A circle has a chord that goes from $\frac{5 π}{4}$ $( 5 pi)/4$ to $\frac{4 π}{3}$ $(4 pi) / 3$ radians on the circle. If the area of the circle is $64 π$ $64 pi$ , what is the length of the chord?

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A circle has a chord that goes from $\frac{5 π}{4}$ $( 5 pi)/4$ to $\frac{4 π}{3}$ $(4 pi) / 3$ radians on the circle. If the area of the circle is $64 π$ $64 pi$ , what is the length of the chord?

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Answer 1 · 2018-04-15T22:49:30

The length of the chord is $= 4 \sqrt{8 - 2 (\sqrt{6} + \sqrt{2})} \approx 2.088$ $=4sqrt(8-2(sqrt(6)+sqrt(2)))~~2.088$

Explanation:

Original Drawing

Calculate the radius of the circle from the area of the circle.

area of circle = $A = π r^{2}$ $A=pir^2$

$r = \sqrt{\frac{A}{π}} = \sqrt{\frac{64 π}{π}} = 8$ $r=sqrt(A/pi)=sqrt((64pi)/pi)=8$

Now calculate the coordinates of the endpoints of the chord.

Let's represent the coordinates for the endpoint at $5 \frac{π}{4}$ $5pi/4$ by

$(x_{1}, y_{1}) = (8 cos (\frac{5 π}{4}), 8 sin (\frac{5 π}{4})) = (- 4 \sqrt{2}, - 4 \sqrt{2})$ $(x_1, y_1)=(8cos((5pi)/4), 8sin((5pi)/4))=(-4sqrt(2),-4sqrt(2))$

, the coordinates for the endpoint at $\frac{4 π}{3}$ $(4pi)/3$ by

$(x_{2}, y_{2}) = (8 cos (\frac{4 π}{3}), 8 sin (\frac{4 π}{3})) = (- 4, - 4 \sqrt{3})$ $(x_2, y_2)=(8cos((4pi)/3), 8sin((4pi)/3))=(-4,-4sqrt(3))$

The length, $l$ $l$ , of the chord can be calculated by the distance formula.

$l = \sqrt{{(y_{1} - y_{2})}_{1}^{2} + {(x_{1} - x_{2})}_{1}^{2}}$ $l=sqrt((y_1-y_2)^2+(x_1-x_2)^2)$

$= \sqrt{{(- 4 \sqrt{2} - (- 4 \sqrt{3}))}^{2} + {(- 4 \sqrt{2} - (- 4))}^{2}}$ $=sqrt((-4sqrt(2)-(-4sqrt(3)))^2+(-4sqrt(2)-(-4))^2)$

$= 4 \sqrt{{(\sqrt{3} - \sqrt{2})}^{2} + {(1 - \sqrt{2})}^{2}}$ $=4sqrt((sqrt(3)-sqrt(2))^2+(1-sqrt(2))^2)$

$= 4 \sqrt{5 - 2 \sqrt{6} + 3 - 2 \sqrt{2}}$ $=4sqrt(5-2sqrt(6)+3-2sqrt(2))$

$= 4 \sqrt{8 - 2 (\sqrt{6} + \sqrt{2})} \approx 2.088$ $=4sqrt(8-2(sqrt(6)+sqrt(2)))~~2.088$

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A circle has a chord that goes from $\frac{5 π}{4}$ $( 5 pi)/4$ to $\frac{4 π}{3}$ $(4 pi) / 3$ radians on the circle. If the area of the circle is $64 π$ $64 pi$ , what is the length of the chord?

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A circle has a chord that goes from $\frac{5 π}{4}$ $( 5 pi)/4$ to $\frac{4 π}{3}$ $(4 pi) / 3$ radians on the circle. If the area of the circle is $64 π$ $64 pi$ , what is the length of the chord?