Points #(2 ,2 )# and #(8 ,1 )# are #(5 pi)/6 # radians apart on a circle. What is the shortest arc length between the points?

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Answer 1 · 2018-01-27T11:54:20

Shorter arc length between the points is #8.25 # unit.

Distance between two points #(2,2) and (8,1) # is

#D= sqrt ((x_1-x_2)^2+(y_1-y_2)^2) =sqrt ((2-8)^2+(2-1)^2# or

#D=sqrt37 ~~ 6.08# unit. So, chord length is #L_c=6.08# unit.

Formula for the length of a chord is #L_c= 2r sin (theta/2)#

where #r# is the radius of the circle and #theta# is the angle

subtended at the center by the chord.

#theta =(5pi)/6=(5*180)/6=150^0 ; theta/2=75^0#

#6.08 = 2*r *sin75 or r = 6.08/(2*sin75)= 3.15# unit.

Arc length is # L_a= 2* pi * r*theta/360=2*pi*3.15*150/360# or

#L_a ~~ 8.25# unit. Shorter arc length between the points is

#8.25 # unit. [Ans]