A circle has a center at $(7, 6)$ $(7 ,6 )$ and passes through $(2, 1)$ $(2 ,1 )$ . What is the length of an arc covering $\frac{π}{8}$ $pi/8$ radians on the circle?

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A circle has a center at $(7, 6)$ $(7 ,6 )$ and passes through $(2, 1)$ $(2 ,1 )$ . What is the length of an arc covering $\frac{π}{8}$ $pi/8$ radians on the circle?

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Answer 1 · 2016-01-27T02:59:51

$\frac{5 \sqrt{2} π}{8}$ $( 5 sqrt(2) pi ) / 8$

Explanation:

We know the center and a point on the circle. The distance between the two points is the radius (draw a picture and convince yourself this).

We know that the distance between two points $(x_{1}, y_{1})$ $(x_1, y_1)$ and $(x_{2}, y_{2})$ $(x_2, y_2)$ on the Euclidean plane is given by $d = \sqrt{{(x_{1} - x_{2})}_{1}^{2} + {(y_{1} - y_{2})}_{1}^{2}}$ $d = sqrt( (x_1 - x_2)^2 + (y_1 - y_2)^2 )$ .
Thus, radius $r = \sqrt{{(7 - 2)}^{2} + {(6 - 1)}^{2}} = \sqrt{5^{2} + 5^{2}} = \sqrt{2 \cdot 5^{2}} = 5 \sqrt{2}$ $r = sqrt( (7 - 2)^2 + (6 - 1)^2 ) = sqrt(5^2 + 5^2) = sqrt(2 * 5^2) = 5 sqrt(2)$ .

By definition, a radian is the angle subtended by an arc of length equal to the radius. Thus, an arc which subtends an angle of $θ$ $theta$ has a length of $θ r$ $theta r$ . In this case, $θ = \frac{π}{8}$ $theta = pi / 8$ , so arc length = $\frac{5 \sqrt{2} π}{8}$ $( 5 sqrt(2) pi ) / 8$ .

Hint to remember
Note that the circumference subtends and angle of $2 π$ $2 pi$ at the center, and has a length of $2 π r$ $2 pi r$ . By unitary method, an arc which subtends an angle of $θ$ $theta$ has a length of $θ r$ $theta r$ .

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A circle has a center at $(7, 6)$ $(7 ,6 )$ and passes through $(2, 1)$ $(2 ,1 )$ . What is the length of an arc covering $\frac{π}{8}$ $pi/8$ radians on the circle?

1 Answer

Explanation:

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A circle has a center at (7,6)(7 ,6 ) and passes through (2,1)(2 ,1 ). What is the length of an arc covering π8pi/8 radians on the circle?

1 Answer

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A circle has a center at $(7, 6)$ $(7 ,6 )$ and passes through $(2, 1)$ $(2 ,1 )$ . What is the length of an arc covering $\frac{π}{8}$ $pi/8$ radians on the circle?