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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?

# Circles - Radius and Diameter and Circumference

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 $$d$$?

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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