2. Geometry 9.2 The Pythagorean Theorem

3. October 10, 2015Geometry 9.2 The Pythagorean Theorem2 Goals Prove the Pythagorean Theorem. Solve triangles using the theorem. Solve problems using the theorem.

4. October 10, 2015Geometry 9.2 The Pythagorean Theorem3 This is ancient history. The Egyptian Pyramid builders used it to make square corners.

5. October 10, 2015Geometry 9.2 The Pythagorean Theorem4 Terminology Leg Hypotenuse

6. October 10, 2015Geometry 9.2 The Pythagorean Theorem5 Hypotenuse = “stretched against” 3 5 4

7. October 10, 2015Geometry 9.2 The Pythagorean Theorem6 Proof Proofs of the Pythagorean Theorem are numerous – well over 300 known. Discovered in many ancient cultures. We can use what we learned about the altitude of right triangles to prove the Pythagorean Theorem, too.

8. October 10, 2015Geometry 9.2 The Pythagorean Theorem7 From last lesson: D a b mn h c

9. October 10, 2015Geometry 9.2 The Pythagorean Theorem8 Chinese Proof ab a a a b b b c c c c

10. October 10, 2015Geometry 9.2 The Pythagorean Theorem9 Chinese Proof ab a a a b b b c c c c Area of the square: A = c 2 Area of one triangle: A = (½) ab Area of 4 triangles: A = 2ab

11. October 10, 2015Geometry 9.2 The Pythagorean Theorem10 Chinese Proof ab a a a b b b c c c c Area of the square: A = c 2 Area of 4 triangles: A = 2ab Area Sum c 2 + 2ab

12. October 10, 2015Geometry 9.2 The Pythagorean Theorem11 Chinese Proof ab a a a b b b c c c c Area Sum c 2 + 2ab ? a + b ?

13. October 10, 2015Geometry 9.2 The Pythagorean Theorem12 Chinese Proof ab a a a b b b c c c c Area Sum c 2 + 2ab Area another way:

14. October 10, 2015Geometry 9.2 The Pythagorean Theorem13 Chinese Proof ab a a a b b b c c c c Area Sum c 2 + 2ab or a 2 + 2ab + b 2 These areas are equal.

15. October 10, 2015Geometry 9.2 The Pythagorean Theorem14 Chinese Proof ab a a a b b b c c c c

16. October 10, 2015Geometry 9.2 The Pythagorean Theorem15 President Garfield (1876) 20 th President of the United States Area of Trapezoid = Sum of area of three triangles

17. October 10, 2015Geometry 9.2 The Pythagorean Theorem16 The Pythagorean Theorem a b c a 2 + b 2 = c 2

18. October 10, 2015Geometry 9.2 The Pythagorean Theorem17 Example 1Solve. 5 6 c

19. October 10, 2015Geometry 9.2 The Pythagorean Theorem18 Example 2Solve. a 2 10

20. October 10, 2015Geometry 9.2 The Pythagorean Theorem19 Example 3Solve. x x 20

21. October 10, 2015Geometry 9.2 The Pythagorean Theorem20 Solve these two triangles. 3 4 c 5 12 c

22. October 10, 2015Geometry 9.2 The Pythagorean Theorem21 Pythagorean Triples 3 4 5 5 12 13 3 – 4 – 5 and 5 – 12 – 13 are Pythagorean Triples. Each side is an integer.

23. October 10, 2015Geometry 9.2 The Pythagorean Theorem22 Example Is 10-10-20 a Pythagorean Triple? 10 2 + 10 2 = 20 2 ? 100 + 100 = 400 ? 200 = 400 ? False! Not a Pythagorean Triple.

24. October 10, 2015Geometry 9.2 The Pythagorean Theorem23 Example Is 20-21-29 a Pythagorean Triple? 20 2 + 21 2 = 29 2 ? 400 + 441 = 841 ? 841 = 841 True It is a Pythagorean Triple. Generating Triples

25. October 10, 2015Geometry 9.2 The Pythagorean Theorem24 Area of a Triangle h b

26. October 10, 2015Geometry 9.2 The Pythagorean Theorem25 Find the area. 12 15 h

27. October 10, 2015Geometry 9.2 The Pythagorean Theorem26 Find the area. 12 15 h A Pythagorean Triple 9

28. October 10, 2015Geometry 9.2 The Pythagorean Theorem27 Find the area. 12 15 9

29. October 10, 2015Geometry 9.2 The Pythagorean Theorem28 Problem The distance between bases on a baseball diamond is 90 feet. A catcher throws the ball from home base to 2 nd base. What is the distance?

30. October 10, 2015Geometry 9.2 The Pythagorean Theorem29 Problem 90 c

31. October 10, 2015Geometry 9.2 The Pythagorean Theorem30 Find the diagonal measure of the LCD screen to the nearest inch. 36.8 in. 20.7 in.

32. October 10, 2015Geometry 9.2 The Pythagorean Theorem31 Find the diagonal measure of the LCD screen to the nearest inch. 36.8 in. 20.7 in. c

33. October 10, 2015Geometry 9.2 The Pythagorean Theorem32 Find the diagonal measure of the LCD screen to the nearest inch. 36.8 in. 20.7 in. c

34. October 10, 2015Geometry 9.2 The Pythagorean Theorem33 Find the diagonal measure of the LCD screen to the nearest inch. 36.8 in. 20.7 in. 42 in. About 42 inches

35. October 10, 2015Geometry 9.2 The Pythagorean Theorem34 Summary In a right triangle, the hypotenuse is the longest side. The sum of the squares of the legs is equal to the square of the hypotenuse. If the three sides are all integers, they form a Pythagorean Triple.

36. October 10, 2015Geometry 9.2 The Pythagorean Theorem35 True or False? a b c  a +  b =  c? http://www.youtube. com/watch?v=DUC ZXn9RZ9s

37. October 10, 2015Geometry 9.2 The Pythagorean Theorem36 False. It should have been… The sum of the squares of the two legs of a right triangle is equal to the square of the remaining side. Oh joy! Rapture! I have a brain!

38. October 10, 2015Geometry 9.2 The Pythagorean Theorem37 Homework How to Generate Pythagorean Triples

39. October 10, 2015Geometry 9.2 The Pythagorean Theorem38 Generating Pythagorean Triples Find two positive integers a & b which are relatively prime and a > b. That is, they have no factors in common other than 1. Then the triples are: a 2 + b 2, 2ab and a 2 – b 2.

40. October 10, 2015Geometry 9.2 The Pythagorean Theorem39 Generating Pythagorean Triples Example: Choose a = 4 and b = 3. a 2 + b 2 = 4 2 + 3 2 = 25. 2ab = 2(4)(3) = 24. a 2 – b 2 = 4 2 – 3 2 = 7. 7, 24, 25 is a Pythagorean Triple.

41. October 10, 2015Geometry 9.2 The Pythagorean Theorem40 Generating Pythagorean Triples 7, 24, 25 is a Pythagorean Triple. Check: 7 2 + 24 2 = 25 2 ? 49 + 576 = 625 ? 625 = 625 That’s a triple!

42. October 10, 2015Geometry 9.2 The Pythagorean Theorem41 Pythagorean Triples a and b are relatively prime. a > b a 2 + b 2 2ab a 2 – b 2

43. October 10, 2015Geometry 9.2 The Pythagorean Theorem42 Try it. Using a = 8 and b = 3, find the Pythagorean Triple. Answer: 8 2 – 3 2 = 64 – 9 = 55 2(8)(3) = 48 8 2 + 3 2 = 73 55 2 + 48 2 = 73 2 ? 3025 + 2304 = 5329 ? 5329 = 5329 checks. Area