A circle's center is at $(2, 4)$ $(2 ,4 )$ and it passes through $(3, 1)$ $(3 ,1 )$ . What is the length of an arc covering $\frac{13 π}{12}$ $(13pi ) /12$ radians on the circle?

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A circle's center is at $(2, 4)$ $(2 ,4 )$ and it passes through $(3, 1)$ $(3 ,1 )$ . What is the length of an arc covering $\frac{13 π}{12}$ $(13pi ) /12$ radians on the circle?

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Answer 1 · 2016-02-18T22:04:44

arc length = radius x arc angle
radius = $\sqrt{10}$ $sqrt(10)$
arc angle = $\frac{13 π}{12}$ $(13pi)/12$

arc length = $\sqrt{10} (\frac{13 π}{12}) ≅ 10.76$ $sqrt(10)((13pi)/12) ~= 10.76$

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 (2,4), so $x_{0} = 2 and y_{0} = 4$ $x_0=2 and y_0=4$
The circle passes through (3,1), so at this point x=3 and y=1.

Inputting all this into the circle equation:
${(3 - 2)}^{2} + {(1 - 4)}^{2} = r^{2}$ $(3-2)^2+(1-4)^2=r^2$
${(1)}^{2} + {(- 3)}^{2} = r^{2}$ $(1)^2+(-3)^2=r^2$
$1 + 9 = r^{2} = 10$ $1+9=r^2=10$
Therefore: r = $\sqrt{10}$ $sqrt(10)$

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

arc angle = $\frac{13 π}{12}$ $(13pi)/12$

arc length = $\sqrt{10} (\frac{13 π}{12}) ≅ 10.76$ $sqrt(10)((13pi)/12) ~= 10.76$

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A circle's center is at $(2, 4)$ $(2 ,4 )$ and it passes through $(3, 1)$ $(3 ,1 )$ . What is the length of an arc covering $\frac{13 π}{12}$ $(13pi ) /12$ radians on the circle?

1 Answer

Explanation:

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A circle's center is at (2,4)(2 ,4 ) and it passes through (3,1)(3 ,1 ). What is the length of an arc covering 13π12(13pi ) /12 radians on the circle?

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A circle's center is at $(2, 4)$ $(2 ,4 )$ and it passes through $(3, 1)$ $(3 ,1 )$ . What is the length of an arc covering $\frac{13 π}{12}$ $(13pi ) /12$ radians on the circle?