Question

The length of the diagonal of a square is $10\sqrt{2}\mathrm{cm}.$ Its area is
(a) 200 cm²
(b) 100 cm²
(c) 150 cm²
(d) 100$\sqrt{2}$ cm²

Answer 1

(b) 100 cm²

A diagonal of a square forms the hypotenuse of a right-angled triangle with base and height equal to side a.

$$\begin{gathered} {\mathrm{Diagonal}}^2=a^2+a^2 \\ \Rightarrow {\mathrm{Diagonal}}^2=2a^2 \\ \Rightarrow a=\frac{1}{\sqrt{2}}\mathrm{Diagonal} \\ =\frac{1}{\sqrt{2}}\times 10\sqrt{2} \\ =10\mathrm{cm} \end{gathered}$$

∴ Area of the square$=a^2=10\times 10=100{\mathrm{cm}}^2$