Question

The area of the square inscribed in a semicircle is $2c{m}^2$. If the semi-circle is extended to a full circle, the area of the square inscribed in this circle is

• A. $6c{m}^2$
• B. $7c{m}^2$
• C. $4c{m}^2$
• D. $5c{m}^2$

Answer 1

The correct option is D

$5c{m}^2$

Area of square = $2c{m}^2$

Side of square = $\sqrt{2}cm$

Let r be the radius of the semicircle.

Then from the figure,

$r^2=(\sqrt{2}{)}^2+{\left(\frac{\sqrt{2}}{2}\right)}^2$

$\Rightarrow r^2=2+\left(\frac{2}{4}\right)$

$\Rightarrow r^2=\frac{5}{2}$

$\Rightarrow r=\sqrt{\left(\frac{5}{2}\right)}$

When the semicircle is extended to form a full circle, the diagonal of the square inscribed in it will be the diameter of the circle.

$r=\sqrt{\left(\frac{5}{2}\right)}$

So, diameter $=2\times \sqrt{\left(\frac{5}{2}\right)}$

Diagonal of square $=\text{side}\times \sqrt{2}$

$\Rightarrow \text{side}\times \sqrt{2}=2\times \sqrt{\left(\frac{5}{2}\right)}$

$\Rightarrow \text{side}\times 2=2\times \sqrt{5}$

$\Rightarrow \text{side}=\sqrt{5}$

Area of square $=sid{e}^2$

$\Rightarrow$ Area $=(\sqrt{5}{)}^2$

$=5c{m}^2$