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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.
Means-Extremes property of
proportions
• The product of the extremes equals the
product of the means.
a
b
c
=
d
ad = cb
The Geometric Mean
“x” is the geometric mean between “a” and “b” if:
a x

x b
Take Notice: The term said to be the
geometric mean will always be crossmultiplied w/ itself.
Take Notice: In a geometric mean problem,
there are only 3 variables to account for,
instead of four.
2
x
= ab
2
√x = √ab
x = +/- √ab
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
(pg. 287)
• No “party people” under the radical
• No fractions under the radical
4
4
2


3
3
3
• No radicals in the denominator
2  3 2 3

 
3
3 3
Party People are
perfect square #’s
which are?
White Board Practice
• Simplify
3 50
15 2
White Board Practice
• Simplify
7  14
7 2
White Board Practice
• Simplify
12
3
4 3
Find the Geometric Mean
• 2 and 3
– √6
• 2 and 6
– 2√3
• 4 and 25
– 10
White Board Practice
• Simplify
8
2 2
2 2
Warm-up
• Simplify
45  5
3
4
• Find Geometric Mean of 7 and 12
White Board Practice
• Simplify
3
4
3
2
Similarity and Geometric Mean
• Similar Triangle Example
• What is special about a geometric mean
proportion?
• We are now going to combine the idea of
similarity with a geometric mean
proportion.
SHMOOP VID
• http://www.shmoop.com/video/geometricmean
Theorem
If the altitude is drawn to the hypotenuse of a right
triangle…..
– 2 additional right triangles are created
– The 3 triangles are all similar
• Their sides are in proportion to one another
Note: What one color side
represents to one triangle,
represents something
different in another!
b
g
1
y
p
2
o
Fill in the table with the letter of the color that
represents each part of each different triangle.
Hypotenuse
Big Leg
Small Leg
OG
Triangle
Medium
Small
PARTNERS: Find all of similarity proportions that
would create geometric mean problems.
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.
y pp

p o
Easier way to remember…
create the proportion of
the legs of both smaller
triangles.
b
g
p
y
o
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.)
yo g

g
o
y  o b b

b
y
b
g
p
y
o
Group Practice
• Pg. 288 #17
• a. √14
• b. 3√ 2
• c. 3 √ 7
Group Practice
• If RS = 2 and SQ = 8 find PS
• PS = 4
Group Practice
• If RP = 10 and RS = 5 find RQ
• RQ = 20
Group Practice
• If RS = 4 and PS = 6, find SQ
• SQ = 9
8.2 The Pythagorean Theorem
Objectives
• State and apply the
Pythagorean Theorem.
• Examine proofs of the
Pythagorean Theorem.
WARM - UP
• Label the triangle with 4 letters
• Re-draw the 3 similar triangles, lining them up
so that their corresponding parts are in the same
position
• Write down 1 of the 3 proportions that create a
geometric mean
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 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 – Pair - 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 – Pair - 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.
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
Brightstorm - proof
Find the value of each variable
1. x  13
x
2
3
Find the value of each variable
2. y  2 5
y
4
6
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 5
4
8
Find the value of each variable
3.
x2 2
4
x
x
Find the value of each variable
5.
4
x
X+2
Find the value of each variable
5.
X2 + (x+2) 2 = 10
X2 + x2 + 4x + 4 = 100
2x2 + 4x – 96 = 0
X2 + 2x – 48 = 0
(x + 8)(x – 6) = 0
X = -8 ; x = 6
10
x
X+2
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.
Given the side lengths of a
triangle….
• Can we tell what type of triangle we have?
YES!!
• How?
– We use c2
a2 + b2
– c always represents the longest side
• Lets try… what type of triangle has sides
lengths of 3, 4, and 5?
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
Right Triangle
2
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
This column should
12,16,20
be memorized!!
15,20,25
Theorem
(pg. 296)
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= 6 , b = 7, c = 8
Is it a right triangle?
a
c  a b
2
2
2
b
Triangle is acute
Theorem
(pg. 296)
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?
c
a
c  a b
2
b
2
2
Triangle is obtuse
Review
• We use c2
a2 + b2
2
•C
= then we a right triangle
2
•C < then we have acute triangle
2
•C > then we have obtuse triangle
• Always make ‘c’ the largest number!!
The sides of a triangle have the
lengths given. Is the triangle acute,
right, or obtuse?
1. 20, 21, 29
•
right
The sides of a triangle have the
lengths given. Is the triangle acute,
right, or obtuse?
2. 5, 12, 14
•
obtuse
The sides of a triangle have the
lengths given. Is the triangle acute,
right, or obtuse?
3. 6, 7, 8
•
acute
The sides of a triangle have the
lengths given. Is the triangle acute,
right, or obtuse?
4. 1, 4, 6
–
Not possible
The sides of a triangle have the
lengths given. Is the triangle acute,
right, or obtuse?
5.
3, 4, 5
• acute
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?
WARM - UP
• Solve for x, y, and z
4
16
x
y
z
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
a leg.
Hypotenuse = √2 ∙ leg
45
x√2
x
45
x
45º - x
45º - x
90º - x 2
Look for the pattern..
USE THIS SET UP EVERY TIME YOU HAVE
ONE OF THESE PROBLEMS!!!
• The sides
opposite the 45◦
angles are
congruent.
• The side opposite the
90◦ angle is the
length of the leg
multiplied by √2
45º - x
45º - x
90º - x 2
Look for the pattern..
USE THIS SET UP EVERY TIME YOU HAVE
ONE OF THESE PROBLEMS!!!
45º - x
45º - x
6

90º - x 2 
Look for the pattern
45º - x  6
45º - x  6
90º - x 2  6 2
Look for the pattern
45º - x
45º - x


90º - x 2  10
Look for the pattern
45º - x  5 2
45º - x  5 2
10
90º - x 2 
White Board Practice
6
x
Hypotenuse = √2 * leg
6
= √2 x
x3 2
x
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
x√2
x
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
White Board Practice
• A square has a perimeter of 20cm, what is
the length of each diagonal?
• Diagonal = 5√2 cm
30º-60º-90º Triangle
30 30
60
A 30º-60º-90º
triangle is half
an equilateral
triangle
60
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.
Hypotenuse = 2 ∙ short leg
60
Long leg = √3 ∙ short leg
2x
30º - x
x
30
x 3
60º - x 3
90º - 2x
Look for the pattern..
USE THIS SET UP EVERY TIME YOU HAVE
ONE OF THESE PROBLEMS!!!
Short leg
30º - x
Long leg
60º - x 3
hypotenuse
90º - 2x
Look for the pattern
30º - x
6
60º - x 3 
90º - 2x

Look for the pattern
30º - x
6
60º - x 3  6 3
90º - 2x  12
Look for the pattern
30º - x
60º - x 3  8
90º - 2x
Look for the pattern
8 3
30º - x

3
60º - x 3  8
90º - 2x
16 3

3
White Board Practice
x
5
y
60º
x5 3
Hypotenuse = 2 ∙ short leg
Long leg = √3 ∙ short leg
y  10
White Board Practice
9
30º
y
x
60º
y = 3√3
x = 6√3
White Board Practice
• Find the length of an altitude of a
equilateral triangle if the side lengths are
16cm.
•8√3 cm
Quiz Review Sec. 1 - 4
8.1
• Geometric mean / simplifying radical expressions
• Corollary 1 & 2 - ** #32 p. 289 **
8.2
• Pythag. Thm – rectangle problems - pg. 292 #10, 13, 14
– Isosceles triangle problems pg. 304 #7
8.3
• Use side lengths to determine the type of triangle (right, obtuse, acute)
– Pg. 297 1 – 5
8.4
• 45-45-90 triangles (problems using squares)
• 30-60-90 triangles (problems using equilateral triangles )
WARM-UP
• What is the one piece of
information we need to
prove 2 RIGHT triangles
are similar? Explain in
complete sentences why.
8.5 The Tangent Ratio
Objectives
• Define the tangent ratio for a right triangle
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
Trigonometry
“Triangle measurement”
Sides are named relative to
an acute angle.
Opposite leg
B
C
A
Adjacent leg
Trigonometry
Adjacent leg
B
C
Sides are named relative to
the acute angle.
What never changes?
A
Opposite leg
The Tangent Ratio
The tangent of an acute angle is defined as the
ratio of the length of the opposite leg divided by
the adjacent leg of the right triangle.
Tangent LA =
length of opposite leg
length of adjacent leg
opposite
B
C
Tan A
Adjacent
A
Opp

Adj
Find Tan A
A
2
Tan A 
7
7
C
2
B
Find Tan B
A
7
Tan B 
2
7
C
2
B
How do we use it?
1. If we know the ratio we can use it to
determine the measurement of the angle
– We either look up the value of the ratio in the book
on page 311
– Or we use a scientific calculator by entering the ratio
and then pressing inverse TAN (TAN-1)
Find  A
A
2
Tan A 
7
7
C
2
B
Tan A ≈ .2857
- pg. 311
-.2857 (TAN-1)
 A 16
A
7
C
2
B
Find  B
A
7
C
 B  74
2
B
Find  A
A
 A  28
17
B
8
C
How do we use it?
2. If we know the angle degree measure we can
use it to find a missing side length
– Look it up in the table (pg. 311) by finding the
degree and then looking under Tangent
– Or we use scientific calculator by entering the degree
measure and then pressing TAN
Find the value of x to the nearest
tenth
10
35º
x
x
Tan 35º 
10
x
.7002 
10
x  7.0
Find the value of x to the nearest
tenth
x  78.1
30
21º
x
Find the measure of angle y
8
5
yº
y  58
Page 306
#7
Brightstorm
Find the value of x to the nearest
tenth
x  8.9
X
20
24º
Find the measurement of angle x
x  37
6
8
Xº
10
WARM-UP
ON PG. 311…
WHY IS THE TANGENT RATIO
◦
FOR 45 1.000?
WHY IS THE TANGENT RATIO
FOR 60◦ 1.7321?
8.6 The Sine and Cosine Ratios
Objectives
• Define the sine and cosine ratio
Sine and Cosine Ratios
• Both of these ratios involve the length of
the hypotenuse
The Cosine Ratio
The cosine of an acute angle is defined as the ratio
of the length of the adjacent leg to the hypotenuse
of the right triangle.
Cosine LA =
length of adjacent leg
length of hypotenuse
opposite
B
C
Cos A
Adjacent
A
Adj

Hyp
Find Cos A
A
9
C
9
Cos A 
15
15
12
B
Find  A
A
9
C
9
Cos A 
15
15
12
B
cos A ≈ .6
- pg. 311
-.3 (COS-1)
 A ≈ 53▫
The Sine Ratio
The sine of an acute angle is defined as the ratio of
the length of the opposite leg to the hypotenuse of
the right triangle.
Sine LA =
length of opposite leg
length of hypotenuse
opposite
B
C
sin A
Adjacent
A
opp

Hyp
Find Sin A
12
B
C
15
9
A
12
Sin A 
15
Find  A using sine
12
B
C
15
9
A
12
sin A 
15
sin A ≈ .8
- pg. 311
-.3 (SIN-1)
 A ≈ 53▫
SOH-CAH-TOA
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.
S
O
H
C
A
H
T
O
A
With a partner
try to come up
with a new
saying.
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.
Whiteboards
• Page 313
– #7, 9
White boards - Example 2
• Find xº correct to the nearest degree.
x ≈ 37º
xº
18
30
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
Brightstorm
Brightstorm
8.7 Applications of Right Triangle
Trigonometry
Objectives
• Apply the trigonometric ratios to solve
problems
• Every problem involves a diagram
of a right triangle
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
Angle of depression
2º
2º
Angle of elevation
Horizontal
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?
Horizontal
2º
X ≈ 716m
88º
25m
88º
2º
Distance to light house (X)
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
WHITE BOARDS
• 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
40º
x
x
Sin 40 
80
x
.6428 
80
51.4  x
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!!
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
38º
3km
x
Tan38 
3
x
.7813 
3
2.34  x
• http://www.brightstorm.com/math/geometry
/basic-trigonometry/angle-of-elevation-anddepression-problem-1/
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