Geometric Mean in Right Triangles

Before applying it to geometry, let's define the geometric mean. For any two positive numbers $a$ and $b$, the geometric mean is the positive number $x$ such that: $\frac{a}{x} = \frac{x}{b}$ Solving for $x$, we get $x^2 = ab$, or $x = \sqrt{ab}$.

The Right Triangle Altitude Theorems

When you draw an altitude from the right angle of a right triangle to its hypotenuse, it divides the original triangle into two smaller triangles. Because they share complementary angles, all three triangles are similar to each other.

This similarity leads to two important geometric mean theorems:

1. The Altitude Rule The altitude drawn to the hypotenuse is the geometric mean of the two segments of the hypotenuse. $\text{Altitude} = \sqrt{\text{Segment 1} \times \text{Segment 2}}$

2. The Leg Rule Each leg of the right triangle is the geometric mean of the entire hypotenuse and the segment of the hypotenuse adjacent to that leg. $\text{Leg} = \sqrt{\text{Whole Hypotenuse} \times \text{Adjacent Segment}}$

Example Problems

Example 1: Find the geometric mean of $3$ and $12$.

Using the formula for the geometric mean: $x = \sqrt{3 \cdot 12} = \sqrt{36} = 6$

Example 2: In right $\triangle ABC$ with altitude $CD$ to hypotenuse $AB$, $AD = 4$ and $DB = 9$. Find $CD$, $AC$, and $BC$.

First, find the altitude $CD$ using the Altitude Rule. $CD$ is the geometric mean of the hypotenuse segments $AD$ and $DB$: $CD = \sqrt{4 \cdot 9} = \sqrt{36} = 6$

Next, find the total length of the hypotenuse $AB$: $AB = AD + DB = 4 + 9 = 13$

Now, find leg $AC$ using the Leg Rule. $AC$ is the geometric mean of the whole hypotenuse ($AB$) and its adjacent segment ($AD$): $AC = \sqrt{13 \cdot 4} = 2\sqrt{13}$

Finally, find leg $BC$ using the Leg Rule. $BC$ is the geometric mean of the whole hypotenuse ($AB$) and its adjacent segment ($DB$): $BC = \sqrt{13 \cdot 9} = 3\sqrt{13}$