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. From the figure, the radius $r = 8$. Given $A = 8\pi$.

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 $8\pi=\frac{\theta}{360^{\circ}}\times\pi\times8^{2}$.
First, simplify the right - hand side: $\frac{\theta}{360^{\circ}}\times\pi\times64$. Then, since $8\pi=\frac{\theta}{360^{\circ}}\times64\pi$.
We can cancel out $\pi$ from both sides of the equation, and we have $8=\frac{\theta}{360^{\circ}}\times64$.

Step3: Solve for $\theta$

Cross - multiply: $8\times360^{\circ}=64\theta$.
So, $2880^{\circ}=64\theta$.
Then, $\theta=\frac{2880^{\circ}}{64}=45^{\circ}$.

Answer:

B. $45^{\circ}$