Leg

The legs (or catheti; singular: cathetus) are the sides in a right triangle that form its right angle (90°). The side opposite the right angle is called the hypotenuse. The legs (or catheti) are the two shorter sides of the triangle.

This two legs are defined according to the acute reference angle α:

• Adjacent leg: side that forms the right angle and is adjacent to angle α.
• Opposite leg: side that forms the right angle and is opposite to angle α.

Keep in mind that the labels “opposite” and “adjacent” depend on which angle you are talking about. If the reference angle is not angle α but angle β (the upper one), the adjacent and opposite legs are reversed.

The hypotenuse is the longest side of a right triangle opposite the right angle (90°). There are several theorems that relate the legs and the hypotenuse.

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).

Leg Rule

The leg rule is a theorem that relates the segments projected by the legs on the hypotenuse with the legs they touch.

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).