Points $(2, 9)$ $(2 ,9 )$ and $(1, 5)$ $(1 ,5 )$ 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 · 2017-07-12T09:39:47

The arc length is $= 5.26$ $=5.26$

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The angle $θ = \frac{3}{4} π$ $theta=3/4pi$

The distance between the points is

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

$= \sqrt{1 + 16}$ $=sqrt(1+16)$

$= \sqrt{17}$ $=sqrt17$

This is the length of the chord

So,

$\frac{d}{2} = r sin (\frac{θ}{2})$ $d/2=rsin(theta/2)$

$r = \frac{d}{2 sin (\frac{θ}{2})}$ $r=d/(2sin(theta/2))$

$= \frac{\sqrt{17}}{2 sin (\frac{3}{8} π)}$ $=sqrt17/(2sin(3/8pi))$

$= 2.23$ $=2.23$

The length of the arc is

$L = r θ = 2.23 \cdot \frac{3}{4} π = 5.26$ $L=rtheta=2.23*3/4pi=5.26$

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