Question

prove that the area of a quadrilateral formed by joining the mid point of the sides of a parallelogram is half the area of the parallelogram

Answer 1

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.

In triangle ABC, P and Q are midpoints of AB and BC respectively.

So by mid point theorem,
PQ ll AC and PQ = 1/2 AC.

Now,
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, join AR which divides triangle ACD into two equal area.

∴ar ( AED ) = 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 ).



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Answer 2

Dear friend, please refer to the image.
If you have any more doubts, please don't hesitate to ask.


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