A circle has a chord that goes from #( pi)/4 # to #(13 pi) / 8 # radians on the circle. If the area of the circle is #32 pi #, what is the length of the chord?

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Answer 1 · 2017-04-27T20:13:58

#c ~~ 9.4" "larr# the length of the chord.

We can use the area of the circle, #32pi#, to compute the radius:

#"Area" = pir^2#

#32pi = pir^2#

#r = sqrt(32) = 4sqrt2#

Two radii and the chord form an isosceles triangle where "c" is the unknown length of the chord.

The lengths of the other two sides are:

#a = b = r = 4sqrt(2)#

The angle between the radii is:

#theta = (13pi)/8-pi/4 = (11pi)/8#

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)#

#c = sqrt(32+32-64cos((11pi)/8)#

#c ~~ 9.4" "larr# the length of the chord.