Question

the area of the shaded sector is 8π. what is the measure of ∠sot, which corresponds to the minor arc st? a 34° b 45° c 48° d 50°

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, $A = 8\pi$ and $r = 8$.

Step2: Substitute values into the formula

Substitute $A = 8\pi$ and $r = 8$ into the formula $A=\frac{\theta}{360^{\circ}}\times\pi r^{2}$. We get $8\pi=\frac{\theta}{360^{\circ}}\times\pi\times8^{2}$.

Step3: Simplify the equation

First, simplify the right - hand side: $\frac{\theta}{360^{\circ}}\times\pi\times64$. The $\pi$ on both sides of the equation cancels out. So we have $8=\frac{\theta\times64}{360^{\circ}}$.

Step4: Solve for $\theta$

Cross - multiply to get $8\times360^{\circ}=\theta\times64$. Then $\theta=\frac{8\times360^{\circ}}{64}$. Calculate $\frac{8\times360}{64}=\frac{2880}{64} = 45^{\circ}$.

Answer:

B. $45^{\circ}$