Question

A circle's center is at `(2 ,7 )` and it passes through `(3 ,1 )`. What is the length of an arc covering `(2pi ) /3` radians on the circle?

Answer 1

Arc length: `(2sqrt(37)pi)/3`

Explanation:

Part 1:
If the circle has center at `(2,7)` and passes through `(3,1)`, its radius is:
`color(white)("XXX")r=sqrt((2-3)^2+(7-1)^2) = sqrt((-1)^2+6^2) = sqrt(37`

Part 2:
For a circle with radius `r`
`color(white)("XXX")"angle" = 2pi rarr "arc length" = 2pir` (circumference of circle)
`color(white)("XXX")"angle" = pi rarr "arc length" = pir`
`color(white)("XXX")"angle" =2/3pi rarr "arc length" = 2/3 pir`

Solution
A arc with radius `sqrt(37)` (from Part 1) and an angle of `(2pi)/3`
will have an arc length of `2/3pisqrt(37)= (2sqrt(37))/3pi`