A circle's center is at $(4, 2)$ $(4 ,2 )$ and it passes through $(6, 2)$ $(6 ,2 )$ . What is the length of an arc covering $\frac{5 π}{3}$ $(5pi ) /3$ radians on the circle?

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A circle's center is at $(4, 2)$ $(4 ,2 )$ and it passes through $(6, 2)$ $(6 ,2 )$ . What is the length of an arc covering $\frac{5 π}{3}$ $(5pi ) /3$ radians on the circle?

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Answer 1 · 2017-01-07T15:36:06

The arc length is the fraction of the circle ( $\frac{θ}{2} π$ $theta/2pi$ ) times the circumference. The radius is needed to find the circumference, so find the radius by finding the distance between the center and $(6, 2)$ $(6,2)$ , a point on the circle.

We could use the distance formula, or picture the distance between $(4, 2)$ $(4,2)$ and $(6, 2)$ $(6,2)$ as 2 units, because the y values of the points are the same.
$r = 2$ $r=2$

Let arc length $= s$ $= s$
$s = (\frac{θ}{2 π}) (2 π r)$ $s=(theta/(2pi))(2pir)$
$s = θ r$ $s=thetar$

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

$s = \frac{10 π}{3} units$ $s=(10pi)/3 "units"$

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A circle's center is at $(4, 2)$ $(4 ,2 )$ and it passes through $(6, 2)$ $(6 ,2 )$ . What is the length of an arc covering $\frac{5 π}{3}$ $(5pi ) /3$ radians on the circle?

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A circle's center is at $(4, 2)$ $(4 ,2 )$ and it passes through $(6, 2)$ $(6 ,2 )$ . What is the length of an arc covering $\frac{5 π}{3}$ $(5pi ) /3$ radians on the circle?