A circle's center is at #(5 ,9 )# and it passes through #(2 ,3 )#. What is the length of an arc covering #(7pi ) /8 # radians on the circle?

1 Answer
Alan P.
Jan 21, 2016

#(21sqrt(5)pi)/8#

Explanation:

By definition
#color(white)("XXX") "arc length of a circle (i.e. the circumference)"=2pir#
and
#color(white)("XXX")"radian measure of a circle " = 2pi#

Therefore
#color(white)("XXX")(7pi)/8# radians represents an arc length of #7/8picolor(blue)(r)#

A circle with center #(5,9)# that passes through #(2,3)# will have a radius:
#color(white)("XXX")color(blue)(r)=sqrt((5-2)^2+(9-3)^2) = sqrt(45) = 3sqrt(5)#

Therefore
#color(white)("XXX")(7pi)/8# radians for the given circle represents an arc length of
#color(white)("XXXXXXXXXXXX")7/8pixx3sqrt(5) = (21sqrt(5))/8pi#

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