A circle has a chord that goes from #pi/4 # to #(5 pi) / 8 # radians on the circle. If the area of the circle is #96 pi #, what is the length of the chord?

I will explain this without the use of diagrams. However , I suggest that you make sketches as we go.

If you draw the radii from the centre of the circle to both ends of the chord , you will have a triangle.
Now the angle subtended at the centre by the chord will be.

#(5pi)/8 - pi/4 = (5pi)/8 - (2pi)/8 = (3pi)/8 #

The radius of the circle can be found , since we are given it's area , by using the following.

#pir^2 = 96pi → r^2 =(96cancel(pi))/cancel(pi) = 96 rArr r = sqrt96#

Hence , the triangle now has 2 sides (radii)#= sqrt96#
with an angle between them of #(3pi)/8 #
and a third side - the chord.

To find the length of the chord, from the triangle , we require to use the #color(blue)" cosine rule " #

#color(red)(|bar(ul(color(white)(a/a)color(black)( a^2 = b^2 + c^2 - (2bc costheta))color(white)(a/a)|)))#
For this question a, is the chord , b and c , the radii and #theta = (3pi)/8#

#a^2 = (sqrt96)^2 + (sqrt96)^2 - ( 2xxsqrt96xxsqrt96xxcos((3pi)/8))#

= 96 + 96 - ( 73.475) ≈ 118.525

#a^2 ≈ 118.525 rArr a "(length of chord)" = sqrt118.525 ≈ 10.89#