the area of the shaded sector is 63π. what is the measure of ∠eof, which corresponds to the minor arc ef…

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, $r = 9$ and $A=63\pi$.

Step2: Substitute values into the formula

Substitute $A = 63\pi$ and $r = 9$ into the formula $A=\frac{\theta}{360^{\circ}}\times\pi r^{2}$. We get $63\pi=\frac{\theta}{360^{\circ}}\times\pi\times9^{2}$. First, simplify the right - hand side: $\pi\times9^{2}=81\pi$. So the equation becomes $63\pi=\frac{\theta}{360^{\circ}}\times81\pi$.

Step3: Solve for $\theta$

Divide both sides of the equation $63\pi=\frac{\theta}{360^{\circ}}\times81\pi$ by $\pi$ (since $\pi\neq0$), we have $63=\frac{\theta}{360^{\circ}}\times81$. Then, solve for $\theta$: $\theta=\frac{63\times360^{\circ}}{81}$. $63\times360 = 22680$ and $\frac{22680}{81}=280$. But we are looking for the measure of the minor arc, and the sum of the minor and major arcs is $360^{\circ}$. The measure of the major arc corresponding to the given sector is what we just found. The measure of the minor arc $\angle EOF$ (in degrees) is $360^{\circ}-280^{\circ}=80^{\circ}$.

Answer:

D. $80^{\circ}$