Square
The above sketch is a right triangle with a square built off of each of the sides of the right triangle. In my sketch I also have the areas of each square shown. From these sketches I started to look and see if there was a correlation between the areas of the different squares. First I looked at the relationships between the smallest square and the right triangle there did not seem to be any particular relationship between the sides. I recorded the different areas of the three squares and then manipulated the size. I did this a few times to help to see the relationship between the areas of the squares. From the table I can start to see a pattern. We can see that areas of the two smallest squares add to the area of the largest square. In the first set of numbers we see that and the area of the third square is . We can also see that this is true for the second and third sets of square. We have and From this we can make the conjecture that the ares of the squares off of the legs of a right triangle add to the area of the square off of the hypotenuse of the right triangle. This is where we can derive the Pythagorean Theorem. If we let the short leg of the right triangle be a, then the area of the square is a2. If we let the long leg of the right triangle be b, then the area of the square would be b2. Lastly if we let the hypotenuse of the right triangle be c, then the square off of the hypotenuse would have an area of c2. From our conjecture, it should be true that a2+b2=c2. Which is the pythagorean theorem.