The Pythagorean theorem is stated in the classic formula: "a squared plus b squared equals c squared." Many people can recite this formula from memory, but they may not understand how it is used in mathematics. The Pythagorean theorem is a powerful tool for solving values in right angle trigonometry.

### Definition

The Pythagorean theorem states that for any right triangle with legs of length “a” and “b” and a hypotenuse of length “c,” the lengths of the sides always satisfy the relationship, “a^2 + b^2 = c^2.” In other words, the sum of the squares of the lengths of the two legs of a triangle is equal to the square of its hypotenuse. The formula is alternatively written with the hypotenuse length isolated (i.e., c = Sqrt(a^2 + b^2).

### Terms

The two key concepts in the Pythagorean theorem are the terms "leg" and "hypotenuse." The two legs of a right triangle are the sides that join to form the right angle. The side opposite the right angle is called the hypotenuse. Since the sum of the angles of a triangle is always 180 degrees, the right angle of a triangle is always the largest angle. The hypotenuse is therefore always larger than the legs. Another term used with the Pythagorean theorem is "Pythagorean triple," which are values of a, b and c that satisfy the Pythagorean theorem. The values a = 3, b = 4 and c = 5 form a Pythagorean triple because 3^2 + 4^2 = 25 = 5^2.

### Significance

The Pythagorean theorem is one of the most significant concepts in trigonometry. Its main usage is in determining the length of the unknown side of a right triangle when two of the side lengths are already known. For example, if a right triangle has one length of 5 and a hypotenuse of 13, you can use the Pythagorean theorem to solve for the length of the other leg: 5^2 + b^2 = 13^2; 25 + b^2 = 169; b^2 = 144; b = 12.

The Pythagorean theorem is actually a special case of the law of cosines, which applies to all triangles: c^2 = a^2 + b^2 - 2ab cos C. For a right triangle, the value of C is 90 degrees, making the value "cos C" equal to zero, which causes the last term to cancel out, leaving the Pythagorean theorem.

### Applications

The distance formula, which is a fundamental formula in applied geometry, is derived from the Pythagorean theorem. The distance formula states that the distance between two points with coordinates (x1, y1) and (x2, y2) is equal to Sqrt((x2 - x1)^2 + (y2 - y1)^2). This can be proven by imagining a right triangle with the line between the two points as the hypotenuse. The lengths of the two legs of the right triangle are the change in “x” and the change in “y” between the two points. Therefore, the distance is the square root of the sum of the squares of the change in “x” value and the change in “y” value between the two points.