A circle has a chord that goes from #( 5 pi)/4 # to #(4 pi) / 3 # radians on the circle. If the area of the circle is #64 pi #, what is the length of the chord?

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Answer 1 · 2018-04-15T22:49:30

The length of the chord is #=4sqrt(8-2(sqrt(6)+sqrt(2)))~~2.088#

Original Drawing

Calculate the radius of the circle from the area of the circle.

area of circle = #A=pir^2#

#r=sqrt(A/pi)=sqrt((64pi)/pi)=8#

Now calculate the coordinates of the endpoints of the chord.

Let's represent the coordinates for the endpoint at #5pi/4# by

#(x_1, y_1)=(8cos((5pi)/4), 8sin((5pi)/4))=(-4sqrt(2),-4sqrt(2))#

, the coordinates for the endpoint at #(4pi)/3# by

#(x_2, y_2)=(8cos((4pi)/3), 8sin((4pi)/3))=(-4,-4sqrt(3))#

The length, #l#, of the chord can be calculated by the distance formula.

#l=sqrt((y_1-y_2)^2+(x_1-x_2)^2)#

#=sqrt((-4sqrt(2)-(-4sqrt(3)))^2+(-4sqrt(2)-(-4))^2)#

#=4sqrt((sqrt(3)-sqrt(2))^2+(1-sqrt(2))^2)#

#=4sqrt(5-2sqrt(6)+3-2sqrt(2))#

#=4sqrt(8-2(sqrt(6)+sqrt(2)))~~2.088#