A circle has a chord that goes from $\frac{3 π}{8}$ $( 3 pi)/8$ to $\frac{4 π}{3}$ $(4 pi) / 3$ radians on the circle. If the area of the circle is $48 π$ $48 pi$ , what is the length of the chord?

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A circle has a chord that goes from $\frac{3 π}{8}$ $( 3 pi)/8$ to $\frac{4 π}{3}$ $(4 pi) / 3$ radians on the circle. If the area of the circle is $48 π$ $48 pi$ , what is the length of the chord?

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Answer 1 · 2017-10-02T14:27:12

$13.83$ $13.83$

Explanation:

To start, we know that the area of a circle is equal it the radius square times pi.

$A = r^{2} \times π$ $A=r^2xxpi$

We also know the area of the circle is $48 π$ $48pi$ , so using this we know that

$48 π = r^{2} \times π$ $48pi=r^2xxpi$

We can divide through by pi.

$48 = r^{2}$ $48=r^2$

And square root.

$4 \sqrt{3} = r$ $4sqrt3=r$

We have calculated the radius of the circle.

Now to find the angle across our chord we subtract the two angles we have been given.

$θ = \frac{4 π}{3} - \frac{3 π}{8} = \frac{23 π}{24}$ $theta=(4pi)/3-(3pi)/8=(23pi)/24$

enter image source here
Source and image

From the image we can see the angle has been bisected, also bisecting the chord creating two right-angled triangles.

Using trigonometry we can calculate half the length of the chord.

We have the radius/hypotenuse, the angle $\frac{θ}{2}$ $theta/2$ / $\frac{23 π}{48}$ $(23pi)/48$ and we are looking for the opposite, so we are using sin.

$sin θ = \frac{o}{h}$ $sintheta=o/h$

$h sin θ = o$ $hsintheta=o$

$4 \sqrt{3} sin (\frac{23 π}{48}) = 6.91 = o$ $4sqrt3sin((23pi)/48)=6.91=o$

This is the length of half the chord, so the chord length is

$13.83$ $13.83$ to 2 s.f

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A circle has a chord that goes from $\frac{3 π}{8}$ $( 3 pi)/8$ to $\frac{4 π}{3}$ $(4 pi) / 3$ radians on the circle. If the area of the circle is $48 π$ $48 pi$ , what is the length of the chord?

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Explanation:

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A circle has a chord that goes from 3π8( 3 pi)/8 to 4π3(4 pi) / 3 radians on the circle. If the area of the circle is 48π48 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 $\frac{3 π}{8}$ $( 3 pi)/8$ to $\frac{4 π}{3}$ $(4 pi) / 3$ radians on the circle. If the area of the circle is $48 π$ $48 pi$ , what is the length of the chord?