2. Geometry 9.2 The Pythagorean Theorem

3. June 11, 2016Geometry 9.2 The Pythagorean Theorem2 Goals I can prove the Pythagorean Theorem. I can solve triangles using the theorem. I can solve problems using the theorem.

4. June 11, 2016Geometry 9.2 The Pythagorean Theorem3 This is ancient history. The Egyptian Pyramid builders used it to make square corners.

5. June 11, 2016Geometry 9.2 The Pythagorean Theorem4 Terminology Leg Hypotenuse

6. June 11, 2016Geometry 9.2 The Pythagorean Theorem5 Hypotenuse = “stretched against” 3 5 4

7. June 11, 2016Geometry 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. June 11, 2016Geometry 9.2 The Pythagorean Theorem7 From last lesson: D a b mn h c

9. June 11, 2016Geometry 9.2 The Pythagorean Theorem8 Chinese Proof ab a a a b b b c c c c

10. June 11, 2016Geometry 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. June 11, 2016Geometry 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. June 11, 2016Geometry 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. June 11, 2016Geometry 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. June 11, 2016Geometry 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. June 11, 2016Geometry 9.2 The Pythagorean Theorem14 Chinese Proof ab a a a b b b c c c c

16. June 11, 2016Geometry 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. June 11, 2016Geometry 9.2 The Pythagorean Theorem16 The Pythagorean Theorem a b c a 2 + b 2 = c 2

18. June 11, 2016Geometry 9.2 The Pythagorean Theorem17 Example 1Solve. 5 6 c

19. June 11, 2016Geometry 9.2 The Pythagorean Theorem18 Example 2Solve. a 2 10

20. June 11, 2016Geometry 9.2 The Pythagorean Theorem19 Example 3Solve. x x 20

21. June 11, 2016Geometry 9.2 The Pythagorean Theorem20 Solve these two triangles. 3 4 c 5 12 c

22. June 11, 2016Geometry 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. June 11, 2016Geometry 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. June 11, 2016Geometry 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. Generating Pythagorean Triples If you know one side of a triangle, you can find the other two sides to generate a Pythagorean triple. Start with an EVEN integer, such as 22. 22 = 2mn such that mn = (1/2)22 Find all possible whole number values of m and n— 22 = 2(1)(11) will work June 11, 2016Geometry 9.2 The Pythagorean Theorem24

26. Now, the formula---(this is just ONE method you could use!) The 3 sides will be as follows: Side 1 = m 2 – n 2 Side 2 = 2mn (this is our original side) Side 3 = m 2 + n 2 So, our Pythagorean Triple would be: 11 2 – 1 2 = 120 2 11 1 = 22 11 2 + 1 2 = 122 check it out! June 11, 2016Geometry 9.2 The Pythagorean Theorem25

27. June 11, 2016Geometry 9.2 The Pythagorean Theorem26 Area of a Triangle h b

28. June 11, 2016Geometry 9.2 The Pythagorean Theorem27 Find the area. 12 15 h

29. June 11, 2016Geometry 9.2 The Pythagorean Theorem28 Find the area. 12 15 h A Pythagorean Triple 9

30. June 11, 2016Geometry 9.2 The Pythagorean Theorem29 Find the area. 12 15 9

31. June 11, 2016Geometry 9.2 The Pythagorean Theorem30 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?

32. June 11, 2016Geometry 9.2 The Pythagorean Theorem31 Problem 90 c

33. June 11, 2016Geometry 9.2 The Pythagorean Theorem32 Find the diagonal measure of the LCD screen to the nearest inch. 36.8 in. 20.7 in.

34. June 11, 2016Geometry 9.2 The Pythagorean Theorem33 Find the diagonal measure of the LCD screen to the nearest inch. 36.8 in. 20.7 in. c

35. June 11, 2016Geometry 9.2 The Pythagorean Theorem34 Find the diagonal measure of the LCD screen to the nearest inch. 36.8 in. 20.7 in. c

36. June 11, 2016Geometry 9.2 The Pythagorean Theorem35 Find the diagonal measure of the LCD screen to the nearest inch. 36.8 in. 20.7 in. 42 in. About 42 inches

37. June 11, 2016Geometry 9.2 The Pythagorean Theorem36 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.

38. June 11, 2016Geometry 9.2 The Pythagorean Theorem37 True or False? a b c  a +  b =  c? http://www.youtube. com/watch?v=DUC ZXn9RZ9s

39. June 11, 2016Geometry 9.2 The Pythagorean Theorem38 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!

40. June 11, 2016Geometry 9.2 The Pythagorean Theorem39 Homework How to Generate Pythagorean Triples

41. June 11, 2016Geometry 9.2 The Pythagorean Theorem40 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.

42. June 11, 2016Geometry 9.2 The Pythagorean Theorem41 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.

43. June 11, 2016Geometry 9.2 The Pythagorean Theorem42 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!

44. June 11, 2016Geometry 9.2 The Pythagorean Theorem43 Pythagorean Triples a and b are relatively prime. a > b a 2 + b 2 2ab a 2 – b 2

45. June 11, 2016Geometry 9.2 The Pythagorean Theorem44 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