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Properties of Circles

What is pi? If the area of a circle is 49π, what is its circumference? If a circle is inscribed in a square, what is the ratio of the areas of the circle and the square?

Radius and Diameter

In the above diagram, circle \(C\) is inscribed in circle \(A\) and is externally tangent to circle \(B.\) The radius of circle \(A\) is \(11,\) the radius of circle \(B\) is \(6,\) and the radius of circle \(C\) is \(5.\) What is the sum of the lengths of \(\overline{AC}\) and \(\overline{BC}?\)

Note: The above diagram is not drawn to scale.

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Consider two circles with radius \(5\) and \(14,\) respectively. If \(d\) denotes the distance between the circle centers, and the number of lines that are tangent to both circles is \(4,\) what is the range of \(d\)?

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Consider two fixed circles with different radii as shown in the above diagram. If the distance between their centers, denoted \(\lvert \overline{AD} \rvert,\) is \(12,\) then the two circles become externally tangent to each other. If \(\lvert \overline{AD} \rvert=4,\) the smaller circle becomes inscribed in the larger one. What is the radius of the larger circle?

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In the above diagram, \(O\) is the center of the circle and the length of \(\overline{OA}\) is \(4.\) What is the length of \(\overline{BC}?\)

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The above partial yin yang symbol is created from a large semicircle with radius \(14\) by removing one semicircle with radius \(7\) on the left and adding one semicircle with radius \(7\) on the right. What is the circumference of the symbol?

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