Question

The areas of two sectors of two different circles with equal corresponding arc lengths are equal. The given statement is :

• 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

Answer: (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 $C_2.$

Solution:

We know that the length of the arc of a sector of angle $\theta =\frac{\theta }{{360}^o}\times 2\pi r$, where $r$ 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 $\frac{{\theta }_1}{{360}^o}\times 2\pi r=\frac{{\theta }_2}{{360}^o}\times 2\pi \times 2r\Rightarrow {\theta }_1=2{\theta }_2$ ....(i)

$\therefore$ 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 $C_2=\frac{{\theta }_2}{{360}^o}\times \pi {r_2}^2$ ...(iii).

Comparing (ii) & (iii), 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.