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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.
8.1 Similarity in Right Triangles
Objectives
• Determine the
geometric mean
between two numbers.
• State and apply the
relationships that exist
when the altitude is
drawn to the
hypotenuse of a right
triangle.
The Geometric Mean
“x” is the geometric mean between “a” and “b” if:
a x

x b
or x  ab
Example
What is the geometric mean between 3 and 6?
3 x

x 6
or x  3  6  18  3 2
You try it
• Find the geometric mean between 2 and 18.
6
Simplifying Radical Expressions
• No “party people” under the radical
48  3 16  4 3
• No fractions under the radical
4
4
2


3
3
3
• No radicals in the denominator
2  3 2 3

 
3
3 3
Theorem
If the altitude is drawn to the hypotenuse of a right
triangle, then the two triangles formed are similar
to the original triangle and to each other.
a
b
1
m
h
2
n
Corollary
When the altitude is drawn to the hypotenuse of a right
triangle, the length of the altitude is the geometric
mean between the segments on the hypotenuse.
piece1 altitude

altitude piece 2
m h

hh n
h
m
n
Corollary
When the altitude is drawn to the hypotenuse of a
right triangle, each leg is the geometric mean
between the hypotenuse and the segment of the
hypotenuse that is adjacent to that leg (closest to
that leg.)
whole a leg

leg
piece
m
b
n
mn a

aa m
mn b

b
n
White Board Practice
• Simplify
3 50
White Board Practice
15 2
White Board Practice
• Simplify
7  14
White Board Practice
7 2
White Board Practice
• Simplify
12
3
White Board Practice
4 3
White Board Practice
• Simplify
8
2 2
White Board Practice
2 2
White Board Practice
• Simplify
45  5
White Board Practice
15
White Board Practice
• Simplify
3
4
White Board Practice
3
2
Group Practice
• If RS = 2 and SQ = 8 find PS
Group Practice
PS = 4
Group Practice
• If RP = 10 and RS = 5 find SQ
Group Practice
SQ = 15
Group Practice
• If RS = 4 and PS = 6, find SQ
Group Practice
SQ = 9
8.2 The Pythagorean Theorem
Objectives
• State and apply the
Pythagorean Theorem.
• Examine two proofs of
the Pythagorean
Theorem.
• Determine several sets
of Pythagorean
numbers.
The Pythagorean Theorem
In a right triangle, the square of the
hypotenuse is equal to the sum of the
squares of the legs.
c  a b
2
2
2
c
a
b
Proof
Pythagorean Sets
A set of numbers is considered to be
Pythagorean set if they satisfy the
Pythagorean Theorem.
3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25
Movie Time
• We consider the scene from the 1939 film
The Wizard Of Oz in which the Scarecrow
receives his “brain,”
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.”
• We also consider the introductory scene from
the episode “$pringfield (Or, How I Learned
to Stop Worrying and Love Legalized
Gambling)” of The Simpsons in which Homer
finds a pair of eyeglasses in a public
restroom.
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!”
• Homer's recitation is the same as the
Scarecrow's, although Homer receives a
response
Think
1. What are Homer and the Scarecrow
attempting to recite? Identify the error or
errors in their version of this well-known
result. Is their statement true for any
triangles at all? If so, which ones?
Pair
1. What are Homer and the Scarecrow
attempting to recite? Identify the error or
errors in their version of this well-known
result. Is their statement true for any
triangles at all? If so, which ones?
Share
1. What are Homer and the Scarecrow
attempting to recite? Identify the error or
errors in their version of this well-known
result. Is their statement true for any
triangles at all? If so, which ones?
Think
2. 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. Use complete
sentences.
Pair
2. 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. Use complete
sentences.
Share
2. 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. Use complete
sentences.
Find the value of each variable
1.
x
2
3
Find the value of each variable
1. x  13
x
2
3
Find the value of each variable
2.
y
4
6
Find the value of each variable
2. y  2 5
y
4
6
Find the value of each variable
3.
4
x
x
Find the value of each variable
3.
x2 2
4
x
x
Find the length of a diagonal of a
rectangle with length 8 and width 4.
4.
Find the length of a diagonal of a
rectangle with length 8 and width 4.
4.
8
4
4
8
Find the length of a diagonal of a
rectangle with length 8 and width 4.
4.
4
8
Find the length of a diagonal of a
rectangle with length 8 and width 4.
4.
4 5
4
8
8.3 The Converse of the
Pythagorean Theorem
Objectives
• Use the lengths of the sides of a triangle to
determine the kind of triangle.
Theorem
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.
c  a b
2
c
a
b
2
2
Theorem
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.
c  a b
2
c
a
b
2
2
Theorem
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.
c  a b
2
2
2
c
a
b
Sketch
The sides of a triangle have the
lengths given. Is the triangle acute,
right, or obtuse?
1. 20, 21, 29
The sides of a triangle have the
lengths given. Is the triangle acute,
right, or obtuse?
1. right
The sides of a triangle have the
lengths given. Is the triangle acute,
right, or obtuse?
2. 5, 12, 14
The sides of a triangle have the
lengths given. Is the triangle acute,
right, or obtuse?
2. obtuse
The sides of a triangle have the
lengths given. Is the triangle acute,
right, or obtuse?
3. 6, 7, 8
The sides of a triangle have the
lengths given. Is the triangle acute,
right, or obtuse?
3. acute
The sides of a triangle have the
lengths given. Is the triangle acute,
right, or obtuse?
4. 1, 4, 6
The sides of a triangle have the
lengths given. Is the triangle acute,
right, or obtuse?
4. Not possible
The sides of a triangle have the
lengths given. Is the triangle acute,
right, or obtuse?
5.
3, 4, 5
The sides of a triangle have the
lengths given. Is the triangle acute,
right, or obtuse?
5. acute
8.4 Special Right Triangles
Objectives
• Use the ratios of the sides of special right
triangles
45º-45º-90º Theorem
In a 45-45-90 triangle, the hypotenuse is 2
times the length of each leg.
45
45º
a 2
a
a
45
a
45º
90º
a
a 2
Look for the pattern
45º
a
45º
90º
a
a 2
Look for the pattern
45º
45º
90º
a
a
a 2
6
?
?
Look for the pattern
45º
45º
90º
a
a
a 2
6
6
6 2
Look for the pattern
45º
45º
90º
a
a
a 2
?
?
10
Look for the pattern
45º
45º
a
a
5 2
5 2
90º
a 2
10
30º-60º-90º Theorem
In a 30-60-90 triangle, the hypotenuse is
twice as long as the shorter leg and the
longer leg is 3 times the shorter leg.
30º
60
2a
a
a
30
a 3
60º
a
3
90º
2a
Look for the pattern
30º
a
60º
a 3
90º
2a
Look for the pattern
30º
60º
90º
a
a 3
2a
6
?
?
Look for the pattern
30º
60º
90º
a
a 3
2a
6
6 3
12
Look for the pattern
30º
a
?
60º
a 3
8
90º
2a
?
Look for the pattern
30º
a
8 3
3
60º
90º
a 3
2a
8
16 3
3
White Board Practice
6
x
x
White Board Practice
x3 2
6
x
x
White Board Practice
x
5
y
60º
White Board Practice
x
5
y
60º
x5 3
y  10
8.5 The Tangent Ratio
Objectives
• Define the tangent ratio for a right triangle
Trigonometry
Sides are named relative to
an acute angle.
Opposite side
B
C
A
Adjacent side
Trigonometry
Sides are named relative to
the acute angle.
Adjacent side
B
C
A
Opposite side
The Tangent Ratio
The tangent of an acute angle is defined as the
ratio of the length of the opposite side to the
adjacent side of the right triangle that
contains the acute angle.
OppositeSi
de
Tangent Angle A 
AdjacentSide
Opp
Tan A 
Adj
Find Tan A
A
7
C
2
B
2
Tan A 
7
A
7
C
2
B
Find Tan B
A
7
C
2
B
7
Tan B 
2
A
7
C
2
B
Find  A
A
7
C
2
B
 A 16
A
7
C
2
B
Find  B
A
7
C
2
B
 B  74
A
7
C
2
B
Find Tan A
A
17
B
8
C
8
Tan A 
15
A
17
B
8
C
Find Tan B
A
17
B
8
C
15
Tan B 
8
A
17
B
8
C
Find  A
A
17
B
8
C
 A  28
A
17
B
8
C
Find  B
A
17
B
8
C
 B  62
A
17
B
8
C
Find the value of x to the nearest
tenth
10
35º
x
Find the value of x to the nearest
tenth
10
35º
x
x  7.0
Find the value of x to the nearest
tenth
30
21º
x
Find the value of x to the nearest
tenth
x  78.1
30
21º
x
Find the value of x to the nearest
tenth
8
5
yº
Find the value of x to the nearest
tenth
8
5
yº
x  58
Find the value of x to the nearest
tenth
6
8
yº
10
Find the value of x to the nearest
tenth
x  37
6
8
yº
10
8.6 The Sine and Cosine Ratios
Objectives
• Define the sine and cosine ratio
The Cosine Ratio
The cosine of an acute angle is defined as the
ratio of the length of the adjacent side to the
hypotenuse of the right triangle that
contains the acute angle.
adjacent leg
cosine =
hypotenuse
Sketch
The Sine Ratio
The sine of an acute angle is defined as the
ratio of the length of the opposite side to the
hypotenuse of the right triangle that
contains the acute angle.
opposite leg
sine =
hypotenuse
Sketch
SOHCAHTOA
Sine
Opposite
Hypotenuse
Cosine
Adjacent
Hypotenuse
Tangent
Opposite
Adjacent
• 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.
Your Turn - Think
S
O
H
C
A
H
T
O
A
Your Turn - Pair
S
O
H
C
A
H
T
O
A
Your Turn - Share
S
O
H
C
A
H
T
O
A
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.
Example 1
• Find the values of x and y to the nearest
integer.
Example 2
• Find xº correct to the nearest degree.
Example 3
• Find the measures of the three angles of
ABC.
Example 4
• Find the lengths of the three altitudes of
ABC
8.7 Applications of Right Triangle
Trigonometry
Objectives
• Apply the trigonometric ratios to solve
problems
An operator at the top of a lighthouse sees a sailboat
with an angle of depression of 2º
Angle of depression = Angle of elevation
Angle of depression
Angle of elevation
Example 1
• 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.
Example 1
• 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.
80
40
x
x
Sin 40 
80
x
.6428 
80
51.4  x
Example
• 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?
Example
• 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?
x
38
3
x
Tan38 
3
x
.7813 
3
2.34  x
Grade
• Incline of a driveway or a road
• Grade = Tangent
Example
• A driveway has a 15% grade
– What is the angle of elevation?
xº
Example
• Tan = 15%
• Tan xº = .15
xº
Example
• Tan = 15%
• Tan xº = .15
9º
Example
• If the driveway is 12m long, about how
much does it rise?
12
9º
x
Example
• If the driveway is 12m long, about how
much does it rise?
12
9º
1.8