Geometric Mean Theorem

Geometric mean (or mean proportional) appears in two popular theorems regarding right triangles.

The geometric mean theorem (or altitude theorem) states that the altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle. This is because they all have the same three angles as we can see in the following pictures:

This means that the if we take a right triangle (ABC) sitting on its hypotenuse (long side) and then put in an altitude line that divides the triangle into two other smaller right triangles (CAD and CDB), using the geometric mean we have the following ecuation:

This is why geometric mean theorem is also known as right triangle altitude theorem (or altitude rule), because it relates the height or altitude (h) of the right triangle and the legs of two triangles similar to the main ABC, by plotting the height h over the hypotenuse, stating that in every right triangle, the height or altitude (h) relative to the hypotenuse is the geometric mean of the two projections of the legs on the hypotenuse (n and m). Let’s see the following right triangle and the formula we obtain applying the geometric mean theorem:

So the main application of this theorem is to calculate the height (h) of the right triangle from the segments into which the hypotenuse is divided (n and m). And if we know the hypotenuse and the height (h), the area of the right triangle can be calculated.

Leg Geometric Mean Theorem (or Leg Rule)

The leg of a right triangle is the geometric mean between the hypotenuse and the projection of the leg on the hypotenuse, that is to say, each leg of the triangle is the mean proportional between the hypotenuse and the part of the hypotenuse directly below the leg:

In every right triangle, a leg (a or b) is the geometric mean between the hypotenuse (c) and the projection of that leg on it (n or m).

We can see an application of the leg geometric mean theorem or leg rule when we need to find the height of a right triangle given only the legs of the triangle.

How to find the Altitude of a Right Triangle given the Legs?

Applying the leg rule to the altitude formula that we have in the geometric mean theorem or altitude rule, we can obtain the altitude of the right triangle knowing its three sides:

Then, apply the formulas of the leg rule that relate the projections of the legs to the sides:

Plug the values into the altitude formula and we obtain:

Exercise

Find the altitude of this right triangle.

We only know the two line segments into which the altitude (h) divides the hypotenuse (c). These segments are n = 3 cm and m = 12 cm.

Given the segments of the right triangle we apply the geometric mean theorem or altitude rule and we get the altitude (h):

The altitude of the right triangle is h = 6 cm. The hypotenuse is the sum of the segments n and m, so we obtain that c = n+m = 3+12 = 15 cm.

Finally we can find the area of the right triangle because we know the length of its hypotenuse and its height:

Relationship between Legs and Hypotenuse

Pythagorean Theorem

The Pythagorean Theorem, also known as Pythagoras’s theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle (2 legs and hypotenuse). This theorem can be written as the following equation: