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

1 Answer
Douglas K.
Nov 19, 2016

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

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