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

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Answer 1 · 2016-08-20T11:02:17

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

Let the radius be $r$

Let the length of arc be $L$

Let the centre point be $P_1 -> (x_1 ,y_1 ) = (7,5)$

Let the point on the circumference be $P_2->(x_2,y_2)=(2,7)$

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

$r=sqrt((x_2-x_1)^2+(y_2-y_1)^2)$

$=> r=sqrt((2-7)^2+(7-5)^2)$

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

To maintain precision to not convert to decimal at this point.
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

$color(blue)("Determine length of ark.")$

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

So the length of $(3pi)/4$ radians gives:

$L=(3pi)/4xxr" " ->" " (3pi)/4xxsqrt29$

$L~~12.688$ 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 (3pi ) /4 (3pi ) /4 radians on the circle?