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Geometry
Name ________________________
Date _______________ Hour _____
Assignment
Intro to Ch. 8
Worksheet
8.1
Worksheet
8.2 Day 1
Worksheet
8.2 Day 2
Worksheet
8.3 Day 1
Worksheet
8.3 Day 2
Worksheet
Review
Review Worksheet
Quiz
8.4 Day 1
Worksheet
8.4 Day 2
Pg. 567 #16-33, 36-42 even. 57-59
8.4 Extra Practice
Worksheet
8.5
pg 577 # 1-17 odd, 32-37
8.5 In-class project
Worksheet (Be ready for a quiz tomorrow)
8.6 Day 1
Worksheet
8.6 Day 2
Worksheet
Ch. 8 review
Review Worksheet
Geometry
Chapter 8 Learning Targets!
By the end of the chapter, you should be able to:
•Find
the geometric mean between two numbers
•Solve
problems involving relationships between parts of a right triangle and the altitude to its hypotenuse
•Use
the Pythagorean Theorem and its converse
•Use
the properties of 45-45-90 triangles and 30-60-90 triangles
•Find
trigonometric ratios using right triangles, and use these ratios to find angle measures in right triangles
•Identify
•Use
and use Angles of Depression and Angles of Elevation to solve problems and find missing values
the Law of Sines and Law of Cosines to find missing values in triangles
1
Geometry
Name ________________________
Date _______________ Hour _____
Ch. 8 Introduction
L.T.#1: Be able to write radicals in “simplest radical
form”!
L.T.#2: Be able to solve equations involving radicals!
Quick Vocab:
Vocab:
What is a radical?
What is simplest radical form?
Now, let’
let’s fold some socks!
18
150
124
You can multiply numbers that are under
radicals!
2⋅ 8
5 3⋅ 8
3 5 ⋅4 3
16 ⋅ 120
2
Geometry
Name ________________________
Date _______________ Hour _____
You can divide numbers that are under radicals!
But, NEVER leave a radical sign in your
denominator! You must RATIONALIZE it!!
294 ÷ 3
4
12
4
3
5
25
2 20
4
5 ⋅ 30
3
Now, let’s solve some equations using radicals! Leave
your answers in simplest radical form!
x 2 + 4 = 20
2 x 2 = 20
− 4 + x 2 = 140
1
2
x 2 + 4 = 20
3
Geometry
Name ________________________
Date _______________ Hour _____
Did we meet the target?
L.T.#1: Be able to write radicals in “simplest radical
form”!
L.T.#2: Be able to solve equations involving radicals!
Solve this equation and write your
answer in simplest radical form!
2 x 2 − 4 = 36
Section 8.1 ~
Geometric Mean!
L.T.: Be able to find the “geometric mean” and use it to find
unknowns in similar right triangles!
Geometric Mean:
when the values on one diagonal of a proportion are
equal to each other
Ex. 1: Identify the geometric mean in each of the following
proportions.
3 y GM:
3 4 GM:
x 4 GM:
=
4
=
=
x
5
y
x
7
Ex. 2: Find the value of the geometric mean.
4 x
=
x 16
y 4
=
5 y
4
Geometry
Name ________________________
Date _______________ Hour _____
Ex. 3: Find the geometric mean of:
a. 3 and 12
b. 3 and 48
c. 15 and 20
The Geometric Mean in Right ∆s!
Theorem:
In a right triangle, when you draw an altitude to the
hypotenuse, you create three similar triangles!
∆ACD ~ ∆ADB ~ ∆DCB
A
C
B
D
The legs and altitude of a right triangle are the geometric
means between the segments of the hypotenuse that
______________________________.
a
b
c
x
y
z
5
Geometry
Name ________________________
Date _______________ Hour _____
Let’s practice!
Ex. 4: Refer to the picture to complete each proportion.
x
=
z
y
Challenge:
b
z
=
a
a
x
=
c
=
b
=
z
b
y
a
z
y
y
b
=
z
b
z
x
y
c
Ex. 5: Find the values of the variables.
9
x
y
x
4
2
8
10
6
Geometry
Name ________________________
Date _______________ Hour _____
Just a few more!
5
x
9
x
z
3
4
y
1
3
Section 8.2 ~ day 1
The Pythagorean Theorem!!
L.T.: Be able to find unknowns in triangles using the
Pythagorean Theorem!
Quick WarmWarm-up:
x 2 + 3 = 15
22 + x 2 = 62
What do you call the longest side of a right triangle?
(the side opposite the right angle)
7
Geometry
Name ________________________
Date _______________ Hour _____
Pythagorean Theorem:
In a __________ triangle, the sum of the squares of the
lengths of the legs is equal to the square of the length of the
hypotenuse.
a2 + b2 = c2
c
a
b
Find the value of x. Leave your answer in simplest radical form.
8
x
4
x
10
x
3
4
4
Pythagorean Triples!
A Pythagorean Triple is a set of whole numbers that satisfies the
equation a2 + b2 = c2.
Common triples:
(Multiples of these sets are also triples!)
Find the value of x.
12
13
x
8
x
15
6
x
8
8
Geometry
Name ________________________
Date _______________ Hour _____
Did we meet the target?
L.T.: Be able to find unknowns in triangles using the
Pythagorean Theorem!
Find the length of the hypotenuse!
9
12
Section 8.2 ~ day 2
The Pythagorean Theorem!!
L.T.#1: Be able to find unknowns in triangles using
the Pythagorean Theorem, and find the areas of
triangles!
Quick Review:
Find the value of each variable! Leave answers in simplest radical form.
12
12
x
y
16
16
What is an example of a Pythagorean Triple?
9
Geometry
Name ________________________
Date _______________ Hour _____
Using Pythagorean Theorem to find AREAS!
Find the area of the triangle. Do the legs form a triple?
42 m
445 m
A right triangle has a hypotenuse of length 25 and a leg of
length 10. Find the area of the triangle in simplest radical form
and also as a decimal.
If
a 2 + b 2 < c 2 , then the ∆ is obtuse.
If
a 2 + b 2 > c 2 , then the ∆ is acute.
The lengths of the sides of a triangle are given. Is the
triangle right, obtuse, or acute? Explain.
12, 16, 20
11, 12, 15
31, 23, 12
11, 7 , 4
A window-washer leans a 41-foot
ladder against the side of a building.
The base of the ladder is 9 feet from
the base of the building. How high
up the side of the building does the
ladder reach?
10
Geometry
Name ________________________
Date _______________ Hour _____
Did we meet the target?
L.T.#1: Be able to find unknowns in triangles using
the Pythagorean Theorem, and find the areas of
triangles!
Find the area of the triangle.
26m
24m
Section 8.3 ~ day 1
Special Right Triangles!
L.T.: Be able to find sides of 45-45-90 triangles!
Quick Review: Rationalize the following!
24
6
32
2
6 3
2
11
Geometry
Name ________________________
Date _______________ Hour _____
Find the length of the hypotenuse:
3
8
3
8
45-45-90 Triangle Theorem:
In a 45-45-90 triangle, both ______ are congruent (isosceles),
and the length of the hypotenuse is _____ times the length of
each leg.
Find the value of each variable:
5 3
x
x
5
2
y
45°
4 2
x
y
45°
y
x
45°
21 2
4
12
x
45°
12
Geometry
Name ________________________
Date _______________ Hour _____
Find the value of each variable!
7
2
6
x
45°
x
y
45°
Find the area of the triangle!
4 2
6
45°
45°
13
Geometry
Name ________________________
Date _______________ Hour _____
Why do we need to know about
45-45-90 triangles?
They are in the real world! Not to mention, it’s a whole lot easier
than using the Pythagorean Theorem.
Yadier Molina wants to know how far he has to throw the ball to catch
a man stealing second.
90 ft
Before, we had to use the
Pythagorean Theorem.
But now that you know
about 45-45-90 triangles,
you can use the shortcut!
x ft
90 ft
Did we meet the target?
L.T.: Be able to find sides of 45-45-90 triangles!
Find the value of x.
x
14
45°
14
Geometry
Name ________________________
Date _______________ Hour _____
Section 8.3 ~ day 2
Special Right Triangles!
L.T.: Be able to find sides of 30-60-90 triangles!
Quick Review:
y
4 2
x
45°
5
45°
x
y
Question:
In a 30-60-90 triangle, will any sides be the same length?
No!
30-60-90 Triangle Theorem:
In a 30-60-90 triangle,
the length of the hypotenuse is ____ times the length of
the shorter leg (shorty)
the length of the longer leg (longy) is _____ times
the length of the shorter leg (shorty)
“shorty”
60°
hyp
ote
nu
se
30°
“longy”
15
Geometry
Name ________________________
Date _______________ Hour _____
Find the value of each variable!
60°
x
3
7
x
30°
y
y
15
y
x
30°
30°
5
12
y
60°
x
x
y
30°
x
y
3 3
15
y
x
30°
30°
5
y
x
16
Geometry
Name ________________________
Date _______________ Hour _____
Let’s practice some more!
Find the area of each triangle.
x 60°
6
x
3 3
30°
y
A window-washer leans a 40-foot ladder against the side
of a building. The base of the ladder make a 60° angle
with the ground. How high up the side of the building
does the ladder reach?
40 ft
y
60°
x
d
Find the value of each variable.
c
60°
b
7 2
45°
a
17
Geometry
Name ________________________
Date _______________ Hour _____
Did we meet the target?
L.T.: Be able to find sides of 30-60-90 triangles!
Find the value of each variable!
5 2
a
60°
45°
b
d
c
Section 8.4 day 2 ~ Sine and Cosine
Ratios!
L.T.: Be able to use the sine and cosine ratios to
find lengths and angles in triangles!
The Sine Ratio:
The Cosine Ratio:
In a right triangle, the ratio of the
leg OPPOSITE an angle to the
__________________ is a constant.
This is called the sine ratio!
In a right triangle, the ratio of the
leg ADJACENT to an angle to the
__________________ is a constant.
This is called the cosine ratio!
sin ∠ =
opp
hyp
A
cos ∠ =
C
B
adj
hyp
A
C
B
18
Geometry
Name ________________________
Date _______________ Hour _____
Let’s practice!
V
Ex. 1: Write the sine and cosine ratios for ∠ T.
T
5
13
5
V
12
4
T
U
3
U
Ex. 2: Find each of the following using your calculator! Round
decimals to the nearest thousandth.
sin 120 =
sin 15 =
cos 45 =
You can find an angle from a given sine or cosine ratio!
These are called “inverse sine” and “inverse
cosine”and you can use the sin-1 and cos-1 buttons
on your calculator!
Ex. 3: Find the angle with the given ratio using your calculator! Round
decimals to the nearest tenth.
cos x = 0.7986
sin x = 0.866
tan x = 5.6713
Ex. 4: Use the triangle to find each of the following.
a.
sin A =
b.
cos A =
A
5
c.
sin B =
d.
cos B =
C
13
12
B
19
Geometry
Name ________________________
Date _______________ Hour _____
Let’s practice with the calculators!
cos 24 = x
sin 41 = x
tan x = 2.4751
cos 8 = x
sin x =
4
5
cos x = 0.3057
Let’s see if we’ve met the learning target!
Ex. 5: Use the appropriate ratio to find the value of each variable.
24°
20
6
x
x
35°
x°
10
x°
10
6.5
27
20
Geometry
Name ________________________
Date _______________ Hour _____
Section 8.4 day2 ~ The Tangent
Ratio!
L.T.: Be able to use the tangent ratio to find
lengths and angles in triangles!
Quick Review:
How do you find the third side of a right triangle when you already
know two sides?
What are the two types of “special” right triangles we have worked with?
What is trigonometry?
Let’s look at a right triangle!
A
What side is the hypotenuse?
What is the leg opposite ∠ B?
opposite
What is the leg adjacent to ∠ B?
C
hypotenuse
adjacent
B
The Tangent Ratio:
In a right triangle, the ratio of the leg _______________ an angle to the leg
_________________ to the same angle is a constant. This is called the
tangent ratio!
A
tan ∠ =
opp
adj
C
B
21
Geometry
Name ________________________
Date _______________ Hour _____
Let’s practice!
Ex. 1: Write the tangent ratios for ∠ T and ∠ U.
T
5
13
5
V
4
V
12
U
T
3
U
Ex. 2: Find the tangent RATIO of each angle using your calculator!
Round decimals to the nearest thousandth.
tan 120 =
tan 45 =
tan 30 =
27°
You can find an angle from a given tangent ratio!
This is called an “inverse tangent” and you can use
the tan-1 button on your calculator!
Ex. 3: Find the ANGLE with the given tangent ratio using your
calculator! Round decimals to the nearest tenth.
tan A = 2.21
tan B = −1.482
Ex. 4: Use the triangle to find each of the following.
tan A =
A
b. m ∠ A =
6
a.
c.
m ∠B =
C
15
B
22
Geometry
Name ________________________
Date _______________ Hour _____
Let’s practice with the calculators!
Round decimals to the nearest tenth.
tan 24 = x
tan x =
tan x = 2.4751
4
5
tan 41 = x
tan 88 = x
tan x = 0.3057
Let’s see some problems like the HW!
Ex. 5: Use the tangent ratio to find the value of each variable. Round to
the nearest tenth.
20
8
24°
x°
6
x
x
35°
10
6
23
Geometry
Name ________________________
Date _______________ Hour _____
tan 46 =
x
12
tan 12 =
3
x
x°
8
5
Section 8.5 ~ Angles of Elevation & Depression!
L.T.: Be able to use the trig ratios with angles of
elevation and depression!
Angle of Depression:
Angle of Elevation:
1
2
Recall: Alternate interior angles are _____.
So, the angle of depression _____ the angle of elevation.
24
Geometry
Name ________________________
Date _______________ Hour _____
Let’s practice!
Ex. 1: Describe each angle as it relates to
the picture shown.
1
Ex. 2: The angle of depression
from the Goodyear Blimp to
home plate at Busch Stadium
is 65˚. If the ground distance
from the blimp to the plate is
900 feet, what is the altitude of
the blimp?
65˚
2
3
x
4
65˚
900 ft
Angles of depression:
Angles of elevation:
Would it be okay if we just do the
homework now? ☺
1. A slide has an angle of elevation of 25˚. It is 60 feet from the end of the
slide to the stairway beneath the top of the slide. How long is the slide?
x
25˚
60 ft
2. A forester is standing 150 feet away from a tree. She measures the
angle of elevation from where she is to the top of the tree to be 30˚. How
tall is the tree?
x
30˚
150 ft
25
Geometry
Name ________________________
Date _______________ Hour _____
3. A moving sidewalk takes zoo visitors up a hill. The vertical distance of
the sidewalk is 48 feet. Its angle of elevation is 15˚. About how long is the
sidewalk?
x
48 ft
15˚
6. An airplane flying 4000 feet above ground begins a 2˚ descent (angle of
depression) to land at an airport. How many miles from the airport is the
plane when it starts its descent? (Hint: 1 mile = 5280 feet)
2˚
4000 ft.
2˚
x
7. A 100-foot-tall lighthouse stands at the top of a 150-foot-tall cliff.
The angle of depression from the top of the lighthouse to a ship is 27˚.
About how far from the cliff is the ship?
100 ft
250 ft
150 ft
27˚
x
10. A building is 50 feet high. At a distance away from the building, an
observer notices that the angle of elevation to the top of the building is 41º.
How far is the observer from the base of the building?
50 ft
41˚
x
26
Geometry
Name ________________________
Date _______________ Hour _____
11. An airplane is flying at a height of 2 miles above the ground. The
distance along the ground from the airplane to the airport is 5 miles. What is
the angle of depression from the airplane to the airport?
2 mi
5 mi
x˚
13. A campsite is 9.41 miles from a point directly below the mountain
top. If the angle of elevation is 12° from the camp to the top of the
mountain, how high is the mountain?
x
12˚
9.41 mi
8.6a The Law of Sines!
L.T.: Be able to use the Law of Sines to find
unknowns in triangles!
Quick Review:
What does Soh-Cah-Toa stand for?
What kind of triangles do we use this for?
27
Geometry
Name ________________________
Date _______________ Hour _____
The Law of Sines:
B
sin A sin B
sin C
=
=
a
b
c
c
A
a
b
C
Note:
capital letters always stand
for __________!
lower-case letters always
stand for ________!
Use the Law of Sines ONLY when:
you DON’T have a right triangle AND
you know an angle and its opposite side
Let’s do some problems! ☺
Ex. 1: Use the Law of Sines to find each missing angle or side. Round any
decimal answers to the nearest tenth.
63°
a
29
A
79˚
38˚
42
C
28
Geometry
Name ________________________
Date _______________ Hour _____
Ex. 2: Use the Law of Sines to find each missing angle or side. Round any
decimal answers to the nearest tenth.
T
sin A sin B sin C =
=
a
b
c
51˚ r
s
89°
40°
4.8
Ex. 3: Draw ∆ABC and mark it with the given information. Solve the
triangle. Round any decimal answers to the nearest tenth.
B
a.
a = 7, m∠A = 37, m∠B = 76
c
A
76˚ 7
37˚
b
C
29
Geometry
Name ________________________
Date _______________ Hour _____
b.
a = 12, m∠A = 70, c = 3.1
B
3.1
A
12
70˚
C
b
The Law of Cosines!
L.T.: Be able to use the Law of Cosines to find
unknowns in triangles!
B
c
A
a
b
C
Note:
capital letters always stand
for __________!
lower-case letters always
stand for ________!
Use the Law of Cosines ONLY when:
you DON’T have a right triangle AND
you can’t use the Law of Sines
30
Geometry
Name ________________________
Date _______________ Hour _____
Let’s do some problems! ☺
Ex. 1: Use the Law of Cosines to solve each triangle. Round any decimal
B
answers to the nearest tenth.
a.
19
In ∆ABC , m∠A = 35, b = 16, and c = 19.
A
35˚
a
16
C
b.
In ∆CAT , a = 16, t = 20, and m∠C = 40.
A
c
T
16
20
40˚
C
31
Geometry
Name ________________________
Date _______________ Hour _____
c.
In ∆RED, r = 8, e = 20, and d = 16.
16
E
R
8
20
D
32