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Geometry

# Circle Properties: Level 3 Challenges

A quarter-circle with radius $$R$$ is drawn. Inside it, two semicircles and a circle are drawn as shown in the above figure. If the area of the grey colored region is $\frac { \pi { R }^{ 2 } }{ X },$ find the value of $$X.$$

$$\Delta ABC$$ is inscribed in a circle such that $$864 \angle A = \angle B = \angle C$$. If $$B$$ and $$C$$ are adjacent vertices of a regular $$n$$-gon inscribed in the circle, find $$n$$.

Details and Assumptions

• The diagram above is (extremely) not to scale.

In a circle with center $$O$$, a chord $$AB$$ is drawn such that $$\angle AOB = 120^\circ$$. Radius $$AO=10$$. A circle is drawn in the major arc such that it's radius is maximum, with center E, it touches the larger circle at point X as shown in figure.

The area of the shaded region can be expressed as $$\dfrac{a\pi +b\sqrt{3}}{c}$$ for integers $$a,b,c$$ with $$\gcd(a,c)=1$$ What is the value of $$a+b+c$$?

You are a farmer, with a round fenced field of radius $$10$$ meters. You tie your goat to the fence with a rope, and realize that the goat can eat exactly $$\frac{1}{2}$$ of the field. How long is the rope?

Note: You can use a computational solver for the last step of the problem.

Find the area of the blue shaded region in this $$14\times 7$$ rectangle, with two semicircles of radius 7 drawn.

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