Question

A rectangle ABCD is inscribed in a circle. Let PQ be the diameter of the circle parallel the side AB. If $\angle BPC$ = ${30}^{\circ }$ , then the ratio of the area of rectangle to the area of circle is

• A. $\frac{\sqrt{3}}{\pi }$
• B. $\frac{\sqrt{3}}{2\pi }$
• C. $\frac{3}{\pi }$
• D. $\frac{\sqrt{3}}{9\pi }$

Answer 1

The correct option is A $\frac{\sqrt{3}}{\pi }$

$\angle BPC={30}^0$

$\angle BOC=2\angle BPC={60}^0$

Angle made by chord at the center of circle is two times the angle made by it on any point on circumference.

$\therefore$ Angle $AOB={180}^0-{60}^0={120}^0$ & $OE\perp AB$, $\therefore \angle LOB={60}^0$

In $△EOB$, $\angle OEB={90}^0$, $\angle EOB={60}^0$, $\therefore$ $\angle OBE={30}^0$

$\&BE=OBcos{30}^0$, In $△OBC$, $OB=OC=BC=r$, (radius)

$\therefore$ $BE=r\times \frac{\sqrt{3}}{2}=\frac{\sqrt{3}r}{2}\&AB=2BE=2\times \frac{\sqrt{3}r}{2}=\sqrt{3}r$

Required ratio $=\frac{\sqrt{3}r\times r}{\pi r^2}=\frac{\sqrt{3}}{\pi }$