A circle has a chord that goes from #( pi)/6 # to #(5 pi) / 6 # radians on the circle. If the area of the circle is #18 pi #, what is the length of the chord?

1 Answer
Douglas K.
Apr 20, 2017

#c = sqrt(54)#

Explanation:

We can compute the radius from the area of the circle:

#pir^2=18pi#

#r = sqrt18#

To radii and the chord form a triangle. The angle between the two radii is:

#theta = (5pi)/6-pi/6= (2pi)/3#

If we use the angle and the length of the two radii, we can use the Law of Cosines:

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

where #a = b = r = sqrt18 and theta = (2pi)/3#

#c = sqrt((sqrt18)^2+(sqrt18)^2-2(sqrt18)(sqrt18)cos((2pi)/3)#

#c = sqrt(54)#

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