Pythagorean Theorem

What? Pythagoras was a Greek philosopher and mathematician from around 570-495 B.C. Although it’s debatable whether he himself or his followers proved the idea, the idea is named after Pythagoras. The idea was used previously by the Babylonians and the Indians; however, the first written proof (math showing the idea works) was recorded by Pythagoras or his followers. Pythagoras is credited with many ideas in math, music, and philosophy. This is his most famous simply because it bears his name.

… What? Theorem - A general proposition not self-evident but proved by a chain of reasoning. A theorem is an idea in math that can be proven true by logical explanation.

Now we know what it means. An idea in math that was made by some old Greek guy named Pythagoras. But what is the idea? It must have to do with a triangle since there was one at the title…

Let’s look at a right triangle, yes it must be a right triangle. Here is where we start. Let’s look at a right triangle, yes it must be a right triangle. 5 cm 3 cm 4 cm

Let’s draw some squares. Each side has a certain length. What if we draw a square out from each side of the triangle? 5 cm 3 cm 4 cm

Now, let’s split each square into single cm blocks.

Something interesting happens when we count up the number of blocks in each square. 25 + = 5 cm 9 3 cm 4 cm 16

But what is another way of saying the number of blocks in the square? The number of blocks in the two smaller squares add up to the number of blocks in the larger square. But what is another way of saying the number of blocks in the square?

The number of blocks is the area of the square, or the length of the side squared. 3 x 3 = 32 = 9 3 5 cm 3 3 cm 4 cm

A2 + B2 = C2 After all that, the Pythagorean Theorem comes down to this simple equation. The length of each of the shorter sides (legs) squared must equal the length of the longest side (hypotenuse) squared. 32 + 42 = 52  9 + 16 = 25

We can now use this to find the length of a side on a right triangle as long as when know the other two sides. A2 + B2 = C2 But remember, the missing side does not equal It’s a number that equals 100 when it’s squared. 62 + 82 = ?2 ? 6 m 36 + 64 = ?2 100 = ?2 8 m ? = 10

C 5 ft 12 ft A2 + B2 = C2  52 + 122 = C2  25 + 144 = 169 C2 = 169, so C = 169 C = 13

15 in 9 in B When we are missing a leg rather than the hypotenuse, we must be sure to find either A or B in the equation. A2 + B2 = C2  92 + B2 = 152  81 + B2 = 225 225 – 81 = 144  B2 = 144  B = 144 = 12

Pythagorean Triples Some groups of numbers are seen over and over again. These groups are called Pythagorean triples. Some common examples are (3, 4, 5); (5, 12, 13); and (8, 15, 17). When two numbers of a triple are present the third number must be the remaining number from the triple.

Why does this help us? 3, 4, 5 is a triple. Because we are talking about the length of a side on a triangle, we are talking about numbers that must maintain a ratio. In order for the shape to remain a triangle, the ratio must be maintained. When one side grows, the other two must also grow. When one shrinks, so must the other two.

50 mm C 30 mm 40 mm We could use A2 + B2 = C2 to find C. But if we recognize a triple we can save ourselves time. This is a 3, 4, 5 triangle that has been increased by a ratio of 10:1. So it is a 30, 40, 50 triangle.

Try these. 17 in 34 in 8 in 16 in 15 in 30 in