Question #f919f

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Question #f919f

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Answer 1 · 2016-08-11T12:40:17

The length of a chord is $4$ $4$ (answer #2)

Explanation:

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Consider a Cartesian coordinate system on a plane with origin $O$ $O$ , axis $O X$ $OX$ (abscissa) and axis $O Y$ $OY$ (ordinate).

A circle defined by equation $x^{2} + y^{2} = 16$ $x^2+y^2=16$ has a radius $R = \sqrt{16} = 4$ $R=sqrt(16)=4$ and a center at origin $O$ $O$ .

Point $A = (4 cos θ, 4 sin θ)$ $A=(4cos theta, 4sin theta)$ lies on this circle such that angle $∠ X O A = θ$ $/_XOA=theta$ (counterclockwise from axis $O X$ $OX$ to radius $O A$ $OA$ )

Point $B = (4 cos (θ + 60^{o}), 4 sin (θ + 60^{o}))$ $B=(4cos(theta+60^o), 4sin(theta+60^o))$ also lies on this circle making an angle between X-axis and radius $O B$ $OB$ larger than $∠ X O A$ $/_XOA$ by $60^{o}$ $60^o$ counterclockwise.
So, $∠ X O B = ∠ X O A + 60^{o}$ $/_XOB=/_XOA+60^o$

Therefore, the angle between radiuses $O A$ $OA$ and $O B$ $OB$ is $∠ A O B = 60^{o}$ $/_AOB=60^o$

Now it is obvious that triangle $Δ A O B$ $Delta AOB$ is equilateral since $OA=OB$ as radiuses and $/_AOB=60^o$ .

Therefore, $AB=OA=OB=4$ .

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