The midpoint of the sides of a quadrilateral are joined in order, prove that the area of the parallelogram so - Maths - Areas of Parallelograms and Triangles

Question

The midpoint 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 are of given quadrilateral!!

Shobhitanshu · Accepted Answer

Let P,Q,R,S be respectively the midpoints of the sides
AB, BC, CD, DA of quad ABCD and let PQRS be formed by joining the
midpoints
Now, join AC and AR
In triangle ABC, P and Q are midpoints of AB and BC
respectively
So, PQ ll AC and PQ = 1/2 AC
In triangle DAC, S and R are midpoints of AD and DC
respectively
Therefore, SR ll AC and SR = 1/2 AC
Thus, PQ ll SR and PQ = SR
Hence, PQRS is a ll gm
Now, median AR divides triangle ACD into two equal
area
Therefore, ar ( ARD ) = 1/2 ar ( ACD ) ( 1
)
Median RS divides triangle ARD into two triangle of
equal area
Hence, ar ( DSR ) = 1/2 ( ARD ) ( 2 )
From ( 1 ) and ( 2 ), we get : ar ( DSR ) = 1/4 ( ACD
)
Similarly, ar ( BQP ) = 1/4 ( ABC )
Now, ar ( DSR ) + ar ( BQP ) = 1/4 [ ar ( ACD ) + ar (
ABC )]
Therefore, ar ( DSR ) + ar ( BQP ) = 1/4 [ quad ABCD] (
3 )
Similarly, ar ( CRQ ) + ar ( ASP ) = 1/4 ( quad ABCD ) (
4 )
Adding ( 3 ) and ( 4 ) , we get
ar ( DSR ) + ar ( BQP ) + ar ( CRQ ) + ar ( ASP ) = 1/2
( quad ABCD ) ( 5 )
But, ar ( DSR ) + ar ( BQP ) + ar ( CRQ ) + ar ( ASP ) +
ar ( llgm PQRS ) = ar ( quad ABCD ) ( 6 )
Subtracting ( 5 ) from ( 6 ) , we get
ar ( llgm PQRS ) = 1/2 ar ( quad ABCD )

The midpoint 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 are of given quadrilateral!!