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- 1. The Pythagorean Theorem Everything you need to know about the sides of a right triangle
- 2. History <ul><li>Pythagoras (569-500 B.C.E.) was born on the island of Samos in Greece. </li></ul><ul><li>Pythagoras founded the Brotherhood of Pythagoreans, devoted to the study of math. </li></ul><ul><li>believed that "Number rules the universe,"and gave numerical values to many objects and ideas. </li></ul><ul><li>Legend has it that upon completion of his famous theorem, Pythagoras sacrificed 100 oxen </li></ul>
- 3. Cont. <ul><li>Not possible to tell if Pythagoras is the actual author of the Pythagorean Theorem </li></ul><ul><li>Known since the days of the ancient Babylonians and Egyptians </li></ul><ul><li>4000 year old Babylonian tablet (c. 1900 B.C.E.), known as Plimpton 322 , lists columns of numbers showing what we now call Pythagorean Triples </li></ul>
- 4. Plimpton 322
- 5. Pythagorean Triples are sets of numbers that satisfy the equation a 2 +b 2 =c 2
- 6. What does it mean? <ul><li>Certain numbers satisfy this equation, like a=3, b=4, and c=5. </li></ul><ul><li>These numbers may not seem that special at first, but is there a source that they all come from? </li></ul><ul><li>The answers is YES, they come from right triangles, and a, b, and c represent the legs </li></ul><ul><li>Lets see how the relationship is discovered </li></ul>
- 7. The Picture
- 8. The Goal <ul><li>Find two different ways to express the area of the square from the previous slide </li></ul><ul><li>Once we have two ways to express the area, set them equal to each other </li></ul><ul><li>Simplify the equation as much as possible and </li></ul><ul><li>SEE WHAT HAPPENS </li></ul>
- 9. First Way a+b a+b Area = (a+b) 2
- 10. Second Way <ul><li>4 triangles with area </li></ul><ul><li>The center square with area = c 2 </li></ul>So the area of the square is also
- 11. Put them together <ul><li>Set both areas equal to each other and simplify </li></ul>
- 12. Bring it together <ul><li>The numbers are special and they do come from something geometrical </li></ul><ul><li>The Pythagorean Theorem has been known for centuries </li></ul><ul><li>It is still a very useful formula today </li></ul>
- 13. Problems to work on <ul><li>Suppose a=4, b=6, what is c? </li></ul><ul><li>Suppose a=3, c=8, what is b? </li></ul><ul><li>Suppose b=5, c=13, what is a? </li></ul>

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