In geometry, the geometric mean relates to finding a length, area, or volume that represents the "average" size within a set of related geometric figures, often right triangles.
Geometric Mean in Right Triangles
The most common application of geometric mean in geometry involves right triangles and altitudes drawn to the hypotenuse. Specifically:
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Altitude to the Hypotenuse Theorem: If an altitude is drawn to the hypotenuse of a right triangle, then the length of the altitude is the geometric mean between the lengths of the two segments of the hypotenuse.
- For a right triangle ABC with right angle at C, and altitude CD drawn to hypotenuse AB, then: CD = √(AD * DB).
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Leg Rule: If an altitude is drawn to the hypotenuse of a right triangle, then each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.
- In the same triangle ABC, AC = √(AD AB) and BC = √(BD AB).
Examples
Example 1: Altitude
Given a right triangle where the altitude to the hypotenuse divides the hypotenuse into segments of length 4 and 9, find the length of the altitude.
- Altitude = √(4 * 9) = √36 = 6
Example 2: Leg
Given a right triangle where the altitude to the hypotenuse divides the hypotenuse into segments of length 4 and 9. Find the length of the leg adjacent to the segment of length 4. The hypotenuse has length 4 + 9 = 13.
- Leg = √(4 * 13) = √52 = 2√13
Summary
The geometric mean in geometry, especially within the context of right triangles and altitudes, provides a way to relate and calculate lengths within the figure, allowing for the determination of unknown side lengths or altitude lengths when certain segments are known. It's a powerful tool for solving geometric problems involving similarity and proportionality.
