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