A circle has a chord that goes from $( pi)/3$ to $(7 pi) / 12$ radians on the circle. If the area of the circle is $16 pi$ , what is the length of the chord?

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Answer 1 · 2016-10-10T20:35:36

The chord length is $c ~~ 3.06$

Use the equation for the area of the circle to find the radius:

$A = pir^2$

$16pi = pir^2$

$16 = r^2$

$r = 4$

Compute the angle:

$theta = 7pi/12 - pi/3$

$theta = 7pi/12 - 4pi/12$

$theta = 3pi/12 = pi/4$

Because the chord and two radii form a triangle, one can use the Law of Cosines to find the length of the chord, c:

$c^2 = r^2 + r^2 - 2(r)(r)cos(theta)$

$c^2 = 2r^2(1 - cos(theta))$

$c^2 = 2(4)^2(1 - cos(pi/4))$

$c ~~ 3.06$

A circle has a chord that goes from ( pi)/3 ( pi)/3 to (7 pi) / 12 (7 pi) / 12 radians on the circle. If the area of the circle is 16 pi 16 pi , what is the length of the chord?