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

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Answer 1 · 2016-10-10T20:35:36

The chord length is $c \approx 3.06$ $c ~~ 3.06$

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

$A = π r^{2}$ $A = pir^2$

$16 π = π r^{2}$ $16pi = pir^2$

$16 = r^{2}$ $16 = r^2$

$r = 4$ $r = 4$

Compute the angle:

$θ = 7 \frac{π}{12} - \frac{π}{3}$ $theta = 7pi/12 - pi/3$

$θ = 7 \frac{π}{12} - 4 \frac{π}{12}$ $theta = 7pi/12 - 4pi/12$

$θ = 3 \frac{π}{12} = \frac{π}{4}$ $theta = 3pi/12 = pi/4$

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

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

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

$c^{2} = 2 {(4)}^{2} (1 - cos (\frac{π}{4}))$ $c^2 = 2(4)^2(1 - cos(pi/4))$

$c \approx 3.06$ $c ~~ 3.06$

A circle has a chord that goes from π3( pi)/3 to 7π12(7 pi) / 12 radians on the circle. If the area of the circle is 16π16 pi , what is the length of the chord?