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

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Answer 1 · 2016-04-23T11:08:41

Length of the chord is 1

Tony B

Given: $∠ A = \frac{2 π}{3} - \frac{π}{3} = \frac{π}{3}$ $" "/_A = (2pi)/3-pi/3 = pi/3$

Known: $∠ C = ∠ B$ $/_C=/_B$

$∠ A + ∠ B + ∠ C = π radians \to 180^{o}$ $/_A+/_B+/_C = pi" radians" -> 180^o$

Thus $∠ C = ∠ B = \frac{1}{2} (π - \frac{π}{3}) = \frac{π}{3} radians \to 60^{o}$ $color(blue)(/_C=/_B = 1/2( pi-pi/3) = pi/3" radians" ->60^o)$

Thus $Delta ABC ->" equilateral triangle"$
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Given: $" area"=4pi$

But $" area"=pir^2$

Equate through area

$" "4pi="area"=pir^2$

Divide both sides by $pi$

$=>4=r^2$

$=> r=2$

Side $b=c=2$
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Known: $cos(/_C)= a/2 -:r=a/(2r)$

$=>2rcos(/_C)=a$

$=>2(2)cos(pi/3)=a$

but $cos(pi/3)=1/2$

$=>2(2)(1/2)=a$

$=>"chord "= a=1$

A circle has a chord that goes from ( pi)/3 π3( pi)/3 to (2 pi) / 3 2π3(2 pi) / 3 radians on the circle. If the area of the circle is 4 pi 4π4 pi , what is the length of the chord?