Question

Prove that for any quadrilateral with a pair of opposite sides parallel, the area is half the product of the sum of the lengths of the parallel sides and the distance between them.

Answer 1

Let the given quadrilateral be ABCD in which AD is parallel to BC.

Also, AD = a units, BC = b units

Construction: Draw a line DE parallel to AB and height DF on side BC such that DF = h units.

As AD || BE and DE || AB, ADEB is a parallelogram.

The following diagram depicts the given quadrilateral ABCD with all the measurements in units.

Here, BE = AD = a units (As ADEB is a parallelogram)

EC = BC − BE = (b − a) units

The perpendicular distance between two parallel lines is same, therefore the height of the parallelogram ADEB and triangle DEC is same.

Area of the parallelogram ADEB = BE × DF

= a units × h units

= ah sq. units

Area of the ΔDEC =

Area of the quadrilateral ABCD = Area of the parallelogram ADEB + Area of ΔDEC

= ah sq. units +

Hence, we can say that if a pair of opposite sides of a quadrilateral is parallel then its area is half the product of the sum of the lengths of the parallel sides and the distance between them.