The Pythagorean Theorem is a fantastically important theorem in mathematics. Students studying math should spend enough time with it to thoroughly understand it. Many times we study a subject and get lost in the details. We see a lot of different aspects to something but don’t get a sense for the things that are worth our time to study and cogitate about. The Pythagorean theorem is so useful in continuing in mathematics not only for geometric shapes, but also for velocities, physics, forces, not to mention the dreaded trigonometry among others. Regardless of the noise of all the things to study in mathematics, the Pythagorean theorem should be mastered and not glossed over.

Simply, the Pythagorean theorem was developed (and can be proved) for any right triangle (in the Euclidean plane) regardless of its orientation. A right triangle is any triangle with one leg of a triangle perpendicular to another leg of a triangle.

The Pythagorean Theorem says that the length of 2 sides determines the length of the other side for right triangles. There is a fixed relation for the 3rd side given any of the other 2. Here it is:

**Pythagorean Theorem**

**a ^{2} + b^{2} = c^{2}**

The relation between the sides is not found in a linear combination, but amazingly in an expansion to a second dimension to areas. In words this is, the sum of the squares of the two legs (here ‘a’ and ‘b’) equals (is the same number) as the sum of the hypotenuse (the longest leg of a right triangle).

I will show why this relationship is true for all right triangles (in the Euclidean plane) from first principles using concepts only from middle school and below will be posted soon.

This relationship does not hold for triangles that do not have 1 right angle between two of the legs.

math, pythagorean theorem