Circles P and Q have radii 6 and 2 and are tangent to each...

Circles $P$ and $Q$ have radii 6 and 2 and are tangent to each other. Find the length of their common external tangent $\overline{A B}$ . (Hint: Draw $\overline{P Q}, \overline{P A},$ and $\overline{Q B}$ .)

Key Concepts

Circle Tangency

This concept refers to the condition where two circles touch at exactly one point. In problems involving tangency, understanding whether circles are externally tangent (touching from the outside) or internally tangent (one touches from the inside of the other) is crucial since it affects the way distances between the centers relate to the radii.

Common External Tangents

A common external tangent is a line that touches two circles without crossing the line segment joining their centers. In solving problems with common external tangents, the geometry usually involves constructing segments from the centers to the points of tangency and applying properties of similar triangles or the Pythagorean theorem to relate distances.

Perpendicularity of Radii to Tangents

At the point of tangency, a tangent to a circle is perpendicular to the radius drawn to that point. This fundamental property enables the formation of right triangles, which can then be analyzed using the Pythagorean theorem to find unknown side lengths in the configuration.

Pythagorean Theorem

The Pythagorean theorem is a key tool in relating the sides of a right-angled triangle. In the context of circle tangent problems, it is applied to right triangles formed by the radii and tangent segments, allowing the calculation of distances such as the length of the tangent segment.

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