Question

A square with side length 1 is inscribed in a semicircle such that one side of the square is on the diameter of the semicircle The perimeter of the semicircle is

• A. $\pi \sqrt{5}$
• B. $\frac{\pi \sqrt{5}}{2}$
• C. $\sqrt{5}(\frac{\pi }{2}+1)$
• D. $\frac{\sqrt{5}}{2}(\pi +1)$

Answer 1

The correct option is C $\sqrt{5}(\frac{\pi }{2}+1)$

If a square is inscribed in a semicircle such that one side of the square is on the diameter of the semi circle then the centre of the semicircle will be the midpoint of the side.
$\therefore OD=OC=radius$
and $OA=OB=\frac{1}{2}$
Now,
$O{D}^2=O{A}^2+A{D}^2$
$=(\frac{1}{2}{)}^2+1$
$=\frac{5}{4}$
$=>OD=\frac{\sqrt{5}}{2}$
$\therefore$ Perimeter of the semicircle=$\pi r+2r$
$=\pi \times \frac{\sqrt{5}}{2}+2\times \frac{\sqrt{5}}{2}$
$=\sqrt{5}(\frac{\pi }{2}+1)$