A circle's center is at $(1, 5)$ $(1 ,5 )$ and it passes through $(4, 8)$ $(4 ,8 )$ . What is the length of an arc covering $\frac{π}{3}$ $(pi ) /3$ radians on the circle?

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Answer 1 · 2016-01-31T11:21:43

$\sqrt{2} π$ $sqrt2pi$

suppose, the equation of the circle is,

${(x - h)}^{2} + {(y - k)}^{2} = r^{2}$ $(x-h)^2+(y-k)^2=r^2$

by putting the values of $h, k, x, y$ $h,k,x,y$ in the equation, we get,

${(4 - 1)}^{2} + {(8 - 5)}^{2} = r^{2}$ $(4-1)^2+(8-5)^2=r^2$

$or, r^{2} = 9 + 9$ $or,r^2=9+9$

$or, r = 3 \sqrt{2}$ $or, r=3sqrt2$

for the length of an arc, we know,

$s = r θ$ $s=rtheta$

$= 3 \sqrt{2} \cdot \frac{π}{3}$ $=3sqrt2*pi/3$

$=cancel(3)sqrt2*pi/(cancel(3))$

$=sqrt2pi$

A circle's center is at (1 ,5 )(1,5)(1 ,5 ) and it passes through (4 ,8 )(4,8)(4 ,8 ). What is the length of an arc covering (pi ) /3 π3(pi ) /3 radians on the circle?