Question

A circle's center is at `(8 ,7 )` and it passes through `(6 ,2 )`. What is the length of an arc covering `(5 pi ) /6` radians on the circle?

Answer 1

`(5sqrt(29)pi)/6`

Explanation:

The length of an arc with radius `r` covering `theta` radians is
`color(white)("XXX")theta*r`
(Think of this in terms of a complete circle whose circumference is `2pir`).

If the arc has a center at `(8,7)` and passes through `(6,2)` then its radius is
`color(white)("XXX")r=sqrt((8-6)^2+(7-2)^2) = sqrt(2^2+5^2) = sqrt(29)`

and the arc length with `theta= (5pi)/6` is
`color(white)("XXX")(5pi)/6*sqrt(29) = (5sqrt(29)pi)/6`