Question

A circle is divided into three sectors. Points P, Q, and R are on the circumference of the circle. Sector POR has an area of $8\pi$, and sector ROQ has an area of $6\pi$. If the radius of circle O is $4$, calculate the measure of the central angle of sector QOP, in degrees.

• A. $45$
• B. $90$
• C. $135$
• D. $180$

Answer 1

The correct option is A $45$

Circumference of circle$=2\pi r$

Area of circle $=\pi r^2=\pi (4{)}^2=16\pi$

Let $\alpha ,\beta$ be the central angles of sector $POR$ and $QOR$ respectively

Area of sector $POR=8\pi$ ...... [Given]

Since, $\frac{\text{Area of sector}}{\text{Total area}}=\frac{\text{Sector angle}}{{360}^o}$

$\therefore \frac{\pi r^2}{2\pi }\times \alpha =8\pi$

$⟹\frac{16\pi }{{360}^o}\times \alpha =8\pi$

$⟹\alpha ={180}^o$

Area of sector $QOR=6\pi$

$⟹\frac{\pi r^2}{2\pi }\times \beta =6\pi$

$⟹\frac{16\pi }{{360}^o}\times \beta =6\pi$

$\Rightarrow \beta ={135}^o$

$\therefore$ Angle of sector $QOP={360}^o-({180}^o+{135}^o)$

$={360}^o-{315}^o$

$={45}^o$

Hence, the answer is ${45}^o$