Points #(5 ,4 )# and #(2 ,0 )# are #(5 pi)/4 # radians apart on a circle. What is the shortest arc length between the points?

1 Answer
sankarankalyanam
Dec 2, 2017

Shortest arc length = 6.3756#

Explanation:

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Chord #c = sqrt((5-2)^2 + (4-0)^2) = 5#

#c2 = 2.5#

#/_ C = (5pi)/4#
#:./_ (C/2) = (5pi)/8#

Also #(c/2) / r = sin /_ (C/2)#
#r = (c/2) / (sin /_ (C/2))#
#r = 2.5 / (sin ((5pi)/8) = 2.5 / 0.9239 = 2.7059#
Circumference of the circle #= 2*pi*2.7059 = 17.0018#

When center angle is #2pi#, arc length = circumference = 17.0018#

When the center angle is #((5pi)/4)#, shortest arc length

#a = (2pi - ((5pi)/4)*r = (2pi - ((5pi)/4)) * 2.7059 = 6.3756#

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