Points $(5, 2)$ $(5 ,2 )$ and $(1, 4)$ $(1 ,4 )$ are $\frac{2 π}{3}$ $(2 pi)/3$ radians apart on a circle. What is the shortest arc length between the points?

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Answer 1 · 2018-01-07T13:41:45

Shortest arc length $s = 5.4077$ $s = color (blue)5.4077$

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Chord length $l_{c} = \sqrt{{(1 - 5)}^{2} + {(4 - 2)}^{2}} = 4.4721$ $l_c = sqrt((1-5)^2 + (4-2)^2) = 4.4721$

$r = \frac{l_{c}}{2} sin (\frac{θ}{2}) = \frac{4.4721}{2 \cdot sin (\frac{π}{3})} = 2.582$ $r = l_c / 2sin(theta/2) = 4.4721/(2*sin (pi/3)) = 2.582$

Arc length $s = r θ = 2.582 \cdot (\frac{2 π}{3}) = 5.4077$ $s = r theta = 2.582 * ((2pi)/3) = color (blue)(5.4077)$

Points (5,2)(5 ,2 ) and (1,4)(1 ,4 ) are 2π3(2 pi)/3 radians apart on a circle. What is the shortest arc length between the points?