If the mid-points of the sides of a quadrilateral are joined in order, prove that the area of the parallelogram so formed will be half of the area of the given quadrilateral.
Solution.
Given that if the mid-points of the sides of a quadrilateral are joined in order, a parallelogram is formed. We have to find the area of this parallelogram.
To prove: ar (parallelogram PFRS) ar(quadrilateral ABCD)
Let ABCD is a quadrilateral and P, F, R and S are the mid-points of the sides BC, CD, AD and AB respectively and PFRS is a parallelogram.
Construction:- Join BD and BR.
We know that median of a triangle divides it into two triangles of equal area.
So, BR divides into two triangles of equal area.
…(i)
Similarly, median RS divides into two triangles of equal area.
…(ii)
From eq. (i) and (ii)
…(iii)
Similarly,
….(iv)
On adding equations (iii) and (iv), we get
(Quadrilateral BCDA) …(v)
Similarly,
….(vi)
On adding (v) and (vi), we get
(quadrilateral BCDA) …(vii)
But
(quadrilateral BCDA) …(viii)
from subtracting eq. (vii) from eq. (viii) we get
ar(parallelogram PFRS) (quadrilateral BCDA)
Hence proved