OFFICIAL EXPLANATION

$\text { Let } \mathrm{ABCD} \text { be the square whose midpoints are joined to form another square say } \mathrm{PQRS}$




We know that the perimeter of the larger square i.e. $$A B C D$$ is $x$, so we get side of the square ABCD as $$\frac{X}{4}$$ each (Perimeter of square is $$=4 \times$ Side )$.

Using Pythagoras theorem i.e. Hypotenuse $${ }^2=$$ Perpendicular $$^2+$$ Base $$^2$$ in triangle APS, we get

$$\mathrm{PS}=\sqrt{AP^2+AS^2}=\sqrt{ (\frac{x}{8})^2+(\frac{x}{8})^2}=\frac{x \sqrt{2}}{8}$$ (The length of $$\mathrm{AS}=\mathrm{AP}=$$ half of the $\mathrm{AB}=\frac{x}{4}[/m]$ ).

So, the perimeter of the smaller square i.e. $$\mathrm{PQRS}=4 \times \operatorname{Side}$$=4 \times \frac{x\sqrt{2}}{8}=\frac{x\sqrt{2}}{2}=\frac{x}{\sqrt{2}}$$ Column A quantity) which is clearly greater than $$\frac{x}{2}$$ (= column B quantity).

Hence the answer is (A).