A circle has a chord that goes from $\frac{π}{3}$ $( pi)/3$ to $\frac{2 π}{3}$ $(2 pi) / 3$ 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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A circle has a chord that goes from $\frac{π}{3}$ $( pi)/3$ to $\frac{2 π}{3}$ $(2 pi) / 3$ 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 · 2016-11-17T02:49:18

The length of the chord is 4.

Explanation:

Because we are given the area, we can find the radius:

$A r e a = π r^{2}$ $Area = pir^2$

$16 π = π r^{2}$ $16pi = pir^2$

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

$r = 4$ $r = 4$

If we imagine two radii, one the starting point of the chord and the other on the ending point of the chord, then they form a triangle with the chord. The angle between to the two radii is:

$θ = \frac{2 π}{3} - \frac{π}{3}$ $theta = (2pi)/3 - pi/3$

$θ = \frac{π}{3}$ $theta = pi/3$

At this point, one can realize that the triangle formed is equilateral and the length of the chord is 4.

But, if this special case does not exist, then the following is how the length of the chord is computed.

We know lengths of two sides and the angle between those two sides, therefore, we can use the Law of Cosines, to find the length of the third side:

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

$c = r \sqrt{2 (1 - cos (θ))}$ $c = rsqrt(2(1 - cos(theta)))$

$c = 4 \sqrt{2 (1 - cos (\frac{π}{3}))}$ $c = 4sqrt(2(1 - cos(pi/3)))$

$c = 4$ $c = 4$

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A circle has a chord that goes from $\frac{π}{3}$ $( pi)/3$ to $\frac{2 π}{3}$ $(2 pi) / 3$ 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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A circle has a chord that goes from ( pi)/3 π3( pi)/3 to (2 pi) / 3 2π3(2 pi) / 3 radians on the circle. If the area of the circle is 16 pi 16π16 pi , what is the length of the chord?

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A circle has a chord that goes from $\frac{π}{3}$ $( pi)/3$ to $\frac{2 π}{3}$ $(2 pi) / 3$ radians on the circle. If the area of the circle is $16 π$ $16 pi$ , what is the length of the chord?