What is the Length of the Altitude Drawn to the Hypotenuse in a Right Triangle?

In a right triangle, the length of the altitude drawn from the right angle to the hypotenuse is found using a specific geometric relationship.

Based on a key geometric theorem, the length of this altitude has a precise relationship to the segments it creates on the hypotenuse.

Understanding the Altitude and Hypotenuse Segments

When you draw an altitude from the vertex of the right angle perpendicular to the hypotenuse, it performs two main actions:

1. It divides the original right triangle into two smaller right triangles that are similar to the original triangle and to each other.
2. It divides the hypotenuse into two distinct segments.

Let's denote the length of the altitude as h. Let the hypotenuse be divided into two segments with lengths x and y.

The Geometric Mean (Altitude) Theorem

The relationship between the altitude's length and the two hypotenuse segments is described by the Geometric Mean (Altitude) Theorem. As stated in the reference:

Theorem 9.7 Geometric Mean (Altitude) Theorem: In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments of the hypotenuse.

This theorem provides the exact way to determine the altitude's length.

Calculating the Altitude's Length

According to the theorem, the length of the altitude (h) is the geometric mean of the lengths of the two segments of the hypotenuse (x and y).

The geometric mean of two numbers is the square root of their product. Therefore, the length of the altitude h can be calculated using the formula:

h = √(x * y)

Where:

• h is the length of the altitude to the hypotenuse.
• x is the length of one segment of the hypotenuse created by the altitude.
• y is the length of the other segment of the hypotenuse created by the altitude.

Practical Application

To find the length of the altitude:

1. Identify the two segments on the hypotenuse that are created by the altitude.
2. Measure the lengths of these two segments.
3. Multiply the lengths of the two segments.
4. Take the square root of the product.

Example:

If the altitude divides the hypotenuse into segments of length 4 units and 9 units:

• Segment 1 (x) = 4
• Segment 2 (y) = 9
• Product of segments = 4 * 9 = 36
• Length of the altitude (h) = √36 = 6

So, the length of the altitude is 6 units.

In summary, the length of the altitude to the hypotenuse in a right triangle is precisely the geometric mean of the lengths of the two segments into which it divides the hypotenuse.