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

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Answer 1 · 2018-07-26T22:24:48

#2.342\ \text{ unit#

The distance #d# between the given points #(5, 2)# & #(4, 4)# is

#d=\sqrt{(5-4)^2+(2-4)^2}#

#=\sqrt{5}#

Now, the radius #R# of circle is given as

#\sin ({5\pi}/6)=\frac{d/2}{R}#

#\sin(\pi/6)=d/{2R}#

#1/2=\sqrt5/{2R}#

#R=\sqrt5#

hence, the shortest arc length between given points which are #{5\pi}/3# apart

#=(2\pi-{5\pi}/3)R#

#=\pi/3(\sqrt5)#

#=2.342\ \text{ unit#