Question

Find the ratio of the area of a square inscribed in a semi circle of radius $r$ to the area of another square inscribed in the entire circle of radius $r$.

• A. $2:1$
• B. $3:2$
• C. $2:5$
• D. $3:5$

Answer 1

The correct option is C $2:5$

Let $ABCD$ be the square inscribed in the semi-circle with center O, and side CD the diameter of the semi-circle.
Let the point M be the mid-point of AB.
Now, join OB and OM to get the right $△OMB$ with OB as the hypotenuse.
Therefore, $O{M}^2+M{B}^2=O{B}^2$
$OM=$ side of the square inscribed in the semi-circle,
$MB=$ half of the side; and
$OB=$ radius
Let the side of the square be $x$.
$x^2+(\frac{x}{2}{)}^2=r^2$
$x^2=\frac{4r^2}{5}$
Hence, the area of the semi-circle $=\frac{4r^2}{5}$
Now diagonal of the square inscribed in the circle $=2r$
Therefore, its area $=\frac{(2r{)}^2}{2}$ $=2r^2$
Area of the square $=\frac{diagona{l}^2}{2}$
Hence, the required ratio $=\frac{4r^2}{5}:2r^2$$=\frac{2}{5}:1$ $=2:5$