Question

A circle has a chord that goes from `( pi)/2` to `(13 pi) / 6` radians on the circle. If the area of the circle is `5 pi`, what is the length of the chord?

Answer 1

The length of the chord is `sqrt5`.

Explanation:

Let the center of the circle be point `O`. Let the point at `pi/2` be point `A` and let the point at `(13pi)/6` be point `B`. Since `(13pi)/6=2pi+pi/6`, point `B` can be thought of as located at an angle of `pi/6`. Therefore, `m/_AOB=pi/2-pi/6=pi/3`. Since points `A` and `B` are located on the circle, `OA=OB` because they are both radii. Since `OA=OB`, we know that `m/_OAB=m/_OBA` and since `m/_AOB=pi/3`, we know that `m/_OAB=m/_OBA=m/_AOB=pi/3` and that `OA=OB=AB`.

The area of a circle is given by `A=pir^2` so we know `5pi=pir^2` or that `r=sqrt5`. Since `OA=r=sqrt5` and chord `AB=OA`, `AB=sqrt5`.