Question

the area of the shaded sector is 5π. what is the measure of the minor arc st? (not drawn to scale) a 18° b 72° c 9° d 36°

Explanation:

Step1: Recall the 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, $A = 5\pi$ and $r = 10$.

Step2: Substitute the values into the formula

Substitute $A = 5\pi$ and $r = 10$ into the formula $A=\frac{\theta}{360^{\circ}}\times\pi r^{2}$. We get $5\pi=\frac{\theta}{360^{\circ}}\times\pi\times(10)^{2}$.

Step3: Simplify the equation

First, simplify the right - hand side of the equation: $\frac{\theta}{360^{\circ}}\times\pi\times100=\frac{100\pi\theta}{360^{\circ}}$. So, our equation becomes $5\pi=\frac{100\pi\theta}{360^{\circ}}$.
Divide both sides of the equation by $\pi$: $5=\frac{100\theta}{360^{\circ}}$.

Step4: Solve for $\theta$

Cross - multiply: $5\times360^{\circ}=100\theta$. Then $1800^{\circ}=100\theta$.
Divide both sides by 100: $\theta = 18^{\circ}$.

Answer:

A. $18^{\circ}$