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?

1 Answer
Apr 15, 2018

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

Explanation:

Original DrawingOriginal 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