Question

A circle has a chord that goes from `( 5 pi)/6` to `(5 pi) / 4` radians on the circle. If the area of the circle is `39 pi`, what is the length of the chord?

Answer 1

Chord length `= **39/2 or 19.5**`

Explanation:

Angle sub tended at the center `theta` by the chord is
`(5pi)/4 - (5pi) /6 = (15pi - 10pi) / 6 = (10pi)/6 = (5pi)/3`
`theta / 2 = (5pi)/6`

Circumference `= 2pir = 39pi`
`r = (39pi) / (2pi) = 39/2`

If chord length is c,
`sin (theta/2) = (c/2)/r`

`c = 2r sin (theta/2) = 2* (39/2) * sin ((5pi)/6)`
`color(red)(sin ((5pi)/6) = sin (pi - ((5pi)/6) = sin (pi/6)))`

`c = 39 *sin(pi/6) = 39 * (1/2) = 39/2 = 19.5`