1. Pythagorean Theorem

2. Pythagoras of Samos Birth: 570 B.C.E Samos, Greece Death: 495 B.C.E

3. Pythagoras’ Theorem a 2 + b 2 = c 2 A formula for finding the length of the unknown side of a right triangle.

4. Pythagoras' Theorem Years ago, a man named Pythagoras found an amazing fact about triangles: If the triangle had a right angle (90°)...... and you made a square on each of the three sides, then...... the biggest square had the exact same area as the other two squares put together!

5. When you know the lengths of two sides of a right triangle, you can use the formula to find the unknown length of the other side. You can see by this example that the squares are indicated by the grids on the sides of the triangle. The short sides are known as the legs and the long side is known as the hypotenuse. To find the length of the unknown side, first use Pythagoras’ theorem to determine the squares; then find the square root of that unknown side to determine the length. For the hypotenuse length in this example the square root of 25 is 5. The length of the hypotenuse is 5 units. You can see that the length is 5 units on that square. What is the length of side A?

6. Let’s break down the steps to reinforce the new concept. Draw out the following problem. You are presented with a right triangle and the lengths of two sides are given; you need to find the unknown length. The triangle has one leg with a length of 3 units and another leg with a length of 4 units. You must find the length of the longest side, known as the hypotenuse. In the formula a 2 + b 2 = c 2, the a and b are generally considered the short sides and c is identified as the longest side. So square 3 and 4. 3 2 = 9 & 4 2 = 16 9 + 16 = 25

7. Since c 2 = 25 we need to find the square root of 25 to determine the side length of the hypotenuse The square root of 25 is 5 So the side length of the hypotenuse is 5 units Remember that the square root is the opposite of finding the power of a number. The square root of 64 is 8 b/c 8 x 8 is 64 The square root of 144 is 12 b/c 12 x 12 is 144 The square root of 81 is 9 b/c 9 x 9 is 81 The square root of 36 is 6 b/c 6 x 6 is 36 the square root of 100 is 10 b/c 10 x 10 = 100

8. Definition The longest side of the triangle is called the "hypotenuse", so the formal definition is: In a right angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides. So, the square of a (a²) plus the square of b (b²) is equal to the square of c (c²): a 2 + b 2 = c 2 Sure... ? Let's see if it really works using an example. A "3,4,5" triangle has a right angle in it, so the formula should work. Let's check if the areas are the same: 3 2 + 4 2 = 5 2 Calculating this becomes: 9 + 16 = 25 Yes, it works !

9. Example: Solve this triangle. a 2 + b 2 = c 2 5 2 + 12 2 = c 2 25 + 144 = c 2 169 = c 2 c 2 = 169 c = √169 c = 13

10. Example: What is the diagonal distance across a square of size 1? a 2 + b 2 = c 2 1 2 + 1 2 = c 2 1 + 1 = c 2 2 = c 2 c 2 = 2 c = √2 = 1.4142...

11. Example: Does this triangle have a Right Angle? Does a 2 + b 2 = c 2 ? (√3) 2 + (√5) 2 = (√8) 2 3 + 5 = 8 1.732 2 + 2.236 2 = 2.828 2 2.999 + 4.999 = 7.998 the numbers are approximate b/c the decimals continue Yes, it does! So this is a right-angled triangle (√3) 2 = 3 this is just another way of writing it

12. Pythagorean Triples These are simply sets of whole numbers which fit the rule: a 2 + b 2 = c 2whole numbers 3,4,5 Triangle 5, 12, 13 Triangle 9, 40, 41 Triangle 3 2 + 4 2 =5 2 5 2 + 12 2 = 13 2 9 2 + 40 2 = 41 2 9 + 16 = 25 25 + 144 = 169 81 + 1600 = 1681