A circle's center is at $(7, 5)$ $(7 ,5 )$ and it passes through $(2, 7)$ $(2 ,7 )$ . What is the length of an arc covering $\frac{3 π}{4}$ $(3pi ) /4$ radians on the circle?

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Answer 1 · 2016-08-20T11:02:17

Length of arc $\approx 12.688$ $~~12.688$ to 3 decimal places

Let the radius be $r$ $r$

Let the length of arc be $L$ $L$

Let the centre point be $P_{1} \to (x_{1}, y_{1}) = (7, 5)$ $P_1 -> (x_1 ,y_1 ) = (7,5)$

Let the point on the circumference be $P_{2} \to (x_{2}, y_{2}) = (2, 7)$ $P_2->(x_2,y_2)=(2,7)$

$Determine distance from centre to the given point.$ $color(blue)("Determine distance from centre to the given point.")$

$r = \sqrt{{(x_{2} - x_{1})}_{2}^{2} + {(y_{2} - y_{1})}_{2}^{2}}$ $r=sqrt((x_2-x_1)^2+(y_2-y_1)^2)$

$\Rightarrow r = \sqrt{{(2 - 7)}^{2} + {(7 - 5)}^{2}}$ $=> r=sqrt((2-7)^2+(7-5)^2)$

$r = \sqrt{29}$ $color(green)(r=sqrt29)" "$ Note that 29 is a prime number

To maintain precision to not convert to decimal at this point.
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$Determine length of ark.$ $color(blue)("Determine length of ark.")$

Note that the length of arc for 1 radian is $r$ $r$

So the length of $\frac{3 π}{4}$ $(3pi)/4$ radians gives:

$L = \frac{3 π}{4} \times r \to \frac{3 π}{4} \times \sqrt{29}$ $L=(3pi)/4xxr" " ->" " (3pi)/4xxsqrt29$

$L \approx 12.688$ $L~~12.688$ to 3 decimal places

A circle's center is at (7,5)(7 ,5 ) and it passes through (2,7)(2 ,7 ). What is the length of an arc covering 3π4(3pi ) /4 radians on the circle?