The Pythagorean theorem states that for any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The document explains this theorem using diagrams of squares constructed from right triangles. It shows that the total area of two smaller squares formed from the legs of the triangles equals the area of the single larger square formed from the hypotenuse. This demonstrates algebraically that a^2 + b^2 = c^2, where a and b are the legs and c is the hypotenuse.

1. Pythagorean Theorem if a and b represent the legs of a right triangle, and c represents the hypotenuse, then a 2 + b 2 = c 2 But Why? a b c

2. Consider the picture below to the right. What is the area of the white square? C 2 a b c

3. Now let's reposition the four right triangles in the figure. a b c

4. Now we have two smaller white squares, instead of one large one. What is the area of the two white squares? a 2 b 2 a b c

5. Now let's look at those two figures side by side. C 2 a 2 b 2

6. How can we use these pictures to help understand the Pythagorean theorem? a 2 b 2 C 2

7. If you look closely, you may notice that the area of both large squares is exactly the same. a 2 b 2 C 2

8. You may also notice that all of the right triangles are the same size. So if the squares are the same size... and there are 4 triangles in each square, then... a 2 b 2 C 2

9. The area of the white parts in the square on the left, must be the same as the area of the white part in the square on the right... In other words... a 2 + b 2 = c 2 a 2 b 2 C 2

10. Another way to think of this is to express the area of each figure as a sum of the areas of the shapes that make it up. a 2 b 2 C 2

11. The shape on the left is made up of a 2 , b 2 , and 4 right triangles. The shape on the right is made up of c 2 , and 4 right triangles. a 2 b 2 C 2

12. a 2 + b 2 + 4 = c 2 + 4 Combining these gives us the following equation: a 2 b 2 C 2

13. Since there are 4 identical triangles are on each side, they cancel each other out, leaving us with... a 2 + b 2 + 4 = c 2 + 4 a 2 b 2 C 2

14. a 2 + b 2 = c 2 Ta Da! a 2 b 2 C 2

15. a 2 + b 2 = c 2 So now you know why the Pythagorean Theorem works. Doesn't that feel good? a 2 b 2 C 2