Question

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

Answer 1

`(4pi)/3`

Explanation:

The distance between the center and any point on the circle is the radius. From the 2 points given in the description, we can determine that the radius `r=9-5=4`
More formally, the distance between any two points is:
`d=sqrt((y_2-y_1)^2+(x_2-x_1)^2)`
In this case: `r = sqrt((2-2)^2 + (9-5)^2) = sqrt(4^2) = 4`

The circumference of this circle is `C = 2pir = 8pi`

The length of the arc is a portion of the total circumference. Since the total arc around the whole circle is `2pi`, the proportion of the arc is `(pi/3) / (2pi) = 1/6` of the total circumference.

So, the length of the arc covering `pi/3` radians is `(1/6)C`,
that is:
`(1/6) * 8pi = (4pi)/3`