A circle has a center at $(7, 9)$ $(7 ,9 )$ and passes through $(1, 3)$ $(1 ,3 )$ . What is the length of an arc covering $\frac{π}{4}$ $pi /4$ radians on the circle?

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Answer 1 · 2016-02-01T21:56:52

$\frac{π}{4} = 45^{o} and \frac{45^{o}}{360^{o}} = \frac{1}{8} of the circumference$ $pi/4=45^o and (45^o)/(360^o)=1/8 " of the circumference"$

Circumference $= 2 π r$ $=2pir$ so, the length of the arc is $(\frac{1}{8}) (2 π r)$ $(1/8)(2pir)$

The radius is the distance between the center and the point on the circle:

Using the distance formula ...

radius $= \sqrt{{(7 - 1)}^{2} + {(9 - 3)}^{2}} = 6 \sqrt{2}$ $=sqrt[(7-1)^2+(9-3)^2]=6sqrt2$

length of the arc is $= (\frac{1}{8}) (2 π r) = (\frac{1}{8}) [2 π (6 \sqrt{2})] = 1.5 π \sqrt{2}$ $=(1/8)(2pir)=(1/8)[2pi(6sqrt2)]=1.5pisqrt2$

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A circle has a center at (7,9)(7 ,9 ) and passes through (1,3)(1 ,3 ). What is the length of an arc covering π4pi /4 radians on the circle?