Question

Points `(6 ,2 )` and `(5 ,4 )` are `(2 pi)/3` radians apart on a circle. What is the shortest arc length between the points?

Answer 1

`(2sqrt 5)/(3sqrt 3) pi`

Explanation:

distance betwee 2 points `= sqrt((6-5)^2+(2-4)^2)`

`= sqrt((1)^2+(-2)^2) = sqrt 5`

Let say `r` is a radius of a circle.

`(sqrt 5/2)/r = sin(1/2*2/3 pi)`

`r = (sqrt 5/2)/(sin (1/3 pi)` `= (sqrt 5/2)/(sqrt3 /2)=sqrt(5/3)`

The shortest arc `=r theta`, where `theta` is a smallest angle between of them.

`=sqrt(5/3)*(2/3 pi) =(2sqrt 5)/(3sqrt 3) pi`