Construction

The following steps will be helpful before drawing the actual figure.

1. Draw a rough sketch of a figure.
2. Mark the given measurement in it.
3. Analyze the figure and plan the steps.

1. Construction of a parallelogram and a triangle having equal area.

Construction of a parallelogram whose area is equal to the area of given triangle when
(a) One angle (b) one side of the parallelogram are given:

(a) Construct a triangle ABC in which AB = 5cm, BC = 6cm and AC = 7cm and construct a parallelogram whose area is equal to the area of given triangle having one angle 60°.

Steps :

1. DrawΔABC with AB = 5cm, BC = 6cm and AC = 7cm.
2. Draw XY parallel to BC through the point A.
3. Take P as mid-point of BC.
4. Draw an angle of60° at P.
5. Cut PC = QR and join the point R and C.
6. Parallelogram PQRC andΔABC are equal in area.

∴ PQRC is the required parallelogram.

(b) Construct a triangle ABC in which AB = 4cm, BC = 5cm and ∠B = 60° and then construct a parallelogram having a side 5.2 cm and equal area to the triangle.

Steps :

1. DrawΔABC with AB = 4cm, BC = 5cm and∠B = 60°.
2. Draw XY parallel to BC through the point A.
3. Take P as mid-point of BC.
4. From P, cut PQ = 5.2cm on XY.
5. CutPC = PQ and join the point R and C.
6. Parallelogram PQRC andΔABC have equal area.

∴ PQRC is the required parallelogram.

2. Construction of rectangle equals in the area to given triangle.

Construct a triangle ABC in which AB = 6.3 cm, BC = 4.5cm and AC = 3.2ccm then construct a rectangle equal area to the triangle.

Steps:

1. DrawΔABC with AB = 6.3 cm BC = 4.5 cm and AC = 3.2 cm.
2. Through A, draw XY//BC.
3. Draw the perpendicular bisector PQ of BC.
4. Draw BP = RQ and join the points R and B.
5. Rectangle BPQR is the required rectangle equal toΔABC.

∴ BPQR is the required rectangle.

3. Construction of two triangles of equal area on the same base and between the same parallels.

Construct a triangle ABC in which AB = 6.3 cm, BC = 7.8 cm and AC = 7.2 cm and construct another triangle PBC equal area toΔABC.

Steps:

1. Draw ΔABC withAB = 6.3 cm, BC = 7.8 cm and AC = 7.2 cm .
2. Through A, draw XY//BC.
3. Take any point P in XY and join P to B and C.
4. ABC and PBC are the triangles of equal area.

∴ PBC is the required triangle.

4. Construction of two parallelograms of equal area on the same base and between the same parallels.

Construct a parallelogram ABCD in which AB = 5.5 cm, BC = 4.8 cm and∠ABC = 75° and construct another parallelogram equal area to the parallelogram ABCD.

Steps:

1. Draw a parallelogram ABCD having AB = 5.5 cm, BC = 4.8 cm and∠ABC =75°.
2. Take two points R and Q in XY such that BC = RQ.
3. Join R to B and Q to C.
4. BCQR is a parallelogram equal in area to parallelogram ABCD.

∴ BCQR is the required parallelogram.

5. Construction of a triangle equal in area to the given quadrilateral.

Construct a quadrilateral ABCDin which AB = 2.8 cm BC = 3.6 cm, AC = 3 cm, CD = 1.7 cm and AD = 2.3 cm and construct a triangle equal area to the quadrilateral ABCD.

Steps:

1. Draw aquadrilateral ABCDin which BC = 3.6 cm, AB = 2.8 cm, AC = 3 cm,AD = 2.3 cm and CD = 1.7 cm.
2. From D, draw DE parallel to AC.
3. Produce BC to E.
4. Join A to E.
5. ABE is a triangle equal area to the quadrilateral ABCD.

∴ ABE is a required triangle.

6. Construction of a quadrilateral equal in area to the given triangle

Construct a triangle ABC in which a = 7.8cm b =7.2 cm and c = 6.3 cm and construct a quadrilateral having an equal area to the triangle ABC.

Steps:

1. DrawΔABC in which BC = a = 7.8 cm, BA = c = 6.3 cm and AC = b = 7.2 cm.
2. Take any point D on BC.
3. Draw DA//CP.
4. Take any point E on CP.
5. ABDE is a quadrilateral equal area toΔABC.

∴ ABDE is the required quadrilateral.