A circle has a chord that goes from $\frac{3 π}{2}$ $( 3 pi)/2$ to $\frac{7 π}{4}$ $(7 pi) / 4$ radians on the circle. If the area of the circle is $99 π$ $99 pi$ , what is the length of the chord?

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A circle has a chord that goes from $\frac{3 π}{2}$ $( 3 pi)/2$ to $\frac{7 π}{4}$ $(7 pi) / 4$ radians on the circle. If the area of the circle is $99 π$ $99 pi$ , what is the length of the chord?

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Answer 1 · 2016-06-21T14:52:50

7.62 units

Explanation:

First, use a unit circle to determine the end points of the chord on the circle.
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If each endpoint on the chord is connected to the center of the circle, an isosceles triangle is formed whose congruent sides each have a length of $r$ $r$ , the length of the radius.

The angle between the two equivalent sides of the triangle is equal to the difference between the angles given in the problem:

$θ = \frac{7 π}{4} - \frac{3 π}{2} = \frac{π}{4} r a d i a n s$ $theta=(7pi)/4-(3pi)/2=pi/4 radians$

Finally, the law of cosines can be used to determine an equation for the length of the chord:

$c^{2} = a^{2} + b^{2} - 2 a b cos θ$ $c^2=a^2+b^2-2abcostheta$

Since both $a$ $a$ and $b$ $b$ are equal to $r$ $r$ , the formula can be rewritten as:
$c^{2} = r^{2} + r^{2} - 2 \cdot r \cdot r \cdot cos θ$ $c^2=r^2+r^2-2*r*r*costheta$
$c^{2} = 2 r^{2} - 2 r^{2} cos θ$ $c^2=2r^2-2r^2costheta$
$c^{2} = 2 r^{2} \cdot (1 - cos θ)$ $c^2=2r^2*(1-costheta)$

The problem states that the area of the circle is $99 π$ $99pi$ . This allows us to solve for $r^{2}$ $r^2$ :

$A = π r^{2}$ $A=pir^2$
$\frac{A}{π} = r^{2}$ $A/pi=r^2$

$r^{2} = \frac{99 π}{π} = 99$ $r^2=(99pi)/pi=99$

Plug this value into the equation for the chord:
$c^{2} = 2 r^{2} \cdot (1 - cos θ)$ $c^2=2r^2*(1-costheta)$
$c^{2} = 2 \cdot 99 \cdot (1 - cos (\frac{π}{4}))$ $c^2=2*99*(1-cos(pi/4))$
$c^{2} = 198 \cdot (1 - 0.707)$ $c^2=198*(1-0.707)$
$c^{2} = 58.014$ $c^2=58.014$
$c = 7.62$ $c=7.62$

Note: Since the units of length are not provided, just use "units."

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A circle has a chord that goes from $\frac{3 π}{2}$ $( 3 pi)/2$ to $\frac{7 π}{4}$ $(7 pi) / 4$ radians on the circle. If the area of the circle is $99 π$ $99 pi$ , what is the length of the chord?

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Explanation:

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A circle has a chord that goes from $\frac{3 π}{2}$ $( 3 pi)/2$ to $\frac{7 π}{4}$ $(7 pi) / 4$ radians on the circle. If the area of the circle is $99 π$ $99 pi$ , what is the length of the chord?