Pythagorean Theorem

The Pythagorean Theorem describes a special relationship between the legs of a right triangle and its hypotenuse.

A right triangle has one right angle (90°) and two minor angles (<90°).

Let see the right triangle below. The two sides that make up the right angle are legs (a and b). The longest side opposite the right angle is the hypotenuse (c).

The Pythagorean Theorem states that:

In a right triangle the square of the hypotenuse side is equal to the sum of squares of the other two sides. That is to say:

As we see in this representation, we can construct squares based on both legs (a and b) and on the hypotenuse (c).

Geometrically, it can be verified that in any right triangle it is true that the sum of the areas of the squares formed on its legs is equal to the area of the square built on its hypotenuse, that is:

Calculate legs

We can use the Pythagorean Theorem to find the length of a right triangle’s leg if we are given measurements for the lengths of the hypotenuse and the other leg.

That is to say, the value of the length of one leg can be calculated from the other one and the hypotenuse using the following formula:

The Converse of the Pythagorean Theorem

The converse of the Pythagorean Theorem states that for any triangle with sides a, b, c, if a² + b² = c², then the angle between a and b measures 90° and the triangle is a right triangle.

• If c² = a² + b² then is a right triangle because the Pythagorean Theorem is verified.
• If c² < a² + b² then is an acute triangle because the angle facing side c is an acute angle. This inequality applies for all sides: b² < a²+c² and a² < b²+c².
• If c² > a² + b² then is an obtuse triangle. The square of the side opposite the obtuse angle is greater than the sum of the squares of the other two sides.

Generalized Pythagorean Theorem

Can the Pythagoras theorem be applied on all triangles?

The Pythagorean theorem can be extended to all types of triangles. The generalized Pythagorean theorem relates the lengths of the three sides of any triangle.

Let p the segment of the projection of side b onto side c. The formula of the generalized Pythagorean theorem is:

Segment p can change sign depending on the type of triangle:

• α=90°: projection on c p=0. The Pythagorean theorem is applied when the term is canceled. It’s a right triangle.
• α<90°: projection on c p>0. It’s an acute triangle.
• α>90°: projection on c p<0. It’s an obtuse triangle.

This form of the generalized Pythagorean theorem, expressed trigonometrically, is the law of cosines or cosine rule. The law of cosines states:

The 3D Pythagorean Theorem

The Pythagorean theorem can be extended to three-dimensional space by applying it to the length of the longest diagonal of a rectangular prism (or cuboid). This diagonal is the hypotenuse of a right triangle having one side on the edge of the prism and the other along a diagonal of a rectangular face of the prism.

Practice

Let a right triangle with legs a = 4 and b = 3 and hypotenuse c = 5.

In this case the sum of the squares of the two legs equals the square of the hypotenuse. Let’s see it:

In the picture bellow it is observed graphically how the sum of the area of the squares built on the legs is equal to the area of the square built on its hypotenuse.

Relationship between legs and hypotenuse

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).