A circle has a chord that goes from #( 5 pi)/3 # to #(17 pi) / 12 # radians on the circle. If the area of the circle is #18 pi #, what is the length of the chord?

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Answer 1 · 2016-10-13T17:53:26

The length of the chord is #c ~~ 3.25#

We can compute the radius, given the area of the circle:

#A = pir^2#

#18pi = pir^2#

#18 = r^2#

#sqrt18 = r#

Compute the angle:

#theta = (17pi)/12 - (5pi)/3#

#theta = (17pi)/12 - (20pi)/12#

#theta = -(3pi)/12 = -pi/4#

We can compute the length of the chord, c, using the Law of Cosines:

#c^2 = a^2 + b^2 - 2(a)(b)cos(theta)

where # a = b = r# and #theta = -pi/4#:

#c^2 = r^2 + r^2 - 2(r)(r)cos(-pi/4)#

#c^2 = 36(1 - sqrt(2)/2)#

#c ~~ 3.25#