A circle has a chord that goes from $\frac{3 π}{4}$ $( 3 pi)/4$ to $\frac{5 π}{4}$ $(5 pi) / 4$ radians on the circle. If the area of the circle is $72 π$ $72 pi$ , what is the length of the chord?

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A circle has a chord that goes from $\frac{3 π}{4}$ $( 3 pi)/4$ to $\frac{5 π}{4}$ $(5 pi) / 4$ radians on the circle. If the area of the circle is $72 π$ $72 pi$ , what is the length of the chord?

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Answer 1 · 2016-03-24T01:54:42

Length of the chord $l = 12$ $" "l=12" "$ units

Explanation:

The central angle $θ = \frac{5 π}{4} - \frac{3 π}{4} = \frac{π}{2}$ $theta=(5pi)/4-(3pi)/4=pi/2$
We have a triangle with sides $r, r, l$ $r, r, l$ and angle $θ$ $theta$ opposite side $l$ $l$

Area of the circle $A = π r^{2}$ $A=pir^2$
$A = 72 π$ $A=72pi$ is given

Area = Area
$72 π = π r^{2}$ $72pi=pi r^2$
$r^{2} = 72$ $r^2=72$
$r = 6 \sqrt{2}$ $r=6sqrt2$

By the cosine law we can solve for the length $l$ $l$

$l = \sqrt{(r^{2} + r^{2} - 2 \cdot r \cdot r \cdot cos θ)}$ $l=sqrt((r^2+r^2-2*r*r*cos theta))$
$l = \sqrt{(72 + 72 - 2 \cdot 72 \cdot cos (\frac{π}{2}))}$ $l=sqrt((72+72-2*72*cos (pi/2)))$
$l = \sqrt{144}$ $l=sqrt144$
$l = 12$ $l=12" "$ units

God bless....I hope the explanation is useful.

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A circle has a chord that goes from $\frac{3 π}{4}$ $( 3 pi)/4$ to $\frac{5 π}{4}$ $(5 pi) / 4$ radians on the circle. If the area of the circle is $72 π$ $72 pi$ , what is the length of the chord?

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Explanation:

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A circle has a chord that goes from 3π4( 3 pi)/4 to 5π4(5 pi) / 4 radians on the circle. If the area of the circle is 72π72 pi , what is the length of the chord?

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A circle has a chord that goes from $\frac{3 π}{4}$ $( 3 pi)/4$ to $\frac{5 π}{4}$ $(5 pi) / 4$ radians on the circle. If the area of the circle is $72 π$ $72 pi$ , what is the length of the chord?