Leg Rule

The Leg Rule (or Leg geometric mean theorem) relates the length of each leg of a right triangle with the segments projected by them on the hypotenuse.

Divide the right triangle (ABC) by its height (h) into two smaller right triangles, (CAD and CDB).

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).

The main application of the leg rule is to calculate the legs (a and b) of the right triangle from the segments of the projections on the hypotenuse (n and m) and the latter (c). If we know the length of the legs and the length of the hypotenuse we can calculate the perimeter of a right triangle.

Exercise

1. Find the length of the legs of a right triangle ABC, in which the projections of the legs on the hypotenuse are n = 2 cm and m = 8 cm. These are the segments in which the altitude h (or height) divides the hypotenuse.
2. Find the perimeter of this right triangle ABC.

Solution:

1. Applying the Geometric Mean (Leg) Theorem (or Leg Rule) we can find the length of the legs if we know the length of the two segments.

   The hypotenuse is the sum of the two segments: c = n + m = 2 + 8 = 10 cm.

   
2. When we know the length of the legs and the hypotenuse we can find the value of the perimeter:
   

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:

Geometric Mean Theorem

The Geometric mean theorem (or Altitude-on-Hypotenuse Theorem) relates the height (h) of the 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 (h) relative to the hypotenuse is the geometric mean of the two projections of the legs on the hypotenuse (n and m).