Question

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

Accepted Answer

Given: $ABCD$ is a quadrilateral and by joining its midpoints $E, F, G & H$ results the parallelogram $EFGH$ .

To prove: $Area (EFGH) = \frac{1}{2} Area (ABCD)$ .

Construction: Join $AG$ and also join $AC$ .

Proof: From the figure, we have $AG$ is the median of $△ ACD$ , then $Area (△ ADG) = \frac{1}{2} Area (△ ACD)$ ------ $(1)$

And $DH$ is the median of $△ ADG$ , then $Area (△ DHG) = \frac{1}{2} Area (△ ADG)$ ---------- $(2)$

From equations $(1)$ and $(2)$ we get, $A r e a (△ D H G) = \frac{1}{2} [\frac{1}{2} A r e a (△ A C D)]$

i.e $A r e a (△ D H G) = \frac{1}{4} A r e a (△ A C D)$ --------------------------- $(3)$

Similarly $A r e a (△ B F E) = \frac{1}{4} A r e a (△ A B C)$ ------------- $(4)$

Also, $Area (△ AEH) = \frac{1}{4} Area (△ ABD)$ ---------------- $(5)$

And $Area (△ CGF) = \frac{1}{4} Area (△ BCD)$ -------------- $(6)$

By adding equations $(3)$ & $(4)$ , we get- $A r e a (△ D H G) + A r e a (△ B F E) = \frac{1}{4} A r e a (△ A C D) + \frac{1}{4} A r e a (△ A B C)$

$\Rightarrow A r e a (△ D H G) + A r e a (△ B F E) = \frac{1}{4} [A r e a (△ A C D) + A r e a (△ A B C)] = \frac{1}{4} [A r e a (A B C D)] . . . . . . . . . . . (7)$

Now adding equations $(5)$ and $(6)$ , we get, $[A r e a (△ A E H) + A r e a (△ C G F)] = \frac{1}{4} [A r e a (△ A B D) + A r e a (△ B C D)]$

$\Rightarrow [Area (△ AEH) + Area (△ CGF)] = \frac{1}{4} [Area (ABCD)]$ --------- $(8)$ .

Again add equations $(7)$ & $(8)$ , we get, $Area (△ AEH) + Area (△ CGF) + Area (△ DHG) + Area (△ BFE) = \frac{1}{4} [Area (ABCD)] + \frac{1}{4} [Area (ABCD)] = \frac{1}{2} Area (ABCD) . . . . . . . . . (9)$

Now, $A r e a (E F G H) = A r e a (A B C D) - [A r e a (△ A E H) + A r e a (△ C G F) + A r e a (△ D H G) + A r e a (△ B F E)]$ $\Rightarrow Area (EFGH) = Area (ABCD) - \frac{1}{2} Area (ABCD) = \frac{1}{2} Area (ABCD)$

Therefore $A r e a (E F G H) = \frac{1}{2} A r e a (A B C D)$

Hence, proved the result.

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

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