Question

Points `(8 ,5 )` and `(7 ,6 )` are `( pi)/3` radians apart on a circle. What is the shortest arc length between the points?

Answer 1

`s = sqrt(2)pi/3`

Explanation:

The length of the chord is:

`c = sqrt((7 - 8)^2 + (6 - 5)^2)`

`c = sqrt(2)`

Two radii, each drawn from the center to its respective end of the chord, form a triangle with the chord, therefore, we can use a variant of the Law of Cosines where `a = b = r`:

`c^2 = r^2 + r^2 - 2(r)(r)cos(theta)`

Substitute `c = sqrt(2) and theta = pi/3`:

NOTE: At this point, we should realize that an isosceles triangle with the third angle equal to `pi/3` is an equilateral triangle but, let's proceed as an example of how to solve the problem, when it is not this special case:

`(sqrt(2))^2 = 2r^2 - 2r^2cos(pi/3)`

`r^2 = (sqrt(2))^2/(2(1 - cos(pi/3))`

`r^2 = 2`

`r = sqrt(2)`

The arc length is, `s = rtheta`

`s = sqrt(2)pi/3`