A circle's center is at $(2, 6)$ $(2 ,6 )$ 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 $(2, 6)$ $(2 ,6 )$ 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 · 2016-02-21T15:41:26

arc length = radius x arc angle
radius = $\sqrt{26}$ $sqrt(26)$
arc angle = $\frac{π}{3}$ $(pi)/3$

arc length = $\sqrt{26} (\frac{π}{3}) ≅ 5.34$ $sqrt(26)((pi)/3) ~= 5.34$

Explanation:

First find the radius, r, using the equation of a circle:
${(x - x_{0})}_{0}^{2} + {(y - y_{0})}_{0}^{2} = r^{2}$ $(x-x_0)^2+(y-y_0)^2=r^2$

The circle's centre is (2,6), so $x_{0} = 2 and y_{0} = 6$ $x_0=2 and y_0=6$
The circle passes through (3,1), so at this point x=3 and y=1.

Inputting all this into the circle equation:
${(3 - 2)}^{2} + {(1 - 6)}^{2} = r^{2}$ $(3-2)^2+(1-6)^2=r^2$
${(1)}^{2} + {(- 5)}^{2} = r^{2}$ $(1)^2+(-5)^2=r^2$
$1 + 25 = r^{2} = 26$ $1+25=r^2=26$
Therefore: r = $\sqrt{26}$ $sqrt(26)$

Now we can find the arc length using:
arc length = radius x arc angle

arc angle = $\frac{π}{3}$ $(pi)/3$

arc length = $\sqrt{26} (\frac{π}{3}) ≅ 5.34$ $sqrt(26)((pi)/3) ~= 5.34$

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A circle's center is at $(2, 6)$ $(2 ,6 )$ 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 $(2, 6)$ $(2 ,6 )$ 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?