Question

If the perimeter and the area of a sector of a circle are $25\text{cm}$ and $38.5{\text{cm}}^2$ respectively, then the central angle can be equal to

• A. ${30}^{\circ }$
• B. ${45}^{\circ }$
• C. ${60}^{\circ }$
• D. ${90}^{\circ }$

Answer 1

The correct option is D ${90}^{\circ }$

Let $r$ and $\theta$ be the radius and central angle of a sector of a circle.
Perimeter of sector $=25\text{cm}$,
Area of sector $=38.5{\text{cm}}^2$
Area of sector $=\frac{\theta }{{360}^{\circ }}\times \pi r^2$
$\Rightarrow 38.5=\frac{\theta }{{360}^{\circ }}\times \pi r^2$
$\Rightarrow \frac{\pi \theta }{{360}^{\circ }}=\frac{38.5}{r^2}$ ... (i)

Perimeter of sector $=\frac{\theta }{{360}^{\circ }}\times 2\pi r+2r$
$\Rightarrow 25=2r(\frac{\pi \theta }{{360}^{\circ }}+1)$
$\Rightarrow 25=2r(\frac{38.5}{r^2}+1)$ (From (i))
$\Rightarrow 25=\frac{77}{r}+2r$
$\Rightarrow 2r^2-25r+77=0$
$\Rightarrow 2r^2-14r-11r+77=0$
$\Rightarrow 2r(r-7)-11(r-7)=0$
$\Rightarrow (r-7)(2r-11)=0$
$\Rightarrow r=7\text{cm}$ or $r=\frac{11}{2}\text{cm}$

Substituting the value of $r$, we get
$\Rightarrow \theta =\frac{38.5}{7^2}\times \frac{{360}^{\circ }}{\pi }$ or $\theta =\frac{38.5}{{\left(\frac{11}{2}\right)}^2}\times \frac{{360}^{\circ }}{\pi }$
$\Rightarrow \theta =\frac{38.5}{7\times 7}\times \frac{360}{22}\times 7$ or $\theta =\frac{38.5}{{\left(\frac{11}{2}\right)}^2}\times \frac{{360}^{\circ }}{22}\times 7$
$\Rightarrow \theta ={90}^{\circ }$ or $\theta ={145.78}^{\circ }$
Hence, the correct answer is option d.