A circle's center is at $(9, 2)$ $(9 ,2 )$ and it passes through $(5, 2)$ $(5 ,2 )$ . 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 $(9, 2)$ $(9 ,2 )$ and it passes through $(5, 2)$ $(5 ,2 )$ . What is the length of an arc covering $\frac{π}{3}$ $(pi ) /3$ radians on the circle?

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Answer 1 · 2018-05-24T07:49:48

$\frac{4 π}{3}$ $(4pi)/3$

Explanation:

The distance between the center and any point on the circle is the radius. From the 2 points given in the description, we can determine that the radius $r = 9 - 5 = 4$ $r=9-5=4$
More formally, the distance between any two points is:
$d = \sqrt{{(y_{2} - y_{1})}_{2}^{2} + {(x_{2} - x_{1})}_{2}^{2}}$ $d=sqrt((y_2-y_1)^2+(x_2-x_1)^2)$
In this case: $r = \sqrt{{(2 - 2)}^{2} + {(9 - 5)}^{2}} = \sqrt{4^{2}} = 4$ $r = sqrt((2-2)^2 + (9-5)^2) = sqrt(4^2) = 4$

The circumference of this circle is $C = 2 π r = 8 π$ $C = 2pir = 8pi$

The length of the arc is a portion of the total circumference. Since the total arc around the whole circle is $2 π$ $2pi$ , the proportion of the arc is $\frac{\frac{π}{3}}{2 π} = \frac{1}{6}$ $(pi/3) / (2pi) = 1/6$ of the total circumference.

So, the length of the arc covering $\frac{π}{3}$ $pi/3$ radians is $(\frac{1}{6}) C$ $(1/6)C$ ,
that is:
$(\frac{1}{6}) \cdot 8 π = \frac{4 π}{3}$ $(1/6) * 8pi = (4pi)/3$

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A circle's center is at $(9, 2)$ $(9 ,2 )$ and it passes through $(5, 2)$ $(5 ,2 )$ . 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 $(9, 2)$ $(9 ,2 )$ and it passes through $(5, 2)$ $(5 ,2 )$ . What is the length of an arc covering $\frac{π}{3}$ $(pi ) /3$ radians on the circle?