1. Remember slope of the line.
Start with the lower point and count how much you rise and run to get to the other point!
rise 3 = =
run 6 6 3
Notice segments formed by the rise and run form a right triangle.
This takes us to the….

2. The
Pythagorean
Theorem c a b

3. 8.1 The Pythagorean Theorem
& Its Converse
(PAGES 491 – 495)
OBJECTIVE: To use the Pythagorean Theorem and its converse to solve problems related to right triangles
STANDARD ADDRESSED:
Use the triangle angle sum theorem and/or the Pythagorean Theorem and its converse, to solve simple triangle problems and justify results.

4. What is the Pythagorean Theorem used for?
to find the length of a missing side in a right triangle 15 10 C b 6 5 a 9 4

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

6. The measure of a right angle is 90o

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

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

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

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

11. And the length of the hypotenuse is usually labeled c.

12. Where is the Hypotenuse?c
Leg a
Hypotenuse
Always opposite of the 90 degree angle and it is the longest side of a right triangle.
OPPOSITE
Leg b
Click here to view interactive example from Mathopenref.com

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

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

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

16. and add them together, 3 4 5

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

18. Click here to explore an interactive example
Is that correct? ? ?
Click here to explore an interactive example

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

20. The relationship Pythagoras discovered is now called The Pythagorean Theorem:b

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

22. then c a b

23. Pythagorean Theorem Formula a and b represent the legs
Page 491
a2 + b2= c2 c
a and b represent the legs
c represents the
hypotenuse a b

24. You can use The Pythagorean Theorem to solve many kinds of problems.
Suppose a boat travels directly west for 48 miles, 48

25. Then turn south and continue for 36 miles.48 36

26. How far are is the boat from where it started?48 36 ?

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

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

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

30. Then all we need to do is calculate:
To get c by itself and solve. Remember to apply the square root to both sides of the equation (square root removes the square)

31. 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

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

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

34. SO
b = 8
a = 15 c

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

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

37. 5 12 c
= 13

38. 10 b 26

39. = 24 10 b 26
(a)
(c)

40. 12 b 15
= 9

41. Some common Pythagorean Triples are:
Pythagorean Theorem
Pythagorean Triple: A set of nonzero whole numbers, a, b, and, c that makes a2+b2=c2 a true statement.
Some common Pythagorean Triples are:
3, 4, 5
5, 12, 13
8, 15, 17
7,24,25

42. Pythagorean Triples

43. Pythagorean Triples 12 9 15

44. Applying the Pythagorean Theorem
Find the value of the missing length. Do the lengths of sides of
ABC form a Pythagorean Triple? 20 8 x 20 21 x

45. Converse of the Pythagorean Theorem
Theorem 8-2: Converse of the Pythagorean Theorem – If the square of the lengths of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.
If a2+b2=c2, then the triangle is a right triangle.

46. Applying the Converse of the Pythagorean
Theorem
If , complete each statement.
Are the following triangles right triangles? EXPLAIN 85 84 13 21 20 28

47. Classifying Triangles
Theorem 8-3: If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is obtuse.
If c2 > a2+b2, then the triangle is obtuse B c a
Page # 494 A C b

48. Classifying Triangles
Theorem 8-4: If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is acute.
If c2 < a2+b2, then the triangle is acute B c a
Page # 494 A C b

49. Classifying Triangles
If c2 > a2+b2, then the triangle is obtuse
If c2 < a2+b2, then the triangle is acute
If c2 = a2+b2, then the triangle is a right triangle
Given a triangle with sides 7, 8, and 9 how can you classify the triangle?
Given a triangle with sides 6, 11, and 14 how can you classify the triangle?

50. Application
The Parks Department rents paddle boats at docks near each entrance to the park. To the nearest meter how far is it to paddle from one dock to the other?
430 m

51. 8.1 The Pythagorean Theorem and its Converse
Section 8.1; Pg. 495
Complete lesson check #1 – 6 before leaving class. Let Mr. Miller check.
Complete # 7 – 32
DUE MONDAY Feb 13