Question

The areas of two sectors of two different circles are equal. Is it necessary that their corresponding arc lengths are equal? Why?

• A. False, since it requires the condition $2{r_1}^2<{r_2}^2$
• B. True, since given that $2{r_1}^2={r_2}^2.$
• C. False, since it requires the condition $2{r_1}^2={r_2}^2.$
• D. True, since given that $2{r_1}^2>{r_2}^2.$

Answer 1

The correct option is C False, since it requires the condition $2{r_1}^2={r_2}^2.$

Given: The radius of the circle $C_1=r_1$ and the radius of the circle $C_2=r_2.$

The length $l_1$ of an arc of $C_1=$ the length $l_2$ of an arc of $C_2.$

The angle of the corresponding sector of $C_1={\theta }_1$ and the angle of the corresponding sector of $C_2={\theta }_2$

To find out the validity of the statement that the area of the corresponding sector of$C_1=$ the area of the corresponding sector of $C_2.$

We know that the length of the arc of a sector of angle $\theta$ $=\frac{\theta }{{360}^o}\times 2\pi r$, when $r$ is the radius of the circle.

$\therefore l_1=\frac{{\theta }_1}{{360}^o}\times 2\pi r\&l_2=\frac{{\theta }_2}{{360}^o}\times 2\pi \times 2r.$

Since $l_1=l_2,$ we have

$\frac{{\theta }_1}{{360}^o}\times 2\pi r=\frac{{\theta }_2}{{360}^o}\times 2\pi \times 2r⟹{\theta }_1=2{\theta }_2.......\left(i\right)$

$\therefore$ area of corresponding sector of $C_1=\frac{{\theta }_1}{{360}^o}\times \pi {r_1}^2=\frac{2{\theta }_2}{{360}^o}\times \pi {r_1}^2$ (from i).........(ii)

And area of corresponding sector of $C_2=\frac{{\theta }_2}{{360}^o}\times \pi {r_2}^2............\left(iii\right).$

Comparing $\left(ii\right)\&\left(iii\right)$, we have

$\frac{\text{area of corresponding sector of}{C}_1}{\text{area of corresponding sector of}{C}_2}=\frac{\frac{2{\theta }_2}{{360}^o}\times \pi {r_1}^2}{\frac{{\theta }_2}{{360}^o}\times \pi {r_2}^2}=\frac{2{r_1}^2}{{r_2}^2}.$

So the areas will be equal if and only if $2{r_1}^2={r_2}^2.$

So the given statement is not correct.