How to Prove the Pythagorean Theorem

There are many ways to do it. I think using two squares is the simplest way. Drawing your own pictures is an important part of learning math, so go get a pen and paper. Proving theorems is an excellent way to increase your math skills. 

Draw a square and inside it, draw another smaller square. Make it so the corners of the smaller square do not touch the midpoint of the sides of the larger square. When you're done, the larger square will contain the smaller square and four right triangles. Call the length of the sides of the smaller square "c". For the larger square, each side will have two segments. Call one segment "a" and the other "b". Your drawing should look like this:

Each of the four right triangles will have three sides: a, b, and c. The longest side, called the hypotenuse, is c. The formula for the area of a triangle is one half times the base times the height. In this case, the area of one of our triangles is (1/2)ab, and since there are four of them, the total area of the triangles is 4*(1/2)ab which is 2ab. The area of a square is just a side multiplied by itself. The area of the smaller square is c^2 and the area of the larger square is (a+b)^2.

The area of the larger square is the same as the area of the four triangles and the smaller square.

(a+b)^2 = 2ab + c^2

When expanding a binomial like (a+b)^2, we use the rule "square the first, square the last, two times the first times the last". 

If we expand (a+b)^2, we get

a^2 + 2ab + b^2 = 2ab + c^2

Last, we can subtract 2ab from both sides to get the famous formula which relates the lengths of all the sides of a right triangle. 

a^2 + b^2 = c^2

Algebra wasn't around during the time of Pythagoras, so he proved it a different way. Whole numbers that satisfy the theorem are called Pythagorean triples. Some examples of that are: 

3, 4, 5

5, 12, 13

7, 24, 25

Multiples of Pythagorean triples are also Pythagorean triples. 6, 8, 10 is an example of that because it is the same as multiplying each number in 3, 4, 5 by 2. 

Here's a good time to introduce trigonometry. The basic equations of trigonometry are sine, cosine, and tangent. For any given angle, the equations return a value which is the ratio of two sides of a right triangle. 

 Let "A" be the angle opposite side "a".  The sine of A is the ratio of the opposite side over the hypotenuse, which for our triangle is a/c. The cosine of A is the ratio of the adjacent side over the hypotenuse, which is b/c. The tangent is the ratio of the opposite side over the adjacent side, which a/b. 

Here, we can prove that the tangent of A is the same the sine of A divided by the cosine of A. This is true because a/b = (a/c)/(b/c) = (a/c)(c/b). 

The ancient Greek mathematician Eratosthenes used math similar to trigonometry to calculate the circumference of the earth 2,300 years ago. His estimate is very close to the actual value, and the experiment also proved that the world is round.