfrac{1}{2} \times 6 = 3$ cm, produce DO to B such that OB = OD.
4. Join AB, BC, CD and DA. Then ABCD is the required parallelogram.

On measuring shorter sides, we get

AB = CD = 3.1 cm.

Hence, length of shorter side = 3.1 cm.

Using ruler and compasses only, construct a parallelogram ABCD with AB = 6 cm, altitude = 3.5 cm and side BC = 4 cm. Measure the acute angles of the parallelogram.

Answer

Steps of construction:

1. Draw AB = 6 cm.
2. At B, draw BP ⊥ AB.
3. From BP, cut BE = 3.5 cm = height of || gm.
4. Through E draw QR parallel to AB.
5. With B as centre and radius = 4 cm draw an arc which cuts QR at C.
   Since, opposite sides of || gm are equal.
   So, AD = BC = 4 cm
6. With A as centre and radius = 4 cm draw an arc which cuts QR at D.
7. Join AD, BC and CD. Then ABCD is the required parallelogram.

On measuring the acute angle of parallelogram it is equal to 61°.

Hence, acute angle = 61°.

The perpendicular distances between the pairs of opposite sides of a parallelogram ABCD are 3 cm and 4 cm and one of its angles measures 60°. Using ruler and compasses only, construct ABCD.

Answer

Steps of construction:

1. Draw a straight-line PQ, take a point A on it.
2. At A, construct ∠QAF = 60°
3. At A, draw AE ⊥ PQ from AE cut off AN = 3 cm
4. Through N draw a straight line to PQ to meet AF at D.
5. At A, draw AG ⊥ AD, from AG cut off AM = 4 cm.
6. Through M, draw a straight line parallel to AD to meet AQ at B and ND at C.
7. Join AB, BC, CD and DA. Then, ABCD is the required parallelogram.

Using ruler and compasses, construct a rectangle ABCD with AB = 5 cm and AD = 3 cm.

Answer

Steps of construction:

1. Draw a straight-line AB = 5 cm.
2. At A and B construct ∠XAB = ∠YBA = 90°.
3. From XA and YB cut off AD and BC = 3 cm each.
4. Join CD. Then ABCD is the required rectangle.

Using ruler and compasses only, construct a rectangle each of whose diagonals measures 6 cm and the diagonals intersect at an angle of 45°.

Answer

Steps of construction:

1. Draw a line segment AC = 6 cm.
2. Bisect AC at O.
3. At O, draw a ray XY making an angle of 45° at O.
4. From XY, cut off OB = OD = $\dfrac{6}{2}$ = 3 cm each.
5. Join AB, BC CD and DA. Then ABCD is the required rectangle.

Using ruler and compasses only, construct a square having a diagonal of length 5 cm. Measure its sides correct to the nearest millimeter.

Answer

Steps of construction:

1. Draw a line segment AC = 5 cm.
2. Draw its perpendicular bisector XY bisecting it at O.
3. From XY, cut off OB = OD = $\dfrac{5}{2}$ = 2.5 cm.
4. Join AB, BC, CD and DA. Then ABCD is the required square.

On measuring, each side = 3.5 cm.

Hence, length of side of square = 3.5 cm.

Using ruler and compasses only construct a rhombus ABCD, given that AB = 5 cm, AC = 6 cm. Measure ∠BAD.

Answer

Steps of construction:

1. Draw a line segment AB = 5 cm.
2. With centre A and radius 6 cm, with centre B and radius 5cm, draw arcs intersecting each other at C.
3. Join AC and BC.
4. With centre A and C and radius 5cm, draw arcs intersecting each other at D.
5. Join AD and CD. Then ABCD is the required rhombus.

On measuring, ∠BAD = 106°.

Hence, ∠BAD = 106°.

Using ruler and compasses only, construct rhombus ABCD with sides of length 4 cm and diagonal AC of length 5 cm. Measure ∠ABC.

Answer

Steps of construction:

1. Draw a line segment AC = 5 cm.
2. With centre A and C and radius 4 cm, draw arcs intersecting each other above and below AC at D and B.
3. Join AB, BC, CD, DA and BD. The ABCD is the required rhombus.

On measuring, ∠ABC = 78°.

Hence, ∠ABC = 78°.

Construct a rhombus PQRS whose diagonals PR, QS are 8 cm and 6 cm respectively.

Answer

Steps of construction:

1. Draw a line segment PR = 8 cm.
2. Draw its perpendicular bisector XY intersecting it at O.
3. From XY, cut off OQ = OS = $\dfrac{6}{2}$ = 3 cm each.
4. Join PQ, QR, RS and SP. Then PQRS is the required rhombus.

Construct a rhombus ABCD of side 4.6 cm and ∠BCD = 135°, by using ruler and compasses only.

Answer

Steps of construction:

1. Draw a line segment BC = 4.6 cm.
2. At C, draw a ray CX making an angle of 135° and cut off CD = 4.6 cm.
3. With centres B and D, and radius 4.6 cm draw arcs intersecting each other at A.
4. Join BA and DA. Then ABCD is the required rhombus.

Construct a trapezium in which AB || CD, AB = 4.6 cm, ∠ABC = 90°, ∠DAB = 120° and the distance between parallel sides is 2.9 cm.

Answer

Steps of construction:

1. Draw a line segment AB = 4.6 cm.
2. At B, draw a ray BZ making an angle of 90° and cut off BC = 2.9 cm. (distance between AB and CD)
3. At C, draw a parallel line XY to AB.
4. At A, draw a ray making an angle of 120° meeting XY at D.
5. Join AD, DC. Then ABCD is the required trapezium.

Construct a trapezium ABCD when one of parallel sides AB = 4.8 cm, height = 2.6 cm, BC = 3.1 cm and AD = 3.6 cm.

Answer

Step construction:

1. Draw a line segment AB = 4.8 cm.
2. At A, draw a ray AZ making an angle of 90° and cut off AL = 2.6 cm.
3. At L, draw a ray LX parallel to AB.
4. With centre A and radius 3.6 cm and with centre B and radius 3.1 cm, draw arcs intersecting LX at D and C respectively.
5. Join AD and BC. Then ABCD is the required trapezium.

Construct a regular hexagon of side 2.5 cm.

Answer

Steps of construction:

1. With O as centre and radius = 2.5 cm, draw a circle
2. Take any point A on the circumference of circle.
3. With A as centre and radius = 2.5 cm, draw an arc which cuts the circumference of circle at B.
4. With B as centre and radius = 2.5 cm, draw an arc which cuts the circumference of circle at C.
5. With C as centre and radius = 2.5 cm, draw an arc which cuts the circumference of circle at D.
6. With D as centre and radius = 2.5 cm, draw an arc which cuts the circumference of circle at E.
7. With E as centre and radius = 2.5 cm, draw an arc which cuts the circumference of circle at F.
8. Join AB, BC, CD, DE, EF and FA. Then ABCDEF is the required hexagon.