1. The Pythagorean Theoremc a b

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

3. Pythagoras (~ B.C.)
Pythagoras was not only a mathematician, but a philosopher and religious leader, as well.
He was responsible for many important developments in math,
astronomy,
and music.

4. The Secret Brotherhood
His students formed a secret society called the Pythagoreans.
As well as studying math, they were a political and religious organization.
Members could be identified by a five pointed star they wore on their clothes.

5. The Secret Brotherhood
They had to follow some unusual rules. They were not allowed to wear wool, drink wine or pick up anything they had dropped!
The initiation into the secret society asked that for the first 5 years into the brotherhood, they would not speak. Just listen.

6. A Pythagorean Puzzle
A right angled triangle

7. A Pythagorean Puzzle
Draw a square on each side.

8. A Pythagorean Puzzle
Measure the length of each side c b a

9. A Pythagorean Puzzle
Work out the area of each square. C² c b b² a a²

10. A Pythagorean Puzzle c² b² a²

11. Proof

12. Let’s look at it this way…c a c b b c2 a2 b2

13. A Pythagorean Puzzle

14. A Pythagorean Puzzle 1 

15. A Pythagorean Puzzle 1 2 

16. A Pythagorean Puzzle 1  2

17. A Pythagorean Puzzle 1 2  3

18. A Pythagorean Puzzle 1 2  3

19. A Pythagorean Puzzle 1 3 2  4

20. A Pythagorean Puzzle 1 3 2 4

21. A Pythagorean Puzzle 1 3 2 5 4

22. A Pythagorean Puzzle 1
What does this tell you about the areas of the three squares? 3 2 5 4
The red square and the yellow square together cover the green square exactly.
The square on the longest side is equal in area to the sum of the squares on the other two sides.

23. A Pythagorean Puzzle Put the pieces back where they came from. 1 3 5 24

24. A Pythagorean Puzzle Put the pieces back where they came from. 1 3 2 54

25. A Pythagorean Puzzle Put the pieces back where they came from. 1 3 2 54

26. A Pythagorean Puzzle Put the pieces back where they came from. 1 2 5 43

27. A Pythagorean Puzzle Put the pieces back where they came from. 1 5 4 23

28. A Pythagorean Puzzle Put the pieces back where they came from. 5 4 2 31

29. A Pythagorean Puzzle c²=a²+b² c² b² This is the Pythagorean Theorem.

30. Pythagorean Theorem
This is the name of Pythagoras’ most famous discovery.
It only works with right-angled triangles.
The longest side, which is always opposite the right-angle, has a special name:
hypotenuse

31. Pythagorean Theorem c a b
c²=a²+b²

32. Pythagoras realized that if you have a right triangle,3 4 5

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

34. and add them together, 3 4 5

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

36. This is true for any right triangle.8 6 10

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

38. 5 12 c
= 13

39. 10 b 26

40. = 24 10 b 26
(a)
(c)

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

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

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

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

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

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

47. Then all we need to do is calculate:

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

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

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

51. b = 8
a = 15 c

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

53. Check It Out! Example 2
A rectangular field has a length of 100 yards and a width of 33 yards. About how far is it from one corner of the field to the opposite corner of the field? Round your answer to the nearest tenth.

54. Understand the Problem
Check It Out! Example 2 Continued 1
Understand the Problem
Rewrite the question as a statement.
• Find the distance from one corner of the field to the opposite corner of the field.
List the important information:
• Drawing a segment from one corner of the field to the opposite corner of the field divides the field into two right triangles.
• The segment between the two corners is
the hypotenuse.
• The sides of the fields are legs, and they are 33 yards long and 100 yards long.

55. Check It Out! Example 2 Continued
Make a Plan
You can use the Pythagorean Theorem to
write an equation.

56. Check It Out! Example 2 Continued
Solve 3
a2 + b2 = c2
Use the Pythagorean Theorem.
= c2
Substitute for the known variables.
,000 = c2
Evaluate the powers.
11,089 = c2
Add.
 c
Take the square roots of both sides.
105.3  c
Round.
The distance from one corner of the field to the
opposite corner is about yards.

57. Ladder Problem
A ladder leans against a second-story window of a house. If the ladder is 25 meters long, and the base of the ladder is 7 meters from the house, how high is the window?

58. Ladder Problem Solution
First draw a diagram that shows the sides of the right triangle.
Label the sides:
Ladder is 25 m
Distance from house is 7 m
Use a2 + b2 = c2 to solve for the missing side.
Distance from house: 7 meters

59. Ladder Problem Solution
b = 24 m
How did you do?

60. Baseball Problem
A baseball “diamond” is really a square. You can use the Pythagorean theorem to find distances around a baseball diamond.

61. Baseball Problem
The distance between consecutive bases is 90 feet. How far does a catcher have to throw the ball from home plate to second base?

62. Baseball Problem
To use the Pythagorean theorem to solve for x, find the right angle.
Which side is the hypotenuse?
Which sides are the legs?
Now use: a2 + b2 = c2

63. Baseball Problem Solution
The hypotenuse is the distance from home to second, or side x in the picture.
The legs are from home to first and from first to second.
Solution:
x2 = = 16,200
x = ft