Points #(3 ,2 )# and #(8 ,4 )# are #( pi)/3 # radians apart on a circle. What is the shortest arc length between the points?

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Answer 1 · 2016-10-03T14:51:18

#s = (pisqrt29)/3#

The square of distance, d, between the two points is:

#d² = (8 - 3)² + (4 - 2)²#

#d² = (5)² + (2)²#

#d² = 29#

d forms one side of a triangle where the other two sides are radii; this allow us the law of cosines to find the radius:

#29 = r² + r² - 2(r)(r)cos(pi/3)##

#r² = 29/(2 - 2cos(pi/3))#

#r = sqrt29#

The arc length, s, is:

#s = rtheta#

#s = (pisqrt29)/3#