Question

A circle's center is at `(9 ,3 )` and it passes through `(2 ,6 )`. What is the length of an arc covering `(2pi ) /3` radians on the circle?

Answer 1

`color(blue)((2pisqrt(58))/3~~15.95043789)`

Explanation:

Arc length is given by:

`rtheta`

Where `theta` is measured in radians.

We first need to find the radius of the given circle. Since the distance from the centre of a circle to any point on its circumference is the radius, we find the distance from:

`(2,6)` to `(9,3)`

Using the distance formula:

`d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

`|r|=sqrt((9-2)^2+(3-6)^2)=sqrt(58)`

Using `rtheta`

`sqrt(58)((2pi)/3)=(2pisqrt(58))/3~~15.95043789`