Geometry

Properties of Circles

Circles - Radius and Diameter and Circumference

         

In the above diagram, circles CC and BB are both externally tangent to each other and inscribed in circle A.A. The radius of circle AA is 11,11, the radius of circle BB is 6,6, and the radius of circle CC is 5.5. What is the sum of the lengths of AC\overline{AC} and BC?\overline{BC}?

Note: The above diagram is not drawn to scale.

Consider two circles with radius 55 and 14,14, respectively. If dd denotes the distance between the circle centers, and the number of lines that are tangent to both circles is 4,4, what is the range of dd?

Consider two fixed circles with different radii as shown in the above diagram. If the distance between their centers, denoted AD,\lvert \overline{AD} \rvert, is 12,12, then the two circles become externally tangent to each other. If AD=4,\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, OO is the center of the circle and the length of OA\overline{OA} is 4.4. What is the length of BC?\overline{BC}?

The above partial yin yang symbol is created from a large semicircle with radius 1414 by removing one semicircle with radius 77 on the left and adding one semicircle with radius 77 on the right. What is the circumference of the symbol?

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