One of the most well-known and easily recognized results of mathematics is the Pythagorean Theorem. This is the theorem that most students state by claiming that a squared plus b squared equals c squared. The Pythagorean Theorem is a very useful result that's been known for quite some time. In this project, you'll get to know it a little better.

1. After the Scarecrow received his brain in the original movie version of The Wizard of Oz, he thought he knew the Pythagorean Theorem. Check out this portion of the movie to see if the Scarecrow got it right. Correct him if he's wrong. Even though the Scarecrow just got a brain, please be critical of him. (You can borrow the video tape from me if you'd like.)
2. Write a very brief biography (no more than three paragraphs) of Pythagoras.
3. Write a brief history of the Pythagorean Theorem.
4. Find a simple proof of the Pythagorean Theorem that differs from the one presented in class. Make a small poster that illustrates your proof.
5. Any set of three whole numbers a, b, and c that satisfy a²+b² = c² is called a Pythagorean Triple. If you choose any two whole numbers n and m with n > m, you can generate a Pythagorean triple by using the formulas

   a = n2-m2        b = 2nm        c = n2+m2.

   1. Using these formulas, algebraically verify that a²+b² = c².
   2. One well-known Pythagorean triple is (3,4,5). Find three other Pythagorean triples.
6. Write three application problems involving the Pythagorean Theorem: one in which a right triangle's base length is unknown, one in which its height is unknown, and the last in which the hypotenuse is unknown.
7. A 25-foot ladder is leaning against a house, with the base of the ladder 9 feet from the house. If the ladder is moved 6 more feet from the house, how far will the top of the ladder slide down the house? Write your final answer in feet and inches, rounded to the nearest inch.
8. An important formula called the distance formula is useful in many situations. This formula gives the distance between two points, and it is easily derived from the Pythagorean Theorem. On graph paper, plot the following pairs of points, connect them with straight line segments, and use the Pythagorean Theorem to determine the distances between them. Round your results to the nearest one-hundredth.
   1. (3,0) and (0,4)
   2. (-1,2) and (-5,4)
   3. (-2,-3/2) and (5/7,-23/8)
9. Given the following figure, find a reasonable estimate for the distance (as the crow flies) from Plum Grove to Morris. Is your estimate greater than or less than the actual distance? Explain your reasoning.