1. Warm Up
Classify each triangle by its angle measures.
3. Simplify
4. If a = 6, b = 7, and c = 12, find a2 + b2 and find c2. Which value is greater?
acute
right 12
85; 144; c2

2. Objectives
Use the Pythagorean Theorem and its converse to solve problems.
Use Pythagorean inequalities to classify triangles.
Justify and apply properties of 45°-45°-90° triangles.
Justify and apply properties of 30°- 60°- 90° triangles.

3. pythagorean
theorem

4. Chou-pei Suan-king Chinese Book from 1200-600 B.C.
Many years after Chinese in 560 B.C., Pythagoras made a formal proof…the Pythagorean Theorem
that's over 1,445 years ago!

5. Always across from the right angle.
Pythagorean Theorem
HYPOTENUSE
Always across from the right angle.
LEG
LEG

6. Pythagorean Theorem c a b
The square of the hypotenuse is equal to the sum of the square of the other two sides.
ONLY FOR RIGHT TRIANGLES

8. 16 ft
12 ft

9. m
LEG
HYP
LEG
14 m

10. Round to the nearest tenth3m
4 m

11. A set of three nonzero whole numbers a, b, and c such that a2 + b2 = c2 is called a Pythagorean triple.

12. x and y 1 and 2 1 and 3 3 and 4 You can create Pythagorean Triples.
Choose 2 integers, x and y.
x and y
1 and 2
1 and 3
3 and 4
Create your own

13. If c is the measure of the hypotenuse, find each missing measure
If c is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary

14. Converse of the Pythag Thrm
If , then the triangle is a right triangle.
Converse: the hypothesis & conclusion are interchanged
Original Pythag: If you have a right triangle, then

15. The measures of 3 sides for a triangle are given
The measures of 3 sides for a triangle are given. Determine whether each triangle is a right triangle.
1. 20, 21, 28
2. 10, 24, 26
Check for a2 + b2 = c2
The legs are always the 2 smaller sides. no
yes

16. Play Ball!
2nd Base
90 ft
90 ft
90 ft
90 ft
Home Plate
How far does a catcher have to throw when he throws the ball from home plate to second base?

18. Special Right Triangles
And

19. An isosceles right triangle
Each isosceles triangle is half a square, so they show up a lot in math and engineering.

20. Pick any integer for l. Use Pythagorean Theorem for find h.
Let’s look for a shortcut for finding the length of an unknown side in a triangle:
Pick any integer for l. Use Pythagorean Theorem for find h. h l l

22. This is our reference triangle for the 45-45-90.
In an isosceles right triangle, if the legs have length l, then the hypotenuse has length ____. 1 1
This is our reference triangle for the

23. EX: 1 Solve for x x 3 3

24. EX: 2 Solve for x x 5 5

25. EX: 3 Solve for x x 3 45

26. EX: 3 Solve for x x 45

27. EX: 3 Solve for x 45 x

29. If you fold an equilateral triangle along one of its altitudes, the triangles you get are 30-60-90.

31. This is our reference triangle for the 30-60-90 triangle.2 1 30
This is our reference triangle for the triangle.

32. Ex: 1 60 x 8 30 y

33. Ex: 2
Solve for x 60 30 24 x
x = 12

34. Ex: 3 30 14 y 60 x

35. Ex: 3 30 x y 60 20

36. Ex: 4 x 60 30 y
y = 10
x = 5

37. Objectives
Use the Pythagorean Theorem and its converse to solve problems.
Use Pythagorean inequalities to classify triangles.
Justify and apply properties of 45°-45°-90° triangles.
Justify and apply properties of 30°- 60°- 90° triangles.

38. Always across from the right angle.
Pythagorean Theorem
_______________
Always across from the right angle.
____
____

39. 16 ft
12 ft

40. m
LEG
HYP
LEG
14 m

41. Round to the nearest tenth3m
4 m

42. A set of three nonzero whole numbers a, b, and c such that a2 + b2 = c2 is called a ____________ _______.

43. If c is the measure of the hypotenuse, find each missing measure
If c is the measure of the hypotenuse, find each missing measure. Round to the nearest tenth, if necessary

44. Converse of the Pythag Thrm
If , then the triangle is a _______________ triangle.
Converse: the hypothesis & conclusion are interchanged
Original Pythag: If you have a right triangle, then

45. The measures of 3 sides for a triangle are given
The measures of 3 sides for a triangle are given. Determine whether each triangle is a right triangle.
1. 20, 21, 28
2. 10, 24, 26

46. Play Ball!
2nd Base
90 ft
90 ft
90 ft
90 ft
Home Plate
How far does a catcher have to throw when he throws the ball from home plate to second base?

47. Special Right Triangles
And

48. This is our reference triangle for the 45-45-90.
In an isosceles right triangle, if the legs have length l, then the hypotenuse has length ____. 1 1
This is our reference triangle for the

49. EX: 1 Solve for x x 3 3

50. EX: 2 Solve for x x 5 5

51. EX: 3 Solve for x 45 3 x

52. EX: 3 Solve for x 45 x

53. EX: 3 Solve for x 45 x

54. This is our reference triangle for the 30-60-90 triangle.2 1 30
This is our reference triangle for the triangle.

55. Ex: 1 60 8 x 30 y

56. Solve for x
Ex: 2 30 x 24 60

57. Ex: 3 30 14 y 60 x

58. Ex: 3 30 15 y 60 x

59. Ex: 4 x 60 30 y

60. Ex: 4 x 60 30 y