Leg Rule

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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.

Explanation of the Leg Rule

Right triangle divided into two smaller right triangles

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

Drawing of the right triangle for the leg theorem

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).

Leg theorem formula

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

Example of the application of the leg theorem to calculate its legs

  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.

    Calculation of the legs of a triangle by the leg theorem
  2. When we know the length of the legs and the hypotenuse we can find the value of the perimeter:
    Perimeter of a triangle by the leg theorem

Relationship between Legs and Hypotenuse

Pythagorean Theorem

Right triangle

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:

Pythagorean theorem formula

Geometric Mean Theorem

Drawing the right triangle for the height 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).

Height theorem formula

AUTHOR: Bernat Requena Serra

YEAR: 2020


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