A circle's center is at #(2 ,7 )# and it passes through #(3 ,1 )#. What is the length of an arc covering #(2pi ) /3 # radians on the circle?

1 Answer
Alan P.
Jan 27, 2016

Arc length: #(2sqrt(37)pi)/3#

Explanation:

Part 1:
If the circle has center at #(2,7)# and passes through #(3,1)#, its radius is:
#color(white)("XXX")r=sqrt((2-3)^2+(7-1)^2) = sqrt((-1)^2+6^2) = sqrt(37#

Part 2:
For a circle with radius #r#
#color(white)("XXX")"angle" = 2pi rarr "arc length" = 2pir# (circumference of circle)
#color(white)("XXX")"angle" = pi rarr "arc length" = pir#
#color(white)("XXX")"angle" =2/3pi rarr "arc length" = 2/3 pir#

Solution
A arc with radius #sqrt(37)# (from Part 1) and an angle of #(2pi)/3#
will have an arc length of #2/3pisqrt(37)= (2sqrt(37))/3pi#

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