Question

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

• A. 2 : 1
• B. 2 : 3
• C. 2 : 5
• D. None of these

Answer 1

The correct option is C 2 : 5

Let the side of the square inscribed in a semi circle be a.
Then radius $r=\sqrt{a^2+\frac{a^2}{4}}$
$\Rightarrow r^2=a^2+\frac{a^2}{4}=\frac{5a^2}{4}\dots \dots \left(1\right)$

Let the side of the square inscribed in the circle of radius r be b. Then
$=(2r{)}^2=b^2+b^2\Rightarrow r^2=\frac{b^2}{2}\dots \dots (2)$
From (1) and (2)
$\frac{5a^2}{4}=\frac{b^2}{2}\Rightarrow \frac{a^2}{b^2}=\frac{2}{5}$
$\therefore$ The required ratio = 2 : 5.