A circle's center is at $(3, 5)$ $(3 ,5 )$ and it passes through $(2, 8)$ $(2 ,8 )$ . What is the length of an arc covering $\frac{5 π}{4}$ $(5pi ) /4$ radians on the circle?

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A circle's center is at $(3, 5)$ $(3 ,5 )$ and it passes through $(2, 8)$ $(2 ,8 )$ . What is the length of an arc covering $\frac{5 π}{4}$ $(5pi ) /4$ radians on the circle?

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Answer 1 · 2016-11-22T12:52:47

$\frac{5 \sqrt{10} π}{4}$ $(5sqrt(10)pi)/4$

Explanation:

Part 1
If a circle has a center at $(3, 5)$ $(3,5)$ and passes through $(2, 8)$ $(2,8)$
it has a radius of $r = \sqrt{{(3 - 2)}^{2} + {(5 - 8)}^{2}} = \sqrt{1 + 9} = \sqrt{10}$ $r=sqrt((3-2)^2+(5-8)^2)= sqrt(1+9)=sqrt(10)$

Part 2
An arc with an angle of $k$ $k$ radians
has a length of $\frac{k}{2 π} \times circumference of the circle$ $k/(2pi) xx "circumference of the circle"$
but since the $circumference of the circle = 2 π r$ $"circumference of the circle" = 2pir$
the length of an arc with an angle of $k$ $k$ radians is k * r#

Part 3
For the given circle and arc
the arc length is $\frac{5 π}{4} \times \sqrt{10}$ $(5pi)/4xxsqrt(10)$

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A circle's center is at $(3, 5)$ $(3 ,5 )$ and it passes through $(2, 8)$ $(2 ,8 )$ . What is the length of an arc covering $\frac{5 π}{4}$ $(5pi ) /4$ radians on the circle?

1 Answer

Explanation:

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A circle's center is at $(3, 5)$ $(3 ,5 )$ and it passes through $(2, 8)$ $(2 ,8 )$ . What is the length of an arc covering $\frac{5 π}{4}$ $(5pi ) /4$ radians on the circle?