Transcript Chapter 8 - 2015

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 Radicals and Geometric
Mean
Objective
• Determine the geometric mean between two
numbers.
Simplifying Radical Expressions
(Complete Page 280 1- 28)
• If you are “perfect” you can’t be “rad”!!!
• No fractions under the radical
4
4
2


3
3
3
• No radicals in the denominator
2  3 2 3

 
3
3 3
PROPORTIONS… HOW DO
WE SOLVE THEM?
5
8
x
=
10
50 = 8x
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
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
• Complete geo mean problems from
workbook
Find the Geometric Mean
• 2 and 3
– √6
• 2 and 6
– 2√3
• 4 and 25
– 10
Warm-up
• Simplify
45  5
(2 3 )
2
• Find Geometric Mean of 7 and 12
8.2 The Pythagorean Theorem
Objectives
• State and apply the
Pythagorean Theorem.
• Examine proofs of the
Pythagorean Theorem.
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.”
• Write this down as it is shown…
• 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?
•
•
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.
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.
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 demo - cutout
• Complete pyth th worksheet in workbook
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
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
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?
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?
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
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
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
• Solve for x
x 2 7
x 3 9
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
a leg.
times the length of each
l = length of leg
45
x√2
x
45
x
leg  45º : 
leg  45º : 
hypot  90º :  2
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
45º : 
45º : 
90º :  2
Look for the pattern..
USE PATTERN LIKE ITS AN
ALGEBRA PROBLEM
leg  45º :   6
leg  45º :  
hypot  90º :  2 
Look for the pattern
45º :   6
45º :   6
90º :  2  6 2
Look for the pattern
45º : 
45º : 


90º :  2  10
Look for the pattern
45º :   5 2
45º :   5 2
10
90º :  2 
White Board Practice
6
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
short leg - 30º : 
big leg - 60º :  3
hypot - 90º : 2
60
THE MEASUREMENTS OF THE
PATTERN ARE BASED ON THE
2l
l
l
LENGTH OF THE SHORT LEG ( )
(OPPOSITE THE 30 DEGREE ANGLE)
30
l
3
Look for the pattern..
USE THIS SET UP EVERY TIME YOU HAVE
ONE OF THESE PROBLEMS!!!
Short leg
30º : 
Big leg
60º :  3
hypotenuse
90º : 2
Look for the pattern
30º : 
6
60º :  3 
90º : 2

Look for the pattern
30º : 
6
60º :  3  6 3
90º : 2  12
Look for the pattern
30º : 
60º :  3  9
90º : 2
Look for the pattern
30º : 
 3 3
60º :  3  9
90º : 2
6 3
White Board Practice
Big leg
5
hypot
60º
bigleg  5 3
hypot  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
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
• Proving 2 triangles similar…. We had
3 shortcuts.
– Which was the shortest of shortcuts?
– AA
• If you have 2 right triangles what is the only
other piece of info you would need to say
they’re similar?
Trigonometry
Objectives
• Understand the basics of trig and the 3
ratios that relate
Show music vid
Trigonometry basics
Pg. 311
• If 2 right triangles have the same acute
angle they have to be similar, therefore the
ratio of their sides has to be equivalent
• Mathematicians have discovered ratios that
exist for every degree from 1 to 89.
• The ratios exist, no matter what size the
triangle
30
30
Trigonometry basics
“Triangle measurement”
Sides are named relative to
an acute angle.
Opposite leg
B
C
A
Adjacent leg
Trigonometry basics
Adjacent leg
B
C
Sides are named relative to
the acute angle.
What never changes?
A
Opposite leg
The Tangent Ratio
Tangent LA =
length of opposite leg
length of adjacent leg
B
opposite
Tan A
C
Adjacent
A
Opp

Adj
Sine and Cosine Ratios
• Both of these ratios involve the length of
the hypotenuse
The Cosine Ratio
Cosine LA =
length of adjacent leg
length of hypotenuse
B
opposite
Cos A
C
Adjacent
A
Adj

Hyp
The Sine Ratio
Sine LA =
length of opposite leg
length of hypotenuse
sin A
opposite
B
C
Adjacent
A
opp

Hyp
• Complete workbook page 21
• Start problems on page 22
Find Tan A
A
2
Tan A 
7
7
C
2
B
Find Tan B
7
Tan B 
2
Page 306
Learning to use the trig table and/or
you calculator
#7
How do we use it?
1. We use the ratio to determine the
measurement of the angle
– page 311
– (TAN-1)
Find m A
A
WHAT ELEMENTS OF THE TRINALGE TO WE HAVE IN
RELATION TO THE  A?
2
Tan A 
7
7
C
2
B
16
Tan A ≈ .2857
- pg. 311
-.2857 (TAN-1)
Find m B
A
7
C
 B  74
2
B
How do we use it?
2. Use the measure of the angle to find a missing
side length
– page 311
– TAN
Find the value of x to the nearest
tenth
10
35º
x
x
Tan 35º 
10
x
.7002 
10
x  7.0
WHITEBOARDS
Find the value of x to the nearest
tenth
x  78.1
30
21º
x
WHITEBOARDS
Find the measure of angle y
8
5
yº
y  58
Find the value of x to the nearest
tenth
x  8.9
X
20
24º
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
Cosine LA =
length of adjacent leg
length of hypotenuse
B
opposite
Cos A
C
Adjacent
A
Adj

Hyp
The Sine Ratio
Sine LA =
length of opposite leg
length of hypotenuse
sin A
opposite
B
C
Adjacent
A
opp

Hyp
Find Cos A
A
9
C
9
Cos A 
15
15
12
B
Find Sin A
12
B
C
15
9
A
12
Sin A 
15
Using the trig table
• Pg. 313 #7
Find m A – set up using
COS and SIN
A
9
C
15
12
B
9
Cos A 
15
cos A ≈ .6
- pg. 311
-.3 (COS-1)
12
sin A 
15
sin A ≈ .8
- pg. 311
-.3 (SIN-1)
 A ≈ 53▫
• Page 313
– 9 and 10
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.
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.
Find the measures of the missing
sides x and y
x
23º
y
x ≈ 110
y ≈ 47
100
67º
White boards - Example 2
• Find xº correct to the nearest degree.
x ≈ 37º
xº
18
30
Find the measurement of angle x
x  37
6
8
Xº
10
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
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
TEMPLATE
ANGLE OF ELEVATION / DEPRESSION
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
Distance to light house (X)
2º
X ≈ 716m
25m
88º
25m
88º
2º
Distance to light house (X)
• Go to workbook page 29
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
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!!
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