1. a b c

2. Student Expectations 8 th 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 90 o 90 o

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

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

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

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

10. and add them together, 3 4 5

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

12. Is that correct? ? ?

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

14. 1. Square both legs 3 ft 4 ft 3 ft 13 456 78 9 1 2 23 4 5678 910 11 12 1314 15 16

15. 2. Count the total squares 3 ft 4 ft 3 ft 13 456 78 9 1 2 23 4 5678 910 11 12 1314 15 16 9 + 16 = 25

16. 3 ft 4 ft 3 ft 13 456 78 9 1 2 23 4 5678 910 11 12 1314 15 16 9 + 16 = 25 3. Put that number of squares on the hypotenuse 2 4 6 8 10 12 14 16 18 20 22 24 1 3 5 7 9 11 13 15 17 19 21 23 25

17. 3 ft 4 ft 3 ft 13 456 78 9 1 2 23 4 5678 910 11 12 1314 15 16 9 + 16 = 25 4. Count the number of squares that touch the hypotenuse. 2 4 6 8 10 12 14 16 18 20 22 24 1 3 5 7 9 11 13 15 17 19 21 23 25 # = 5

18. 3 ft 4 ft 3 ft 13 456 78 9 1 2 23 4 5678 910 11 12 1314 15 16 9 + 16 = 25 5.That number is the length of the hypotenuse. 2 4 6 8 10 12 14 16 18 20 22 24 1 3 5 7 9 11 13 15 17 19 21 23 25 # = 5 Length = 5

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

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

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

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

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

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

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

28. then a b c

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. 48 2 Using The Pythagorean Theorem, 48 36 c 36 2 +=c2c2

33. Why?Can you see that we have a right triangle? 48 36 c 48 2 36 2 +=c2c2

34. Which side is the hypotenuse?Which sides are the legs? 48 36 c 48 2 36 2 +=c2c2

35. Then all we need to do is calculate:

36. And you end up 60 miles from where you started. 48 36 60 So, since c 2 is 3600, c is60.So, since c 2 is 3600, c is

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

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

39. b = 8 a = 15 c

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

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

42. 5 12 c = 13

43. 10 b 26

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

45. 12 b 15 = 9