# How to Calculate the Geometric Mean in Triangles

Triangles, with their vast array of properties and theorems, have a hidden gem that seamlessly ties algebra and geometry together: the geometric mean. As a bridge between areas and ratios, the concept of the geometric mean in triangles is both fascinating and profoundly useful. Embark with us on an exploration of how the geometric mean interacts with triangles, unlocking deeper understanding and new perspectives.

## Step-by-step Guide: Geometric Mean in Triangles

**1. Definition of Geometric Mean:**

The geometric mean between two numbers, \(a\) and \(b\), is the square root of their product, represented as \(\sqrt{a \times b}\).

**2. Geometric Mean in Right Triangles:**

**Altitude to the Hypotenuse:** When an altitude is drawn to the hypotenuse of a right triangle, it creates two smaller triangles that are similar to the original triangle. The altitude serves as the geometric mean between the two segments it divides the hypotenuse into.

If the hypotenuse is divided into segments \(x\) and \(y\), and \(h\) is the length of the altitude, then:

\( h^2 = x \times y \)

\( h = \sqrt{x \times y} \)

**Legs as Geometric Mean:** In the same scenario, each leg of the right triangle is the geometric mean between the hypotenuse and the segment of the hypotenuse adjacent to the leg.

### Examples

**Example 1:**

In a right triangle, if an altitude drawn to the hypotenuse divides it into segments of \(3 \text{ cm}\) and \(12 \text{ cm}\), find the length of the altitude.

**Solution:**

Using the property of geometric mean:

\( h = \sqrt{3 \times 12} \)

\( h = \sqrt{36} \)

\( h = 6 \text{ cm} \)

**Example 2:**

Given a right triangle with a hypotenuse of \(10 \text{ cm}\) and one segment of the hypotenuse being \(4 \text{ cm}\) (after drawing an altitude), determine the length of the leg adjacent to the \(4 \text{ cm}\) segment.

**Solution:**

Using the geometric mean property for legs:

\( \text{leg} = \sqrt{\text{hypotenuse} \times \text{adjacent segment}} \)

\( \text{leg} = \sqrt{10 \times 4} = \sqrt{40} \approx 6.32 \text{ cm} \)

### Practice Questions:

- If an altitude to the hypotenuse of a right triangle divides the hypotenuse into segments of \(5 \text{ cm}\) and \(20 \text{ cm}\), what is the length of the altitude?
- In a right triangle with a hypotenuse of \(13 \text{ cm}\) and one segment of \(5 \text{ cm}\), find the length of the leg adjacent to the \(5 \text{ cm}\) segment.

**Answers:**

- \( h = \sqrt{5 \times 20} = \sqrt{100} = 10 \text{ cm}\).
- Leg length \(= \sqrt{13 \times 5} \approx 8.06 \text{ cm}\).

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