Properties of Circles

Circle Properties: Level 3 Challenges


A quarter-circle with radius RR 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 πR2X,\frac { \pi { R }^{ 2 } }{ X }, find the value of X.X.

ΔABC\Delta ABC is inscribed in a circle such that 864A=B=C864 \angle A = \angle B = \angle C. If BB and CC are adjacent vertices of a regular nn-gon inscribed in the circle, find nn.

Details and Assumptions

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

In a circle with center OO, a chord ABAB is drawn such that AOB=120\angle AOB = 120^\circ. Radius AO=10AO=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 aπ+b3c\dfrac{a\pi +b\sqrt{3}}{c} for integers a,b,ca,b,c with gcd(a,c)=1\gcd(a,c)=1 What is the value of a+b+ca+b+c?

You are a farmer, with a round fenced field of radius 1010 meters. You tie your goat to the fence with a rope, and realize that the goat can eat exactly 12\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 14×714\times 7 rectangle, with two semicircles of radius 7 drawn.


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