A circle's center is at $(4, 6)$ $(4 ,6 )$ and it passes through $(3, 1)$ $(3 ,1 )$ . What is the length of an arc covering $\frac{π}{3}$ $(pi ) /3$ radians on the circle?

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A circle's center is at $(4, 6)$ $(4 ,6 )$ and it passes through $(3, 1)$ $(3 ,1 )$ . What is the length of an arc covering $\frac{π}{3}$ $(pi ) /3$ radians on the circle?

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Answer 1 · 2016-02-08T13:40:21

The arc length is $\frac{\sqrt{26}}{3} π$ $sqrt(26)/3 pi$ .

Explanation:

First of all, you need to compute the radius.

If you center is at $(4, 6)$ $(4, 6)$ and an arbitrary point on a circle is $(3, 1)$ $(3,1)$ , we can compute the radius as follows:

$r = \sqrt{{(4 - 3)}^{2} + {(6 - 1)}^{2}} = \sqrt{1 + 25} = \sqrt{26}$ $r = sqrt((4-3)^2 + (6-1)^2) = sqrt(1 + 25) = sqrt(26)$

Now, the the length of an arc covering the whole circle would be equivalent to the perimeter of a circle, $2 π r$ $2pi r$ .

In your case, you would like to compute an arc covering $\frac{π}{3}$ $pi/3$ radians instead of the whole $2 π$ $2pi$ (equivalent to $60^{\circ}$ $60^@$ which is $\frac{1}{6}$ $1/6$ of the whole circle).

Thus, your arc length is $\frac{π}{3} r = \frac{\sqrt{26}}{3} π$ $pi/3 r = sqrt(26)/3 pi$ .

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A circle's center is at $(4, 6)$ $(4 ,6 )$ and it passes through $(3, 1)$ $(3 ,1 )$ . What is the length of an arc covering $\frac{π}{3}$ $(pi ) /3$ radians on the circle?

1 Answer

Explanation:

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A circle's center is at $(4, 6)$ $(4 ,6 )$ and it passes through $(3, 1)$ $(3 ,1 )$ . What is the length of an arc covering $\frac{π}{3}$ $(pi ) /3$ radians on the circle?