Three circles of radius 2.00,5.00, and 8.00 inches are tangent to cach other (see the figure). F

step 1 We have three circles tangent to each other. One circle have radius of 2 inches, 2 inches. The s

Three circles of radius 2.00,5.00, and 8.00 inches are...

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Key Concepts

Circle Tangency

When two circles are tangent to each other, they touch at exactly one point. The line connecting their centers passes through this point, and the distance between the centers equals the sum of their radii if the circles are externally tangent. This fundamental property allows one to relate the sizes of the circles to the distances between their centers.

Triangle Formation from Circle Centers

When multiple circles are tangent to each other, the centers of the circles form a triangle where each side is determined by the sum of the radii of the two circles that are tangent. This triangle encapsulates the spatial relationships among the circles and provides a structure for further geometric analysis.

Law of Cosines

The Law of Cosines is a generalization of the Pythagorean theorem applicable to any triangle. It relates the lengths of the sides of a triangle to the cosine of one of its angles. This formula is essential for computing angles in a triangle when the lengths of all three sides are known, which is particularly useful in problems involving tangency where the side lengths of the triangle are derived from the radii of the circles.

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