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

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Answer 1 · 2016-10-12T16:59:17

$s \approx 10.89$ $s ~~ 10.89$

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

$c = \sqrt{{(5 - 2)}^{2} + {(9 - 1)}^{2}}$ $c = sqrt((5 - 2)^2+(9 - 1)^2)$

$c = \sqrt{3^{2} + 8^{2}}$ $c = sqrt(3^2+8^2)$

$c = \sqrt{73}$ $c = sqrt(73)$

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:

$c^{2} = r^{2} + r^{2} - 2 (r) (r) cos (θ)$ $c^2 = r^2 + r^2 - 2(r)(r)cos(theta)$

$r = \sqrt{\frac{c^{2}}{2 - 2 cos (θ)}}$ $r = sqrt(c^2/(2 - 2cos(theta))$

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

$s = r θ$ $s = rtheta$

$s = θ \sqrt{\frac{c^{2}}{2 - 2 cos (θ)}}$ $s = thetasqrt(c^2/(2 - 2cos(theta))$

$s = (3 \frac{π}{4}) \sqrt{\frac{73}{2 - 2 cos (3 \frac{π}{4})}}$ $s = (3pi/4)sqrt(73/(2 - 2cos(3pi/4))$

$s \approx 10.89$ $s ~~ 10.89$

Points (2,1)(2 ,1 ) and (5,9)(5 ,9 ) are 3π4(3 pi)/4 radians apart on a circle. What is the shortest arc length between the points?