The Geometric Mean
The geometric mean of two numbers and is . It appears naturally when an altitude is drawn to the hypotenuse of a right triangle, creating three similar triangles.
Introduction
When you drop an altitude from the right angle to the hypotenuse, you split the original right triangle into two smaller right triangles. All three triangles are similar to each other (by AA). This beautiful relationship produces three geometric mean relationships that are essential for right triangle problems.
Past Knowledge
AA Similarity (7.2.1). Right triangle properties. Proportions (7.1).
Today's Goal
Find the geometric mean and apply the altitude-on-hypotenuse theorems.
Future Success
Pythagorean Theorem proof (8.1.1), trigonometry (Unit 8).
Key Concepts
Geometric Mean
The geometric mean of two positive numbers and is:
The geometric mean is always between and , and equals them when .
Altitude-on-Hypotenuse Theorem
In a right triangle, the altitude to the hypotenuse creates three similar triangles. If the hypotenuse segments are and , and the altitude is :
- Altitude rule: (altitude is the geometric mean of the two hypotenuse segments)
- Leg rule: Each leg is the geometric mean of the hypotenuse and the adjacent segment: and
Theorem & Proof
Two-Column Proof: Altitude-on-Hypotenuse Theorem (Altitude Rule)
Given: Right with right angle at , altitude to hypotenuse
Prove: (i.e., )
Strategy: Show the two smaller triangles are similar using AA, then set up a proportion.
| # | Statement | Reason |
|---|---|---|
| 1 | and | Given |
| 2 | (definition of altitude to hypotenuse) | |
| 3 | (in and ) | Reflexive Property |
| 4 | AA Similarity (each pair shares a right angle and another acute angle) | |
| 5 | In particular, , so | Corresponding sides of similar triangles are proportional |
| 6 | Cross-multiply step 5 |
∎ The altitude to the hypotenuse is the geometric mean of the two segments it creates. The leg rules follow similarly from and .
Worked Examples
Computing a Geometric Mean
Find the geometric mean of 4 and 16.
Check: ✓
Altitude Rule
In a right triangle, the altitude to the hypotenuse creates segments of length 3 and 12. Find the altitude.
Altitude = geometric mean of the segments:
Finding a Leg Using the Leg Rule
In right with altitude to hypotenuse , and . Find leg .
Leg rule:
Common Pitfalls
Confusing Arithmetic and Geometric Mean
The arithmetic mean (average) of 4 and 16 is . The geometric mean is . They're different! The geometric mean is always ≤ the arithmetic mean.
Mixing Up the Altitude and Leg Rules
The altitude is the geometric mean of the two segments of the hypotenuse. A leg is the geometric mean of the whole hypotenuse and the adjacent segment. Don't swap them.
Real-Life Applications
Finance — Average Growth Rate
If an investment grows by 44% one year and by 0% the next, the average annual return is NOT 22%. The geometric mean gives the correct average: , or 20% per year.
Screen Diagonal — Display Sizing
The geometric mean relates screen width and height to the diagonal in certain aspect ratio calculations, which is why monitor sizes use it indirectly when converting between formats.
Practice Quiz
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