Points $(7, 1)$ $(7 ,1 )$ 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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Points $(7, 1)$ $(7 ,1 )$ 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 · 2018-07-15T02:18:14

$Length of shortest arc (A B) = r \cdot θ = 10.5153$ $color(brown)("Length of shortest arc " (AB) = r * theta =10.5153$

Explanation:

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$Point A (7, 1), Point B (5, 9), ˆ A O B = θ = {(\frac{3 π}{4})}^{c}$ $"Point " A (7,1), " Point " B(5,9), hat (AOB) = theta = ((3pi) / 4)^c$

To find arc length S.

$Chord Length ¯ ¯¯¯¯ ¯ A B = \sqrt{{(7 - 5)}^{2} + {(1 - 9)}^{2}} = \sqrt{68}$ $"Chord Length " bar (AB) = sqrt ((7-5)^2 + (1-9)^2) = sqrt 68$

$ˆ A O M = \frac{ˆ A O B}{2} = \frac{3 π}{8}$ $hat (AOM) = hat (AOB) / 2 = (3pi)/8$

$¯ ¯¯¯¯ ¯ O A = r = \frac{A M}{sin (A O M)} = \frac{\frac{\sqrt{68}}{2}}{sin (\frac{3 π}{8})} = 4.4628$ $bar(OA) = r = (AM) / sin (AOM) = (sqrt 68 / 2) / sin ((3pi)/8) = 4.4628$

$Length of shortest arc (A B) = r \cdot θ = 4.4628 \cdot (\frac{3 π}{4}) = 10.5153$ $color(brown)("Length of shortest arc " (AB) = r * theta = 4.4628 * ((3pi)/4) = 10.5153$

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Points $(7, 1)$ $(7 ,1 )$ 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?

1 Answer

Explanation:

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Points $(7, 1)$ $(7 ,1 )$ 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?