Question

The ratio of the area of a square inscribed in a semicircle to that of the area of a square inscribed in the circle of the same radius is

Answer 1

Let $r$ be radius of semi circle.

$h$ be side of square. (Ref. image 1)

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

$\frac{h^2}{4}+h^2=r^2\Rightarrow \frac{5h^2}{4}=r^2$

$\Rightarrow h^2=\frac{4r^2}{5}$ ……$\left(1\right)$

from $\left(1\right)$ $r=\frac{\sqrt{5}h}{2}$

Same radius (Ref. image 2)

$(2r{)}^2=x^2+x^2$

$4r^2=2x^2$

Area of square $\Rightarrow x^2=2r^2$……..$\left(2\right)$

$\therefore$ Ratio $=\frac{\left(1\right)}{\left(2\right)}$ $=\frac{\frac{4r^2}{5}}{2r^2}\Rightarrow \frac{2}{5}$.