Properties of Circles

Circles - Radius and Diameter and Circumference


In the above diagram, circles \(C\) and \(B\) are both externally tangent to each other and inscribed in circle \(A.\) 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.

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 \(d\)?

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?

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

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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