Prove that the area of the quadrilateral formed by joining the midpoints of the adjacent sides of a quadrilateral is half the area of the quadrilateral.

February 14, 2020

Prove that the area of the quadrilateral formed by joining the midpoints of the adjacent sides of a quadrilateral is half the area of the quadrilateral.

Let P, Q, R, S be respectively the midpoints of the sides AB, BC, CD, DA of the quadrilateral ABCD

Now, join AC.

In triangle ABC

P and Q are midpoints of AB and BC respectively.

So by midpoint theorem, we have

PQ ll AC and PQ = 1/2 AC

In triangle DAC

S and R are midpoints of AD and DC respectively.

SR ll AC and SR = 1/2 AC.

Therefore, PQ ll SR and PQ = SR.

Hence, PQRS is a parallelogram

Now, join AR which divides triangle ACD into two equal areas.
ar ( AED ) = 1/2 ar ( ACD ) ...................................( 1 )

Again Median RS divides triangle ARD into two triangles of equal area.

Hence, ar ( DSR ) = 1/2 ar( ARD ) ....................... ( 2 )

From ( 1 ) and ( 2 )

ar ( DSR ) = 1/4 ar( ACD )

Similarly, ar ( BQP ) = 1/4 ar( ABC )

Now, ar ( DSR ) + ar ( BQP ) = 1/4

Therefore, ar ( DSR ) + ar ( BQP ) = 1/4 ar( ABCD)...................( 3 )

Similarly, ar ( CRQ ) + ar ( ASP ) = 1/4 (ABCD ) ......................( 4 )

Adding ( 3 ) and ( 4 ), we get

ar ( DSR ) + ar ( BQP ) + ar ( CRQ ) + ar ( ASP ) = 1/2 (ABCD ) ................................. ( 5 )

But,ar ( DSR ) + ar ( BQP ) + ar ( CRQ ) + ar ( ASP ) + ar ( PQRS ) = ar ( ABCD ).... ( 6 )

By subtracting ( 5 ) from ( 6 )

ar ( PQRS ) = 1/2 ar ( ABCD ).