Question

Prove that the area of a quadrilateral formed by joining the midpoint of the sides of a parallelogram is half the area of athe parallelogram

Answer 1

Because it lies between the same parallel and base there fore it half the area of the parallelogram

Answer 2

Hello,

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 II AC and PQ = 1/2 AC.
In triangle DAC, S and R are midpoints of AD and DC respectively. Therefore,
SR II AC and SR =1/2 AC.
Thus, PQ II SR and PQ = SR.
Hence, PQRS is a II 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(lIgmPQRS)=ar(quad.ABCD)  (6)

Subtracting (5) from (6) , we get
 ar ( lIgm PQRS) = 1/2 ar ( quad. ABCD ). 

bye :-)