Question

What is the area (in $c{m}^2$) of the quadrilateral formed by joining the mid-points of the sides of a square of area 144 cm${}^2$ ?

• A. 72
• B. 36
• C. 108
• D. 144

Answer 1

The correct option is A 72


Let ABCD be the sqaure whose area is $144c{m}^2$.
then, $A{B}^2=144AB=12cm$

PQRS is the quadrilateral formed by joining the mid-points of the sides of a square ABCD.
$AP=AS=6cm$ (P, S are the mid-points of side AB and AD respectively)

In $△PAS$ using pythagoras theorem we get
$A{P}^2+A{S}^2=P{S}^2\Rightarrow P{S}^2=36+36\Rightarrow P{S}^2=72\Rightarrow PS=6\sqrt{2}cm$

Similary, $PS=PQ=QR=RS=6\sqrt{2}cm$
So, quadrilateral PQRS is a square.

Area of quadrilateral PQRS $=P{Q}^2=(6\sqrt{2}{)}^2=72c{m}^2$