A circle has a chord that goes from $\frac{5 π}{3}$ $( 5 pi)/3$ to $\frac{17 π}{12}$ $(17 pi) / 12$ radians on the circle. If the area of the circle is $18 π$ $18 pi$ , what is the length of the chord?

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Answer 1 · 2016-10-13T17:53:26

The length of the chord is $c \approx 3.25$ $c ~~ 3.25$

We can compute the radius, given the area of the circle:

$A = π r^{2}$ $A = pir^2$

$18 π = π r^{2}$ $18pi = pir^2$

$18 = r^{2}$ $18 = r^2$

$\sqrt{18} = r$ $sqrt18 = r$

Compute the angle:

$θ = \frac{17 π}{12} - \frac{5 π}{3}$ $theta = (17pi)/12 - (5pi)/3$

$θ = \frac{17 π}{12} - \frac{20 π}{12}$ $theta = (17pi)/12 - (20pi)/12$

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

We can compute the length of the chord, c, using the Law of Cosines:

#c^2 = a^2 + b^2 - 2(a)(b)cos(theta)

where $a = b = r$ $a = b = r$ and $θ = - \frac{π}{4}$ $theta = -pi/4$ :

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

$c^{2} = 36 (1 - \frac{\sqrt{2}}{2})$ $c^2 = 36(1 - sqrt(2)/2)$

$c \approx 3.25$ $c ~~ 3.25$

A circle has a chord that goes from 5π3( 5 pi)/3 to 17π12(17 pi) / 12 radians on the circle. If the area of the circle is 18π18 pi , what is the length of the chord?