1. Chapter 8 Right Triangles
Determine the geometric mean between two numbers.
State and apply the Pythagorean Theorem.
Determine the ratios of the sides of the special right triangles.
Apply the basic trigonometric ratios to solve problems.

2. 8.1 Radicals and Geometric Mean
Objective
Determine the geometric mean between two numbers.

3. Simplifying Radical Expressions (Complete Page 280 1- 28 3n)
No “party people” under the radical
No fractions under the radical
No radicals in the denominator
Party People are perfect square #’s which are?
These are the three “nevers” for simplifying a radical. Do additional examples as you see fit.

4. Means-Extremes property of proportions
The product of the extremes equals the product of the means. = a b c d
ad = cb

5. The Geometric Mean x2 = ab √x2 = √ab
“x” is the geometric mean between “a” and “b” if:
x2 = ab
√x2 = √ab
The geometric mean comes from a special proportion. It is useful to compare it to the arithmetic mean (average) and show that it is a number between two other numbers. Either definition above can work, but each serves a purpose, and can be used to solve different problems.
Take Notice: The term said to be the geometric mean will always be cross-multiplied w/ itself.
Take Notice: In a geometric mean problem, there are only 3 variables to account for, instead of four.

6. Example What is the geometric mean between 3 and 6?
This example shows that we have to be able to simplify radical expressions if we want to solve geometric mean problems. Most of the problems involving right triangles require us to simplify redical expressions. Lets review the rules now.

7. You try it
Find the geometric mean between 2 and 18. 6

8. Find the Geometric Mean
2 and 3 √6
2 and 6
2√3
4 and 25 10

9. Warm-up
Simplify 2
Find Geometric Mean of 7 and 12

10. 8.2 The Pythagorean Theorem
Objectives
State and apply the Pythagorean Theorem.
Examine proofs of the Pythagorean Theorem.

11. Movie Time

12. We consider the scene from the 1939 film The Wizard Of Oz in which the Scarecrow receives his “brain,”

13. Scarecrow:  “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.”
Write this down as it is shown…

14. We also consider the introductory scene from the episode of The Simpsons in which Homer finds a pair of eyeglasses in a public restroom…

15. Homer:  “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.”
Man in bathroom stall:  “That's a right triangle, you idiot!”
Homer:  “D'oh!”

16.   Homer's recitation is the same as the Scarecrow's, although Homer receives a response

17. Think – Pair - Share
What are Homer and the Scarecrow attempting to recite?
Is their statement true for any triangles at all? If so, which ones?
Identify the error or errors in their version of this well-known result.

18. Think – Pair - Share
Is the correction from the man in the stall sufficient?
Give a complete, correct statement of what Homer and the Scarecrow are trying to recite.
Do this first using only English words…
and a second time using mathematical notation.

19. The Pythagorean Theorem
In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.
While most students know or have heard of the Pythagorean theorem, most do not know why it is true. Many proofs have been done (set of 36 different proofs at and you should demonstrate your two favorite ones to them. c a b
Brightstorm - proof

20. Find the value of each variable1. x 2 3

21. Find the value of each variable2. y 4 6

22. Find the length of a diagonal of a rectangle with length 8 and width 4.

23. Find the length of a diagonal of a rectangle with length 8 and width 4.

24. 3. Find the length of the diagonal of a square with a perimeter of 20
4. Find the length of the altitude to the base of an isosceles triangle with sides of 5, 5, 8

25. Warm – up Create a diagram and label it…
An isosceles triangle has a perimeter of 38in with a base length of 10 in. The altitude to the base has a length of 12in. What are the dimensions of the right triangles within the larger isosceles triangle?

26. 8.3 The Converse of the Pythagorean Theorem
Objectives
Use the lengths of the sides of a triangle to determine the kind of triangle.
Determine several sets of Pythagorean numbers.
                            

27. Given the side lengths of a triangle….
Can we tell what type of triangle we have?
YES!!
How?
We use c a2 + b2
c always represents the longest side
Lets try… what type of triangle has sides lengths of 3, 4, and 5?

28. Theorem Right Triangle
If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.
Right Triangle
This is the converse to the Pythagorean theorem. Any triangle whose sides satisfy these conditions is a right triangle. c a b

29. Pythagorean Sets
A set of numbers is considered to be Pythagorean set if they satisfy the Pythagorean Theorem. WHAT DO I MEAN BY SATISFY THE PYTHAGOREAN THEOREM?
3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25
6,8,10 10,24,26
9,12,15
12,16,20
15,20,25
Although many more sets exist, we should only commit a few to memory.
This column should be memorized!!

30. Theorem (pg. 296) Triangle is acute
If the square of one side of a triangle is less than the sum of the squares of the other two sides, then the triangle is an acute triangle.
a= 6 , b = 7, c = 8
Is it a right triangle? c
See note on next slide. a b
Triangle is acute

31. Theorem (pg. 296) Triangle is obtuse
If the square of one side of a triangle is greater than the sum of the squares of the other two sides, then the triangle is an obtuse triangle.
a= 3 , b = 7, c = 9
Is it a right triangle?
Bundle the three previous theorems together and you get an all-purpose triangle determination tool. If the length of the longest side works with the other two sides, then the triangle is right. If the length of the longest side is too short, then the triangle is acute, and if it is too long, it is obtuse. Use the sketch to demonstrate this. c a b
Triangle is obtuse

32. C2 = then we a right triangle C2 < then we have acute triangle
Review
We use c a2 + b2
C2 = then we a right triangle
C2 < then we have acute triangle
C2 > then we have obtuse triangle
Always make ‘c’ the largest number!!

33. The sides of a triangle have the lengths given
The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse?
20, 21, 29
right

34. The sides of a triangle have the lengths given
The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse?
5, 12, 14
obtuse

35. The sides of a triangle have the lengths given
The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse?
6, 7, 8
acute

36. The sides of a triangle have the lengths given
The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse?
1, 4, 6
Not possible

37. The sides of a triangle have the lengths given
The sides of a triangle have the lengths given. Is the triangle acute, right, or obtuse? 5.
acute

38. 8.4 Special Right Triangles
Objectives
Use the ratios of the sides of special right triangles

39. 45º-45º-90º Theorem a
Hypotenuse = √2 ∙ leg 45
x√2 x 45 x

40. Look for the pattern.. USE THIS SET UP EVERY TIME YOU HAVE ONE OF THESE PROBLEMS!!!
The legs opposite the 45◦ angles are congruent.
Hypotenuse - opposite the 90◦ angle is the length of the leg multiplied by √2

41. Look for the pattern.. USE PATTERN LIKE ITS AN ALGEBRA PROBLEM

42. Look for the pattern 42

43. Look for the pattern 10 43

44. Look for the pattern 10 44

45. White Board Practice
Hypotenuse = √2 * leg
= √2 x 6 x x

46. Partner Discussion
If we know the length of a diagonal of a square, can we determine the length of a side? If so, how?
x√2 x x

47. Perimeter = 8√2cm White Board Practice
If the length of a diagonal of a square is 4cm long, what is the perimeter of the square?
Perimeter = 8√2cm

48. Diagonal = 5√2 cm White Board Practice
A square has a perimeter of 20cm, what is the length of each diagonal?
Diagonal = 5√2 cm

49. A 30º-60º-90º triangle is half an equilateral triangle

50. 30º-60º-90º Theorem
THE MEASUREMENTS OF THE PATTERN ARE BASED ON THE LENGTH OF THE SHORT LEG (OPPOSITE THE 30 DEGREE ANGLE) 60 2x x 30 x

51. Look for the pattern.. USE THIS SET UP EVERY TIME YOU HAVE ONE OF THESE PROBLEMS!!!
Short leg
Long leg
hypotenuse

52. Look for the pattern

53. Look for the pattern

54. Look for the pattern
On this I would want to solve for a by dividing by rad 3. Once I know a, I can solve for the hypotenuse

55. Look for the pattern

56. White Board Practice x 5 Hypotenuse = 2 ∙ short leg y
Long leg = √3 ∙ short leg
60º

57. White Board Practice 9
30º
y = 3√3
x = 6√3 y x
60º

58. 8√3 cm White Board Practice
Find the length of an altitude of a equilateral triangle if the side lengths are 16cm.
8√3 cm

59. Quiz Review Sec
8.1
Geometric mean / simplifying radical expressions
8.2
Pythag. Thm – rectangle problems - pg #10, 13, 14
Isosceles triangle problems pg. 304 #7
8.3
Use side lengths to determine the type of triangle (right, obtuse, acute)
Pg – 5
8.4
triangles (problems using squares)
triangles (problems using equilateral triangles )

60. WARM-UP Proving 2 triangles similar…. We had 3 shortcuts.
AA, SAS, SSS
What is the one additional piece of information we need to prove 2 RIGHT triangles are similar? (look at the shortcuts above)

61. 8.5 The Tangent Ratio Objectives
Define the tangent ratio for a right triangle
Show music vid

63. Trigonometry Pg. 311
When you have a right triangle you always have a 90◦ angle and 2 acute angles
Based on the measurements of those acute angles you can discover the lengths of the sides of the right triangle
Mathematicians have discovered ratios that exist for every degree from 1 to 89.
The ratios exist, no matter what size the triangle

64. Trigonometry “Triangle measurement”B
Sides are named relative to an acute angle.
Hypotenuse
Opposite leg A C
Adjacent leg

65. Trigonometry What never changes?
Sides are named relative to the acute angle. B
What never changes?
Hypotenuse
Adjacent leg A C
Opposite leg

66. The Tangent Ratio Tangent LA = Tan A length of opposite leg
length of adjacent leg
Tangent LA = B
Tan A
Show both how to use the tables on page 311 as well as a calculator. C
opposite A
Adjacent

67. Find Tan A A
Tan A 7
Find Tan B B C 2
Tan B

68. Page 306
Learning to use the trig table and/or you calculator #7

69. How do we use it?
We use the ratio to determine the measurement of the angle
page 311
(TAN-1)

70. Find m A WHAT ELEMENTS OF THE TRINALGE TO WE HAVE IN RELATION TO THE  A?
Tan A 7
Tan A ≈ .2857
- pg. 311
(TAN-1) B B C 2

71. Find m B A 7  B C 2

72. How do we use it?
Use the measure of the angle to find a missing side length
page 311
TAN

73. Find the value of x to the nearest tenth10
35º x
Tan 35º
.7002

74. WHITEBOARDS Find the value of x to the nearest tenth30
21º x

75. WHITEBOARDS Find the measure of angle y8 5 yº

76. Find the value of x to the nearest tenth20
24º

77. 8.6 The Sine and Cosine Ratios
Objectives
Define the sine and cosine ratio

78. Sine and Cosine Ratios
Both of these ratios involve the length of the hypotenuse

79. The Cosine Ratio Cosine LA = Cos A length of adjacent leg
length of hypotenuse
Cosine LA = B
Cos A
Show both how to use the tables on page 311 as well as a calculator. C
opposite
Hypotenuse A
Adjacent

80. The Sine Ratio Sine LA = sin A length of opposite leg
length of hypotenuse
Sine LA =
sin A B
Show both how to use the tables on page 311 as well as a calculator. C
opposite
Hypotenuse A
Adjacent

81. Find Cos A A 15 9
Cos A B C 12

82. Find Sin A 12 B C 9
Sin A 15 A

83. Using the trig table
Pg. 313 #7

84. Find m A – set up using COS and SIN
- pg. 311
-.3 (COS-1)
Cos A A
sin A ≈ .8
- pg. 311
-.3 (SIN-1)
sin A 15 9 B
 A ≈ 53▫ C 12

85. Page 313
9 and 10

86. SOH-CAH-TOA
Sine
Opposite
Hypotenuse
Cosine
Adjacent
Tangent

87. Some Old Horse Caught Another Horse Taking Oats Away.
Sally Often Hears Cats Answer Her Telephone on Afternoons
Sally Owns Horrible Cats And Hits Them On Accident.

88. So which one do I use? Sin Cos Tan
Label your sides and see which ratio you can use. Sometimes you can use more than one, so just choose one.

89. Find the measures of the missing sides x and y
23º y
100
x ≈ 110
y ≈ 47
67º

90. x ≈ 37º White boards - Example 2
Find xº correct to the nearest degree.
x ≈ 37º xº 18 30

91. Find the measurement of angle x6 8 Xº 10

92. White Board
An isosceles triangle has sides 8, 8, and 6. Find the length of the altitude from angle C to side AB.
√55 ≈ 7.4

93. 8.7 Applications of Right Triangle Trigonometry
Objectives
Apply the trigonometric ratios to solve problems
Every problem involves a diagram of a right triangle

94. An operator at the top of a lighthouse sees a sailboat
with an angle of depression of 2º
Angle of depression = Angle of elevation
Horizontal 2º
Angle of depression
Angle of elevation 2º
Horizontal

95. TEMPLATE ANGLE OF ELEVATION / DEPRESSION

96. X ≈ 716m An operator at the top of a lighthouse (25m) sees a
Sailboat with an angle of depression of 2º. How far away is the boat?
Distance to light house (X)
Horizontal 2º
X ≈ 716m
88º
25m
25m
88º 2º
Distance to light house (X)

97. Example 1
You are flying a kite is flying at an angle of elevation of 40º. All 80 m of string have been let out. Ignoring the sag in the string, find the height of the kite to the nearest 10m.
How would I label this diagram using these terms..
Kite, yourself, height (h) , angle of elev.,
80m

98. WHITE BOARDS
An observer located 3 km from a rocket launch site sees a rocket at an angle of elevation of 38º. How high is the rocket?
Use the right triangle to first correctly label the diagram!!

99. Grade
Incline of a driveway or a road
Grade = Tangent

100. Example
A driveway has a 15% grade
What is the angle of elevation? xº

101. Example
Tan = 15%
Tan xº = .15 xº

102. Example
Tan = 15%
Tan xº = .15 9º

103. Example If the driveway is 12m long, about how much does it rise? 12 x9º

104. Example If the driveway is 12m long, about how much does it rise? 12
1.8 9º