Points #(2 ,1 )# and #(5 ,9 )# are #(3 pi)/4 # radians apart on a circle. What is the shortest arc length between the points?

1 Answer
Douglas K.
Oct 12, 2016

#s ~~ 10.89#

Explanation:

The length of the line segment, c, between the two points points is:

#c = sqrt((5 - 2)^2+(9 - 1)^2)#

#c = sqrt(3^2+8^2)#

#c = sqrt(73)#

Because the line segment between the two points and two radii form a triangle, we can use the Law of Cosines to find the radius:

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

#r = sqrt(c^2/(2 - 2cos(theta))#

The arc length, s, is found using the following:

#s = rtheta#

#s = thetasqrt(c^2/(2 - 2cos(theta))#

#s = (3pi/4)sqrt(73/(2 - 2cos(3pi/4))#

#s ~~ 10.89#

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