A sector of a circle has a central angle of 150° and radius 10 cm. Another sector of the same circle has a central angle of $\frac{5\pi }{6}$ radians. What is the ratio of the area of the first sector to that of the second?

Correct Answer: Option 1 : 1

Step-by-step Explanation:

Step 1: Understand the given information

We are given a circle of radius $r=10\text{cm}$.

The first sector has a central angle of ${\theta }_1=150°$.

The second sector of the same circle has a central angle of ${\theta }_2=\frac{5\pi }{6}$ radians.

Step 2: Convert the central angle of the second sector to degrees (or the first to radians)

To compare the two sectors easily, let us convert the angle of the second sector from radians to degrees using the relation $1\text{radian}=\frac{180°}{\pi }$:

${\theta }_2=\frac{5\pi }{6}\times \frac{180°}{\pi }$

${\theta }_2=5\times 30°=150°$

Thus, both sectors have the exact same central angle, ${\theta }_1={\theta }_2=150°$.

Step 3: Calculate the ratio of the areas

The area of a sector of a circle with radius $r$ and central angle $\theta$ (in degrees) is given by:

$\text{Area}=\frac{\theta }{360°}\times \pi r^2$

Since both sectors belong to the same circle (same radius $r$) and have equal central angles (${\theta }_1={\theta }_2$), their areas must also be equal.

Therefore, the ratio of the area of the first sector to that of the second sector is:

$\text{Ratio}=\frac{\text{Area of First Sector}}{\text{Area of Second Sector}}=\frac{1}{1}$

Hence, the ratio is 1 : 1.