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Right Triangle Trigonometry

Proving and Using the Pythagorean Theorem

In right triangles, there is a fundamental relationship between the lengths of the legs and the length of the hypotenuse. This relationship is presented in the Pythagorean Theorem.
Concept

Right Triangle

A right triangle is a specific type of triangle that contains a right angle. Its sides are commonly called the legs and the hypotenuse.

In every right triangle, the hypotenuse lies opposite the right angle and is always the longest side of the triangle.
Rule

Pythagorean Theorem

In a right triangle, the length of the hypotenuse squared equals the sum of the squares of the lengths of the legs.

Right triangle

Proof

Using Area
Four copies of a right triangle with legs and and hypotenuse will be drawn. These triangles will be arranged in such a way to create a square.
Arranging four right triangles into a square
Notice that the side lengths of the outer square are equal to Conversely, the side lengths of the inner square are equal to The area of both squares and the area of the four triangles are written below.
Area of the Inner Square Area of the Outer Square Area of the Four Triangles
In the diagram, it can be seen that the area of the outer square equals the sum of the area of the inner square and the area of the four triangles. The above equation can be simplified by expanding the square of the binomial on the left-hand side.
fullscreen
Exercise

Find the length of

Show Solution
Solution

Since the triangle is a right triangle, we can use the Pythagorean Theorem to find the length of

The hypotenuse is the side opposite the right angle, for this triangle it measures  units. The two other sides are the legs, and We can now form an equation with the Pythagorean Theorem, that can be solved for .
Since is a length, it must be a positive value. Therefore, measures units.
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