Points $(6, 7)$ $(6 ,7 )$ and $(7, 5)$ $(7 ,5 )$ 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 $(6, 7)$ $(6 ,7 )$ and $(7, 5)$ $(7 ,5 )$ 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 · 2016-02-17T19:14:00

$\frac{2}{9} \cdot \sqrt{15} \cdot π ≅ 2.704$ $2/9*sqrt(15)*pi~=2.704$

Explanation:

Refer to the figure below

I created this figure using MS Excel

$α = \frac{2 π}{3} r a d i a n s$ $alpha=(2pi)/3 radians$
Or
$α = \frac{2 \cdot 180^{\circ}}{3} = 120^{\circ}$ $alpha=(2*180^@)/3=120^@$

$A B = \sqrt{{(7 - 6)}^{2} + {(5 - 7)}^{2}} = \sqrt{1 + 4} = \sqrt{5}$ $AB=sqrt((7-6)^2+(5-7)^2)=sqrt(1+4)=sqrt(5)$

Applying Law of Cosines in $△_{A B C}$ $triangle_(ABC)$ :

$A B^{2} = r^{2} + r^{2} - 2 r \cdot r \cdot {cos 180}^{\circ}$ $AB^2=r^2+r^2-2r*r*cos 180^@$
$2 r^{2} - 2 r^{2} \cdot (- \frac{1}{2}) = {(\sqrt{5})}^{2}$ $2r^2-2r^2*(-1/2)=(sqrt(5))^2$
$3 r^{2} = 5$ $3r^2=5$ => $r = \sqrt{\frac{5}{3}}$ $r=sqrt(5/3)$

Length of the arc

$a r c A B = r \cdot α$ $arc AB=r*alpha$ , where alpha is given in radians
$a r c A B = \sqrt{\frac{5}{3}} \cdot \frac{2 \cdot π}{3} \cdot (\frac{\sqrt{3}}{\sqrt{3}}) = \frac{2}{9} \cdot \sqrt{15} \cdot π$ $arc AB=sqrt(5/3)*(2*pi)/3*(sqrt(3)/sqrt(3))=2/9*sqrt(15)*pi$

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Points $(6, 7)$ $(6 ,7 )$ and $(7, 5)$ $(7 ,5 )$ 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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Points (6,7)(6 ,7 ) and (7,5)(7 ,5 ) 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 $(6, 7)$ $(6 ,7 )$ and $(7, 5)$ $(7 ,5 )$ are $\frac{2 π}{3}$ $(2 pi)/3$ radians apart on a circle. What is the shortest arc length between the points?