Points $(6, 2)$ $(6 ,2 )$ and $(5, 4)$ $(5 ,4 )$ 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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Points $(6, 2)$ $(6 ,2 )$ and $(5, 4)$ $(5 ,4 )$ 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 · 2017-01-31T04:35:43

$\frac{2 \sqrt{5}}{3 \sqrt{3}} π$ $(2sqrt 5)/(3sqrt 3) pi$

Explanation:

distance betwee 2 points $= \sqrt{{(6 - 5)}^{2} + {(2 - 4)}^{2}}$ $= sqrt((6-5)^2+(2-4)^2)$

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

Let say $r$ $r$ is a radius of a circle.

$\frac{\frac{\sqrt{5}}{2}}{r} = sin (\frac{1}{2} \cdot \frac{2}{3} π)$ $(sqrt 5/2)/r = sin(1/2*2/3 pi)$

$r = \frac{\frac{\sqrt{5}}{2}}{sin (\frac{1}{3} π)}$ $r = (sqrt 5/2)/(sin (1/3 pi)$ $= \frac{\frac{\sqrt{5}}{2}}{\frac{\sqrt{3}}{2}} = \sqrt{\frac{5}{3}}$ $= (sqrt 5/2)/(sqrt3 /2)=sqrt(5/3)$

The shortest arc $= r θ$ $=r theta$ , where $θ$ $theta$ is a smallest angle between of them.

$= \sqrt{\frac{5}{3}} \cdot (\frac{2}{3} π) = \frac{2 \sqrt{5}}{3 \sqrt{3}} π$ $=sqrt(5/3)*(2/3 pi) =(2sqrt 5)/(3sqrt 3) pi$

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Points $(6, 2)$ $(6 ,2 )$ and $(5, 4)$ $(5 ,4 )$ are $\frac{2 π}{3}$ $(2 pi)/3$ radians apart on a circle. What is the shortest arc length between the points?

1 Answer

Explanation:

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Science

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Points (6,2)(6 ,2 ) and (5,4)(5 ,4 ) are 2π3(2 pi)/3 radians apart on a circle. What is the shortest arc length between the points?

1 Answer

Explanation:

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Points $(6, 2)$ $(6 ,2 )$ and $(5, 4)$ $(5 ,4 )$ are $\frac{2 π}{3}$ $(2 pi)/3$ radians apart on a circle. What is the shortest arc length between the points?