Points $(3, 2)$ $(3 ,2 )$ and $(8, 4)$ $(8 ,4 )$ are $\frac{π}{3}$ $( pi)/3$ radians apart on a circle. What is the shortest arc length between the points?

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Answer 1 · 2016-10-03T14:51:18

$s = \frac{π \sqrt{29}}{3}$ $s = (pisqrt29)/3$

The square of distance, d, between the two points is:

$d² = (8 - 3)² + (4 - 2)²$

$d² = (5)² + (2)²$

$d² = 29$

d forms one side of a triangle where the other two sides are radii; this allow us the law of cosines to find the radius:

$29 = r² + r² - 2(r)(r)cos(pi/3)$ #

$r² = 29/(2 - 2cos(pi/3))$

$r = sqrt29$

The arc length, s, is:

$s = rtheta$

$s = (pisqrt29)/3$

Points (3 ,2 )(3,2)(3 ,2 ) and (8 ,4 )(8,4)(8 ,4 ) are ( pi)/3 π3( pi)/3 radians apart on a circle. What is the shortest arc length between the points?