A circle has a chord that goes from #( pi)/3 # to #(7 pi) / 12 # radians on the circle. If the area of the circle is #16 pi #, what is the length of the chord?

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Answer 1 · 2016-10-10T20:35:36

The chord length is #c ~~ 3.06#

Use the equation for the area of the circle to find the radius:

#A = pir^2#

#16pi = pir^2#

#16 = r^2#

#r = 4#

Compute the angle:

#theta = 7pi/12 - pi/3#

#theta = 7pi/12 - 4pi/12#

#theta = 3pi/12 = pi/4#

Because the chord and two radii form a triangle, one can use the Law of Cosines to find the length of the chord, c:

#c^2 = r^2 + r^2 - 2(r)(r)cos(theta)#

#c^2 = 2r^2(1 - cos(theta))#

#c^2 = 2(4)^2(1 - cos(pi/4))#

#c ~~ 3.06#