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