The 3-4-5 triangle

The Pythagorean Theorem is often stated as


a² + b² = c²

means that the sum of the squares of the lengths of the two short sides (the ones that meet to create the right angle) is equal to the square of the length of the long side opposite the 90°, the side known as the hypotenuse.

If you pick two whole numbers at random to be the short sides, also known as the legs, the hypotenuse will be a square root of a whole number.

For example: If a = 1 and b = 3, 1² + 3² = 10, which means c² = 10. 10 is not a perfect square, so c is equal to the irrational number the square root of 10, which I will write as sqrt(10).

If we are looking at one digit numbers only, we have just one pair of legs that will add up to a perfect square. 3² + 4² = 9 + 16 = 25 = 5².

Three whole numbers that satisfy a² + b² = c² are called a Pythagorean triple. If we find such a triple, we can create infinitely more by multiplying all the sides by the same whole number.  Here are some examples.

Multiply 3-4-5 by 2: 6² + 8² = 10²
Multiply 3-4-5 by 3: 9² + 12² = 15²
Multiply 3-4-5 by 4: 12² + 16² = 20²
...

All of these triangles are similar and when discussing Pythagorean triples, the ones that are relatively prime are a special case.

The Pythagorean Theorem is named for the ancient Greek mathematician Pythagoras (570-495 BCE), but as often happens in math, this does not mean he was the first person ever to notice the pattern. There is strong evidence that the ancient Egyptians who built the pyramids understood the 3-4-5 triangle at the very least, and they lived thousands of years before Pythagoras. In many archeological digs, among the building tools are three sticks of lengths with the ratio 3:4:5. It is assumed that the builders used these when constructing walls to make sure the walls and the floors met at 90° angles.

There are other relatively prime Pythagorean triples; in fact, there are infinitely many. The next two smallest ones are 5² + 12² = 13² and 8² + 15² = 17².

Tomorrow, we will discuss how to create all the Pythagorean triples, both those that are relatively prime and those that aren't.