Question

the area of the shaded sector is 10π. what is the measure of ∠aob, which corresponds to the minor arc ab? (not drawn to scale) a 36° b 25° c 42° d 30°

Explanation:

Step1: Recall sector - area formula

The formula for the area of a sector of a circle is $A=\frac{\theta}{360^{\circ}}\times\pi r^{2}$, where $A$ is the area of the sector, $\theta$ is the central - angle measure in degrees, and $r$ is the radius of the circle. Here, $r = 10$ and $A=10\pi$.

Step2: Substitute values into formula

Substitute $A = 10\pi$ and $r = 10$ into the formula $A=\frac{\theta}{360^{\circ}}\times\pi r^{2}$. We get $10\pi=\frac{\theta}{360^{\circ}}\times\pi\times(10)^{2}$.

Step3: Simplify the equation

First, cancel out $\pi$ on both sides of the equation. The equation becomes $10=\frac{\theta}{360^{\circ}}\times100$. Then, we can rewrite it as $\frac{\theta}{360^{\circ}}=\frac{10}{100}=\frac{1}{10}$.

Step4: Solve for $\theta$

Cross - multiply to find $\theta$. $\theta=\frac{1}{10}\times360^{\circ}=36^{\circ}$.

Answer:

A. $36^{\circ}$