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# What is the circumference of the circle above?

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Re: What is the circumference of the circle above? [#permalink]
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Q: What is the radius of circle.
$$Circumference=2*\pi*radius$$
$$\pi$$ is a constant.

1. $$\stackrel{\frown}{XYZ}=18$$
$$\stackrel{\frown}{XYZ}=(\theta/360)*2*\pi*radius$$
$$\stackrel{\frown}{XYZ}$$=length of the arc XYZ
$$18=(\theta/360)*2*\pi*radius$$
$$\theta$$ angle subtended by major arc $$\stackrel{\frown}{XYZ}$$ at the center of the circle
$$\theta$$ unknown
NOT SUFFICIENT.

2. $$\angle r = \angle s$$
The triangle is an equilateral triangle and $$\angle r = 60^{\circ}$$
Minor arc $$\stackrel{\frown}{XZ}$$ makes an angle of $$60^{\circ}$$ with the point Y, which lies on the circumference.
Theorem,The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.
So if the minor arc XZ is making an angle of r with a point on the circumference of the circle, minor arc XZ will make an angle of 2r with the centre of the circle. Now we know angle of minor arc XZ is $$2r=120^{\circ}$$
And angle subtended by major arc XYZ at the center is $$\theta=360-120=240$$
However, with this information also we won't be able to find the radius of the circle because the length of the arc is not known. NOT SUFFICIENT.

Combining both of the above;
We know, length of major arc $$\stackrel{\frown}{XYZ}$$ and angle made by the arc from the center i.e. $$\theta$$

Put these into the formula now;
$$\stackrel{\frown}{XYZ}=(\theta/360)*2*\pi*radius$$
$$\theta=240^{\circ} \hspace \hspace \stackrel{\frown}{XYZ}=18$$
$$18=(240/360)*2*\pi*radius$$
$$radius=27/(2*\pi)$$

$$Circumference=2*\pi*radius=(2*\pi)*27/(2*\pi)=27$$
SUFFICIENT.

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Re: What is the circumference of the circle above? [#permalink]
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Re: What is the circumference of the circle above? [#permalink]
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