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)

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. Assume the radius of the circle is $r$. Here, $A = 10\pi$. Let the measure of $\angle AOB=\theta$. So, $10\pi=\frac{\theta}{360^{\circ}}\times\pi r^{2}$.

Step2: Simplify the equation

Since we are not given the radius, we can assume for a unit - circle ($r = 1$) (the ratio of the sector area to the circle area is independent of the radius value). Canceling out $\pi$ from both sides of the equation $10\pi=\frac{\theta}{360^{\circ}}\times\pi r^{2}$, we get $10=\frac{\theta}{360^{\circ}}\times r^{2}$. If $r = 1$, then $10=\frac{\theta}{360^{\circ}}$.

Step3: Solve for $\theta$

Cross - multiply to find $\theta$. We have $\theta=10\times36^{\circ}=36^{\circ}$.

Answer:

A. $36^{\circ}$