A circle has a chord that goes from #( 2 pi)/3 # to #(17 pi) / 12 # radians on the circle. If the area of the circle is #12 pi #, what is the length of the chord?

1 Answer
Douglas K.
Dec 23, 2016

The length of the chord is #6.4#

Explanation:

Given: The area of the circle is #12pi#

The area of a circle is:

#Area = pir^2#

#12pi = pir^2#

#r = sqrt(12)#

If you draw a radius from the center to one end of the chord and a radius from the center to the other end of the chord, you have a triangle. The angle, #theta#, between the two radii is:

#theta = (17pi)/12 - (2pi)/3 = (3pi)/4#

We can use the Law of Cosines to find the length of the chord:

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

where #a = b = r = sqrt(12) and theta = (3pi)/4#

#c = sqrt(12 + 12 - 2(12)cos((3pi)/4))#

#c = 6.4#

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