Hypotenuse

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Drawing the elements of a right triangle

The hypotenuse of a right triangle (or right-angled triangle) is the side opposite its right angle. It is the longest side of a right triangle.

We can find the hypotenuse of a triangle using the Pythagorean Theorem if we know the length of the two legs (a and b). The formula is:

Formula for the calculation of the hypotenuse by the Pythagorean theorem

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.

Triangle-total.rar         or   Triangle-total.exe      

Note. Courtesy of the author: José María Pareja Marcano. Chemist. Seville, Spain.

Exercise

Find the hypotenuse of the triangle shown below:

Example of a right triangle to calculate the hypotenuse

We apply the equation:

Example of calculating the hypotenuse of a right triangle

So the hypotenuse is 5 cm.

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

Leg Rule

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

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

AUTHOR: Bernat Requena Serra

YEAR: 2020


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