The

1. The Pythagorean Theorem c a b

2. Student Expectations 8th Grade: • 8.3.7C Use pictures or models to demonstrate the Pythagorean theorem. • 8.4.9A Use the Pythagorean theorem to solve real-life problems.

3. This is a right triangle:

4. We call it a right triangle because it contains a right angle.

5. The measure of a right angle is 90o 90o

6. in the The little square angle tells you it is a right angle. 90o

7. About 2,500 years ago, a Greek mathematician named Pythagoras discovered a special relationship between the sides of right triangles.

8. 5 3 4 Pythagorus realized that if you have a right triangle,

9. 5 3 4 and you square the lengths of the two sides that make up the right angle,

10. 5 3 4 and add them together,

11. 5 3 4 you get the same number you would get by squaring the other side.

12. Is that correct? ? ?

13. Pythagorean Theorem You can demonstrate the Pythagorean Theorem geometrically…

14. 1. Square both legs 1 2 3 4 5 6 7 8 4 ft 11 9 10 12 13 14 16 15 3 ft 4ft 1 2 3 4 5 6 3 ft 7 8 9

15. 2. Count the total squares 1 2 3 4 5 6 7 8 4 ft 11 9 10 12 13 14 16 15 3 ft 4ft 1 2 3 9 + 16 = 25 4 5 6 3 ft 7 8 9

16. 21 16 22 1 2 3 4 23 17 11 18 24 12 6 5 6 7 8 13 19 7 25 1 4 ft 11 8 14 20 9 10 12 2 3 15 9 13 14 16 15 4 3 ft 10 4ft 5 1 2 3 9 + 16 = 25 4 5 6 3 ft 7 8 9 3. Put that number of squares on the hypotenuse

17. 21 16 22 1 2 3 4 23 17 11 18 24 12 6 5 6 7 8 13 19 7 25 1 4 ft 11 8 14 20 9 10 12 2 3 15 9 13 14 16 15 4 3 ft 10 4ft 5 1 2 3 9 + 16 = 25 4 5 6 3 ft 7 8 9 4. Count the number of squares that touch the hypotenuse. # = 5

18. 21 16 22 1 2 3 4 23 17 11 18 24 12 6 5 6 7 8 13 19 7 25 1 4 ft 11 8 14 20 9 10 12 2 3 15 9 13 14 16 15 4 3 ft 10 4ft 5 1 2 3 9 + 16 = 25 4 5 6 3 ft 7 8 9 5.That number is the length of the hypotenuse. # = 5 Length = 5

19. 10 8 6 It is. And it is true for any right triangle.

20. The two sides which come together in a right angle are called

21. The two sides which come together in a right angle are called

22. The two sides which come together in a right angle are called legs.

23. The lengths of the legs are usually called a and b. a b

24. The side across from the right angle is called the hypotenuse. a b

25. And the length of the hypotenuse is usually labeled c. c a b

26. The relationship Pythagorus discovered is now called The Pythagorean Theorem: c a b

27. The Pythagorean Theorem says, given the right triangle with legs a and b and hypotenuse c, c a b

28. then c a b

29. You can use The Pythagorean Theorem to solve many kinds of problems. Suppose you drive directly west for 48 miles, 48

30. Then turn south and drive for 36 miles. 48 36

31. How far are you from where you started? 48 36 ?

32. Using The Pythagorean Theorem, 48 482 + 362 = c2 36 c

33. 48 482 + 362 = c2 36 c Why? Can you see that we have a right triangle?

34. 48 482 + 362 = c2 36 c Which side is the hypotenuse? Which sides are the legs?

35. Then all we need to do is calculate:

36. So, since c2 is 3600, c is 60. And you end up 60 miles from where you started. So, since c2 is 3600, c is 48 36 60

37. 15" 8" Find the length of a diagonal of the rectangle: ?

38. 15" 8" Find the length of a diagonal of the rectangle: ? c b = 8 a = 15

39. c b = 8 a = 15

40. 15" 8" Find the length of a diagonal of the rectangle: 17

41. Practice using The Pythagorean Theorem to solve these right triangles:

42. c 5 12 = 13

43. b 10 26

44. b 10 26 = 24 (a) (c)

45. 12 b 15 = 9