### Instruction: In a polar coordinate system, the midpoint of the line segment whose endpoints are $\left( 8, \frac{5 \pi}{12} \right)$ and $\left( 8, -\frac{3 \pi}{12} \right)$ is the point $(r, \theta).$  Enter $(r, \theta),$ where $r > 0$ and $0 \le \theta < 2 \pi.$.### Response: I want to find the midpoint of a line segment in polar coordinates, so I need to average the radii and the angles of the endpoints.
The radii are both 8, so the average radius is 8 as well.
The angles are $\frac{5 \pi}{12}$ and $-\frac{3 \pi}{12},$ which have a difference of $\frac{2 \pi}{3}.$
To find the average angle, I can add half of that difference to the smaller angle, or subtract half of that difference from the larger angle.
Adding half of $\frac{2 \pi}{3}$ to $-\frac{3 \pi}{12}$ gives me $-\frac{3 \pi}{12} + \frac{\pi}{3} = \frac{\pi}{12}.$
Subtracting half of $\frac{2 \pi}{3}$ from $\frac{5 \pi}{12}$ gives me $\frac{5 \pi}{12} - \frac{\pi}{3} = \frac{\pi}{12}.$
So the average angle is $\frac{\pi}{12},$ which is between 0 and $2 \pi.$
Therefore, the midpoint of the line segment in polar coordinates is $\left( 8, \frac{\pi}{12} \right).$