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:
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:
How to find the Altitude of a Right Triangle given the Legs?
Plug the values into the altitude formula and we obtain:
Download this calculator to get the results of the formulas on this page. Choose the initial data and enter it in the upper left box. For results, press ENTER.
Note. Courtesy of the author: José María Pareja Marcano. Chemist. Seville, Spain.
Relationship between Legs and Hypotenuse
AUTHOR: Bernat Requena Serra