Question

A circle's center is at `(3 ,5 )` and it passes through `(2 ,8 )`. What is the length of an arc covering `(5pi ) /4` radians on the circle?

Answer 1

`(5sqrt(10)pi)/4`

Explanation:

Part 1
If a circle has a center at `(3,5)` and passes through `(2,8)`
it has a radius of `r=sqrt((3-2)^2+(5-8)^2)= sqrt(1+9)=sqrt(10)`

Part 2
An arc with an angle of `k` radians
has a length of `k/(2pi) xx "circumference of the circle"`
but since the `"circumference of the circle" = 2pir`
the length of an arc with an angle of `k` radians is k * r#

Part 3
For the given circle and arc
the arc length is `(5pi)/4xxsqrt(10)`