Question

A circle's center is at `(5 ,9 )` and it passes through `(2 ,3 )`. What is the length of an arc covering `(7pi ) /8` radians on the circle?

Answer 1

`(21sqrt(5)pi)/8`

Explanation:

By definition
`color(white)("XXX") "arc length of a circle (i.e. the circumference)"=2pir`
and
`color(white)("XXX")"radian measure of a circle " = 2pi`

Therefore
`color(white)("XXX")(7pi)/8` radians represents an arc length of `7/8picolor(blue)(r)`

A circle with center `(5,9)` that passes through `(2,3)` will have a radius:
`color(white)("XXX")color(blue)(r)=sqrt((5-2)^2+(9-3)^2) = sqrt(45) = 3sqrt(5)`

Therefore
`color(white)("XXX")(7pi)/8` radians for the given circle represents an arc length of
`color(white)("XXXXXXXXXXXX")7/8pixx3sqrt(5) = (21sqrt(5))/8pi`