Download Geometry Student Text Chapter 5

C H A P T E R
Similarity
© 2010 Carnegie Learning, Inc.
5
A scale model is a representation of an object that is either smaller or larger
than the actual object. Small-scale models of houses can be used to help
builders and buyers visualize the full-sized house before it is actually built. You
will use ratios and proportions to calculate the scale and dimensions of models.
5.1
Ace Reporter
Review of Ratio and Proportion | p. 257
5.5
Geometric Mean
Similar Right Triangles | p. 287
5.2
Picture Picture on the Wall...
Similar Polygons | p. 263
5.6
Indirect Measurement
Application of Similar Triangles | p. 293
5.3
To Be or Not To Be Similar?
Similar Triangle Postulates | p. 273
5.4
Triangle Side Ratios
Angle Bisector/Proportional Side
Theorem | p. 281
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Introductory Problem for Chapter 5
Washington Monument Problem
The Washington Monument is a tall obelisk
built between 1848 and 1884 in honor of the
first President of the United States, George
Washington. It is the tallest
free-standing masonry structure in the world
and has 897 steps from bottom to top.
The Washington National Monument Society,
formed by Congress in 1833, wanted to
construct the largest monument in the world
with dimensions and magnificence that
would be proportionate to the greatness of
George Washington. They wanted to show
the gratitude that the people of United States
felt toward him.
Washington Monument
The official dedication ceremony for the
memorial took place the day before Washington’s birthday in 1885. The final cost of
the project was $1,187,710.
• Your eyes are 6 feet above ground level.
• The reflecting pool is located between the Lincoln Memorial and the
Washington Monument.
• As you stand facing the Washington Monument, you can see in the reflection
pool the very top of the monument.
• The distance from the spot where you are standing to the spot where you see
the top of the monument in the reflecting pool is 12 feet.
• The distance from the location in the reflecting pool where you see the top of
the monument to the base of the monument is 1110 feet.
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It is possible to determine the height of the Washington Monument using only a
simple tape measure and these few known facts:
Lincoln Memorial
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1. Draw a diagram of the situation and
calculate the height of the Washington
Monument.
The reflecting pool
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Be prepared to share your solutions and methods.
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5.1
Ace Reporter
Review of Ratio and Proportion
OBJECTIVES
KEY TERMS
In this lesson you will:
● Write and simplify ratios.
● Compare ratios.
● Write and solve proportions.
● Use survey results to make predictions.
PROBLEM 1
●
●
●
●
●
ratio
probability
proportion
means
extremes
Survey Says
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You are a reporter for your school’s
newspaper. You are writing an article
about the order of classes during the
school day and are interviewing students
to get their opinions.
From the information you have gathered,
it seems the students have a strong opinion
on when the gym class should occur. You
have surveyed many students and recorded
the results in the table shown.
When Do You Think Gym Classes Should Be Held?
Beginning of Day
End of Day
Any Time
8
14
2
1. How many students did you survey?
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2. What can you conclude from your survey results?
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3. You want to compare the results of the survey in your article. One way
you could compare the results is by writing the statement: “Eight out of
24 students prefer to have gym class at the beginning of the day.”
Which two responses from the survey results are being compared?
4. Write a comparison sentence using the results of your survey. Include the
number of students who prefer to have gym class at the end of the day.
5. Write a comparison sentence using the results of your survey. Include the
number of students who have no preference when gym class is held.
A ratio is a comparison of two or more numbers that uses division.
You can mathematically compare the results in the table by using ratios. You can
write a ratio as a fraction, or by using a colon. For instance, you can write “Eight out
of 24 students prefer to have gym class at the beginning of the day” in two ways.
8 students
As a fraction: ___________
24 students
Using a colon : 8 students : 24 students
When you use a colon, you read the colon as the word “to.” So, the ratio
“8 students : 24 students” is read as “8 students to 24 students.”
7. Suppose that you have only surveyed students in your own grade. A friend
offers to help you out and surveys students from another grade in your school.
Your friend’s results are shown in the table.
When Do You Think Gym Classes Should Be Held?
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Beginning of Day
End of Day
Any Time
15
18
3
a. How many students did your friend survey?
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6. Write each comparison sentence you wrote for Questions 4 and 5 as a ratio.
Write each ratio as a fraction. If possible, simplify your fractions.
b. Write three different ratios using the table, and then describe what each ratio
represents. Simplify fractions if possible.
8. Do a larger portion of the students in your survey or your friend’s survey prefer:
a. Gym class at the beginning of the day?
b. Gym class at the end of the day?
c. No preference when gym class is held?
© 2010 Carnegie Learning, Inc.
9. When are two different ratios equivalent?
10. Complete the table to show the results of your survey and your friend’s survey
together. Then write two equivalent ratios for each statement. Write your ratios
as fractions.
When Do You Think Gym Classes Should Be Held?
Beginning of Day
End of Day
Any Time
a. Students who prefer gym class at end of day : Students who prefer gym class
at beginning of day
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b. Students who have no preference : Students who prefer gym class at end
of day
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PROBLEM 2
Making Predictions
1. Use the combined results of the surveys in Problem 1 to write the following
ratios. Write each ratio as a fraction in simplest form.
a. Students who prefer gym class at beginning of day : All students surveyed
b. Students who prefer gym class at end of day : All students surveyed
c. Students with no preference : All students surveyed
Ratios are used to describe probability. The probability of an outcome is the ratio of
the number of successful outcomes to the number of possible outcomes.
2. Suppose each student was required to submit a questionnaire to participate in
the survey in Problem 1. All of the completed questionnaires were put in a box
and one questionnaire was selected at random.
a. What is the probability that the questionnaire selected was from a student
who preferred to take gym class at the end of the day?
c. What is the probability that the questionnaire selected was from a student
who had no preference when gym class is held?
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3. Suppose that you want to interview students from the other grades in your
school. Would you expect the results you would gather from other grades to be
very different from the results you already have? Why or why not?
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b. What is the probability that the questionnaire selected was from a student
who preferred to take gym class at the beginning of the day?
4. Suppose that you interviewed 30 students in a different grade. How many
students would you expect to respond that they prefer to have gym class at the
end of the day?
When two ratios that compare the same quantities are equal, you can write them as
a proportion. A proportion is an equation that states that two ratios are equivalent,
or equal. You write a proportion by placing an equals sign between two equivalent
ratios or by using a double colon in place of the equals sign. For instance, you could
have used a proportion to answer Question 4:
8 students ___________
? students
___________
15 students
30 students
5. What is the value of the unknown quantity in the proportion? Explain.
When you calculated the unknown quantity, you were solving the proportion.
Another way to solve a proportion is by using the proportion’s means and
extremes.
a
__
b
c
__
d
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means
extremes
6. What are the means and extremes of the solved proportion in Question 4?
7. Calculate the product of the means and the product of the extremes from
Question 4. What do you notice?
8. Solve the proportion for x.
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4 __
__
3
x
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9. Suppose that there are 480 students in your school. Use the combined survey
results from Problem 1, Question 10, to predict how many students in your
school would prefer to have gym class at the beginning of the day, how many
students would prefer to have gym class at the end of the day, and how many
students have no preference.
10. Suppose the 480 students in your school submitted a questionnaire
in response to your survey. All questionnaires were put in a box. One
questionnaire was selected at random from the box. Use the combined survey
results from Problem 1, Question 10, to calculate each probability.
a. What is the probability that the questionnaire selected was from a student
who preferred to take gym class at the beginning of the day?
b. What is the probability that the questionnaire selected was from a student
who preferred to take gym class at the end of the day?
Be prepared to share your solutions and methods.
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c. What is the probability that the questionnaire selected was from student who
had no preference when gym class is held?
5.2
Picture Picture on the Wall...
Similar Polygons
OBJECTIVES
KEY TERMS
In this lesson you will:
●
●
●
●
●
●
●
Identify similar polygons.
Identify corresponding angles and corresponding
sides in similar polygons.
Calculate unknown measures in similar polygons.
Calculate unknown measures in a scale model.
Compare side lengths, perimeters, and areas of
similar polygons.
Compare the ratios of side lengths, perimeters,
and areas of similar polygons.
PROBLEM 1
●
●
similar
scale model
scale
The Perfect Picture
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When you frame a picture, it is not unusual to put a mat inside the frame. A mat is
a piece of paperboard that is used to provide a transition between a picture and the
picture frame.
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You are creating your own collage of pictures. You bought a large frame and will cut
out rectangular holes in the mat as shown.
4 in.
5 in.
3 in.
A
6 in.
7 in.
2 in.
3 in.
2 in.
D
C
B
1. What are the interior angle measures of each mat opening?
2. Write a ratio that compares the length of rectangle A to the length of
rectangle B. Then, write a ratio that compares the width of rectangle A
to the width of rectangle B. What do you notice?
4. Write a ratio that compares the length of rectangle A to the length of
rectangle C. Then, write a ratio that compares the width of rectangle A
to the width of rectangle C. What do you notice?
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3. Write a ratio that compares the length of rectangle A to the length of
rectangle D. Then, write a ratio that compares the width of rectangle A
to the width of rectangle D. What do you notice?
PROBLEM 2
Similarity
Two polygons are similar when the corresponding angles are congruent and the
ratios of the measures of the corresponding sides are equal.
1. Which rectangles from Problem 1 are similar?
2. The two triangles shown are similar. You can write 䉭UVW ⬃ 䉭XYZ, where the
symbol ⬃ means “is similar to.”
Y
V
X
Z
U
W
Again, the order in which you write the vertices in a similarity statement indicates
the corresponding angles and the corresponding sides.
© 2010 Carnegie Learning, Inc.
a. List the corresponding angles and the corresponding sides.
b. Write a ratio that compares a side length of 䉭UVW to a corresponding side
length of 䉭XYZ.
c. Write a ratio that compares a side length of 䉭XYZ to a corresponding
side length of 䉭UVW.
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d. Are the two ratios equal from Questions 2(b) and (c)? Why or why not?
e. When you write a proportion relating the corresponding side lengths of two
similar polygons, what must be true about both of the ratios?
3. In the figure shown, 䉭GHI ⬃ 䉭KLM.
H
G
L
I
K
M
b. Suppose that GH 3 feet, KL 9 feet, and HI 5 feet. Write a proportion
to calculate LM. Solve the proportion.
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a. Write proportions that relate the ratios side lengths using the corresponding
sides of 䉭GHI and 䉭KLM.
c. Suppose that you also know that KM 12 feet. Calculate GI.
d. Calculate the ratio of the height of 䉭GHI to the height of 䉭KLM.
e. Calculate the ratio of the length of the base of 䉭GHI to the length of
the base of 䉭KLM.
© 2010 Carnegie Learning, Inc.
f. Compare the ratios of the lengths and heights.
g. Calculate the areas of the triangles. Write the ratio of the area of 䉭GHI to
the area of 䉭KLM. Write your ratio as a fraction in simplest form.
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h. How is the ratio of the areas related to the ratio of the heights and to the
ratio of the lengths of the bases?
Lesson 5.2
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Picture Picture on the Wall...
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A scale model (or model) is a replication that is similar to an actual object, but
it is either larger or smaller. The ratio of a dimension of the actual object to a
corresponding dimension in the model is called the scale of the model.
5. A scale model of a framed picture is being created to put inside a dollhouse.
The actual rectangular picture is 4 inches wide and 8 inches long. The scale of
the model is 4 : 1. Calculate the length and width of the dollhouse picture.
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4. A wall mural is being painted from a picture that is 6 inches long and 4 inches
wide. The wall mural should be 48 inches long.
a. Complete the statement to calculate the scale of the model. Write your answer
as a fraction in simplest form.
length
of picture
_______________
length of mural
b. Now use the scale to complete the proportion to calculate the width of
the mural.
width
of picture
_______________
width of mural
c. Calculate the width of the mural.
6. Determine the ratio of the perimeter of the actual picture and the perimeter of
the dollhouse picture.
7. Determine the ratio of the area of the actual picture and the area of the
dollhouse picture.
PROBLEM 3
Ratios of Area and Perimeter of
Similar Rectangles
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1. Draw a rectangle on the grid. Each square on the grid represents a square that
is one foot long and one foot wide.
2.
3.
4.
5.
Calculate the perimeter and area of the rectangle you drew.
Draw a similar rectangle on the grid by multiplying each side by a scale factor.
Calculate the perimeter and the area of the second rectangle.
Repeat steps 3 and 4 two more times.
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Picture Picture on the Wall...
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6. Record the length, height, perimeter, and area of each rectangle in the table.
Length
(in feet)
Height
(in feet)
Perimeter
(in feet)
Area
(in square feet)
First Rectangle
Second Rectangle
Third Rectangle
Fourth Rectangle
7. Complete the table by calculating the ratios of side lengths, perimeters, and
areas for each rectangle comparison. Simplify all fractions.
Ratio of Rectangles
Ratio of Side
Ratio of
Lengths
Perimeters
Ratio of
Areas
First Rectangle : Second Rectangle
Second Rectangle : Third Rectangle
Third Rectangle : Fourth Rectangle
Compare your table with others in your group or class. Look for patterns.
8. What is the relationship between the ratio of side lengths of two similar
rectangles and the ratio of perimeters of the two similar rectangles?
a.
10. The ratio of side lengths of two similar rectangles is __
b
a. What is the ratio of perimeters of the two similar rectangles?
b. What is the ratio of areas of the two similar rectangles?
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4 . What is the ratio of
11. The ratio of side lengths of two similar rectangles is __
5
areas of the two similar rectangles?
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© 2010 Carnegie Learning, Inc.
9. What is the relationship between the ratio of side lengths of two similar
rectangles and the ratio of areas of the two similar rectangles?
PROBLEM 4
Area and Perimeter Ratios of Triangles
Is the relationship between the ratio of side lengths of two similar rectangles
and the ratio of areas of two similar rectangles the same for all polygons?
Let’s consider triangles.
© 2010 Carnegie Learning, Inc.
1. Draw a right triangle on grid. Each square on the grid represents a square that is
one foot long and one foot wide. For easier calculations, use Pythagorean triples.
2.
3.
4.
5.
6.
Calculate the perimeter and area of the triangle you drew.
Draw a similar triangle on the grid by multiplying each side by a scale factor.
Calculate the perimeter and the area of the second triangle.
Repeat steps 3 and 4 two more times.
Record the length of the base, height, perimeter, and area of each triangle
in the table.
Base
(in feet)
Height
(in feet)
Perimeter
(in feet)
Area
(in square feet)
First Triangle
5
Second Triangle
Third Triangle
Fourth Triangle
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Picture Picture on the Wall...
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7. Complete the table by calculating the ratios of side lengths, perimeters, and
areas for each triangle comparison. Simplify all fractions.
Ratio of Triangles
Ratio of Side
Lengths
Ratio of
Perimeters
Ratio of
Areas
First Triangle : Second Triangle
Second Triangle : Third Triangle
Third Triangle : Fourth Triangle
Compare your table with others in your group or class. Look for patterns.
8. What is the relationship between the ratio of side lengths of two similar
triangles and the ratio of perimeters of the two similar triangles?
9. What is the relationship between the ratio of two side lengths of similar
triangles and the ratio of areas of the two similar triangles?
a.
10. The ratio of side lengths of two similar triangles is __
b
a. What is the ratio of perimeters of the two similar triangles?
4 . What is the
11. The ratio of side lengths of two similar triangles is expressed as __
5
ratio of areas of the two similar triangles?
12. What might you conclude about the relationships of similar polygons with
respect to their ratios of side lengths, ratios of perimeters, and ratios of areas?
5
Be prepared to share your solutions and methods.
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b. What is the ratio of areas of the two similar triangles?
5.3
To Be or Not To Be Similar?
Similar Triangle Postulates
OBJECTIVE
In this lesson you will:
●
Use constructions and given
information to determine
whether two triangles are
similar.
KEY TERMS
●
●
●
●
●
Angle-Angle Similarity Postulate
Side-Side-Side Similarity Postulate
Side-Angle-Side Similarity Postulate
included angle
included side
© 2010 Carnegie Learning, Inc.
An art projector is a piece of equipment that artists use to create exact copies of
artwork, to enlarge artwork, or to reduce artwork. A basic art projector uses a light
bulb and a lens within a box. The light rays from the art being copied are collected
onto a lens at a single point. The lens then projects the image of the art onto a
screen as shown.
If the projector is set up properly, the triangles will be similar polygons. You can
show that these triangles are similar without measuring all of the side lengths and
all of the interior angles.
Lesson 5.3
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To Be or Not To Be Similar?
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PROBLEM 1
Using Two Angles
Two polygons are similar if all corresponding angles are congruent and the ratios of
the measures of all corresponding sides are equal.
1. State all corresponding congruent angles and all corresponding proportional
sides using the similar triangles shown.
䉭RST ⬃ 䉭WXY
R
W
S
X
Y
T
Are there any shortcuts that can be taken? Can you use fewer pairs of angles or
fewer pairs of sides to show triangles are similar? Constructions can be used to
answer these questions.
2. Construct triangle DEF using only ⬔D and ⬔E in triangle DEF as shown.
Be sure that the side lengths of triangle DEF are different than the
side lengths of triangle DEF.
D
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E
F
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You can conclude that two triangles are similar if you are able to prove that three
pairs of corresponding angles are congruent and three pairs of corresponding
sides are proportional.
3. Did everyone in your group and class construct the same triangle? If not,
describe the difference between the triangles that were constructed.
4. Measure the angles and sides of triangle DEF and triangle DEF. Is the
third pair of angles congruent? Are the three pairs of corresponding sides
proportional?
5. Are the two triangles similar?
6. Would your answers to Questions 3 through 5 change if you had constructed
triangle DEF using a different side length for DE?
7. Two pairs of corresponding angles are congruent. Is this sufficient information
to conclude that the triangles are similar?
The Angle-Angle Similarity Postulate states: “If two angles of one triangle are
congruent to two angles of another triangle, then the triangles are similar.”
© 2010 Carnegie Learning, Inc.
B
E
F
C
D
A
If m⬔A m⬔D and m⬔C m⬔F, then 䉭ABC ⬃ 䉭DEF.
8. The triangles shown are isosceles triangles. Do you have enough information to
show that the triangles are similar? Explain your reasoning.
Q
M
5
L
N
R
P
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To Be or Not To Be Similar?
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9. The triangles shown are isosceles triangles. Do you have enough information to
show that the triangles are similar? Explain your reasoning.
T
W
S
U
V
PROBLEM 2
X
Using Two and Three Proportional Sides
___
___
1. Construct triangle DEF by doubling the lengths of sides DE and EF . Be sure
to first construct the new DE and EF separately and then construct the
triangle; this will ensure a ratio of 2 : 1. Do not duplicate angles.
D
F
2. Did everyone in your group and class construct the same triangle? If not,
describe the difference between the triangles that were constructed.
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3. Measure the angles and sides of triangle DEF and triangle DEF. Are the
corresponding angles congruent? Are the three pairs of corresponding
sides proportional?
Chapter 5
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Similarity
© 2010 Carnegie Learning, Inc.
E
4. Are the two triangles similar?
5. Two pairs of corresponding sides are proportional. Is this sufficient information
to conclude that the triangles are similar?
6. Construct triangle DEF by doubling the lengths of sides DE, EF, and FD. Be
sure to first construct the new side lengths separately, and then construct the
triangle. Do not duplicate angles.
D
E
© 2010 Carnegie Learning, Inc.
F
7. Did everyone in your group and class construct the same triangle? If not,
describe the difference between the triangles that were constructed.
8. Measure the angles and sides of triangle DEF and triangle DEF. Are the
corresponding angles congruent?
5
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To Be or Not To Be Similar?
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9. Are the two triangles similar?
10. Three pairs of corresponding sides are proportional. Is this sufficient
information to conclude that the triangles are similar?
The Side-Side-Side Similarity Postulate states: “If the corresponding sides of two
triangles are proportional, then the triangles are similar.”
E
B
F
C
D
A
BC ___
AC , then 䉭ABC ⬃ 䉭DEF.
AB ___
If ___
DE
EF
DF
11. If the corresponding sides of two triangles are proportional, what makes the
triangles similar?
12. Determine whether 䉭UVW is similar to 䉭XYZ. If so, use symbols to write a
similarity statement.
Z
V
16 meters
Y
33 meters
W
36 meters
U
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Similarity
24 meters
22 meters
X
© 2010 Carnegie Learning, Inc.
24 meters
The Side-Angle-Side Similarity Postulate states: “If two of the corresponding sides
of two triangles are proportional and the included angles are congruent, then the
triangles are similar.”
B
E
C
F
D
A
AC and ⬔A 艑 ⬔D, then 䉭ABC ⬃ 䉭DEF.
AB ___
If ___
DE
DF
PROBLEM 3
Using Two Proportional Sides and an Angle
An included angle is an angle formed by two consecutive sides of a figure.
An included side is a line segment between two consecutive angles of a figure.
1. Construct triangle DEF by doubling the lengths of any two sides and
duplicating the included angle. Be sure to first construct the new side lengths
separately, and then construct the triangle.
© 2010 Carnegie Learning, Inc.
D
E
F
2. Measure the angles and sides of triangle DEF and triangle DEF. Are the
corresponding angles congruent? Are the corresponding sides proportional?
Lesson 5.3
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To Be or Not To Be Similar?
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3. Are the two triangles similar?
4. Two pairs of corresponding sides are proportional and the corresponding
included angles are congruent. Is this sufficient information to conclude that the
triangles are similar?
PROBLEM 4
Guess My Triangle
Be prepared to share your solutions and methods.
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Similarity
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1. Gaelin is thinking of a triangle and he wants everyone in his class to draw a
similar triangle. Which combinations of sides and angles could he provide?
5.4
Triangle Side Ratios
Angle Bisector/Proportional Side Theorem
OBJECTIVES
KEY TERM
In this lesson you will:
●
●
●
Angle Bisector/Proportional Side Theorem
Prove the Angle Bisector/
Proportional Side Theorem.
Apply the Angle Bisector/
Proportional Side Theorem.
PROBLEM 1
When an interior angle of a triangle is bisected, proportional relationships involving
the sides of the triangle occur. You will be able to prove that these relationships
apply to all triangles. To do this, it will be necessary to extend a side of the triangle
and add an auxiliary line parallel to one side of the triangle.
© 2010 Carnegie Learning, Inc.
The Angle Bisector/Proportional Side Theorem states: “A bisector of an angle in a
triangle divides the opposite side into two segments whose lengths are in the same
ratio as the lengths of the sides adjacent to the angle.”
To prove the Angle Bisector/Proportional Side Theorem, consider the statements and
figure shown.
___
Given: AD bisects ⬔BAC
AC
AB ___
Prove: ___
BD
CD
A
5
B
D
C
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Triangle Side Ratios
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___
___
1. Draw an auxiliary line parallel to AB through point C. Extend AD until it
intersects the auxiliary line. Label the point of intersection, point E.
A
B
D
C
E
2. Complete the two-column proof.
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Reasons
1.
1. Given
2.
2. Construction
3.
3. Definition of angle bisector
4. ⬔BAE ⬵ ⬔CEA
4.
5.
5. Transitive Property of 艑
6.
6. If two angles of a triangle are congruent, then
the sides opposite the angles are congruent.
7.
7. Definition of congruent segments
8.
8. Alternate Interior Angle Theorem
9. 䉭DAB ⬃ 䉭DEC
9.
AB ___
BD
10. ___
EC
CD
10.
11.
11. Rewrite as an equivalent proportion
AC
AB ___
12. ___
BD
CD
12.
Similarity
© 2010 Carnegie Learning, Inc.
Statements
PROBLEM 2
Applying the Angle Bisector/Proportional Side Theorem
Apply the Angle Bisector/Proportional Side Theorem to solve each problem.
1. On the map shown, North Craig Street bisects the angle formed between
Bellefield Avenue and Ellsworth Avenue.
• The distance from the ATM to the Coffee Shop is 300 feet.
• The distance from the Coffee Shop to the Library is 500 feet.
• The distance from your apartment to the Library is 1200 feet.
Determine the distance from your apartment to the ATM.
© 2010 Carnegie Learning, Inc.
___
2. CD bisects ⬔C. Solve for DB.
A
D
8
24
B
30
5
C
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Triangle Side Ratios
283
___
3. CD bisects ⬔C. Solve for AC.
A
9
11
D
B
22
C
___
4. AD bisects ⬔A. AC AB 36. Solve for AC and AB.
B
A
14
D
7
© 2010 Carnegie Learning, Inc.
C
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___
5. BD bisects ⬔B. Solve for AC.
A
14
B
6
D
16
C
© 2010 Carnegie Learning, Inc.
Be prepared to share your solutions and methods.
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5.5
Geometric Mean
Similar Right Triangles
OBJECTIVES
KEY TERMS
In this lesson you will:
●
●
Construct an altitude to the hypotenuse
in a right triangle.
●
Explore the relationships created when
an altitude is drawn to the hypotenuse
of a right triangle.
PROBLEM 1
●
●
●
Right Triangle/Altitude
Similarity Theorem
geometric mean
Right Triangle Altitude Theorem 1
Right Triangle Altitude Theorem 2
Right Triangles
© 2010 Carnegie Learning, Inc.
A bridge is needed to cross over a canyon. The dotted line segment connecting
points S and R represents the bridge. The distance from point P to point S is
45 yards. The distance from point Q to point S is 130 feet. How long is the bridge?
5
To determine the length of the bridge, you must first explore what happens when an
altitude is drawn to the hypotenuse of a right triangle.
When an altitude is drawn to the hypotenuse of a right triangle, it forms two smaller
triangles. All three triangles have a special relationship. Let’s explore this relationship.
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1. Construct an altitude to the hypotenuse in the right triangle ABC. Label the
altitude CD.
C
D
A
B
2. Name all right triangles in the figure.
© 2010 Carnegie Learning, Inc.
3. Trace each of the triangles on separate pieces of paper and label all
the vertices on each triangle. Cut out each triangle. Label the vertex
of each triangle.
4. Arrange the triangles so that all of the triangles have the same orientation. The
hypotenuse, the shortest leg, and the longest leg should all be in corresponding
positions. You may have to flip triangles over to do this.
5. Draw each triangle in the same orientation.
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6. Is 䉭ABC ⬃ 䉭ACD? Explain.
7. Is 䉭ABC ⬃ 䉭CBD? Explain.
8. Is 䉭ACD ⬃ 䉭CBD? Explain.
9. Write the corresponding sides of 䉭ABC and 䉭ACD as proportions.
10. Write the corresponding sides of 䉭ABC and 䉭CBD as proportions.
11. Write the corresponding sides of 䉭ACD and 䉭CDB as proportions.
You have just proven a theorem!
The Right Triangle/Altitude Similarity Theorem states: “If an 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.”
© 2010 Carnegie Learning, Inc.
You can now use this theorem as a valid reason in proofs.
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289
PROBLEM 2
Geometric Mean
When an altitude of a right triangle is constructed from the right angle to
the hypotenuse, three similar right triangles are created. This altitude is a
geometric mean.
The geometric mean of two positive numbers a and b is the positive number x such
a __
x.
that __
x
b
Two theorems are associated with the altitude to the hypotenuse of a right triangle.
You will prove these theorems in the Assignments for this lesson.
The Right Triangle Altitude Theorem 1 states: “The measure of the altitude drawn
from the vertex of the right angle of a right triangle to its hypotenuse is the geometric
mean between the measures of the two segments of the hypotenuse.”
1. Which proportion(s) in Problem 1 demonstrate(s) this theorem?
The Right Triangle Altitude Theorem 2 states: “If the altitude is drawn to the
hypotenuse of a right triangle, each leg of the right triangle is the geometric mean of
the hypotenuse and the segment of the hypotenuse adjacent to the leg.”
2. Which proportion(s) in Problem 1 demonstrate(s) this theorem?
3. In each triangle, solve for x.
a.
x
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9
© 2010 Carnegie Learning, Inc.
4
b.
4
x
8
c.
x
4
20
d.
© 2010 Carnegie Learning, Inc.
x
2
8
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291
4. In the triangle shown, solve for x, y, and z.
5
10
z
x
y
PROBLEM 3
Bridge Over the Canyon
Be prepared to share your solutions and methods.
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1. Solve for the length of the bridge in Problem 1 using the geometric mean.
5.6
Indirect Measurement
Application of Similar Triangles
OBJECTIVES
KEY TERM
In this lesson you will:
● Identify similar triangles to calculate indirect
measurements.
● Use proportions to solve for unknown
measurements.
PROBLEM 1
●
indirect measurement
How Tall is That Flagpole?
At times, situations occur when measuring something directly is impossible, or
physically undesirable. When these situations arise, indirect measurement, the
technique that uses proportions to calculate measurement, can be implemented.
Your knowledge of similar triangles can be very helpful in these situations.
Use the following steps to measure the height of the school flagpole or any other tall
object outside. You will need a partner, a tape measure, a marker, and a flat mirror.
© 2010 Carnegie Learning, Inc.
Step 1: Use a marker to create a dot near the center of the mirror.
Step 2: Face the object you would like to measure and place the mirror
between yourself and the object. You, the object, and the mirror
should be collinear.
Step 3: Focus your eyes on the dot on the mirror and walk backward until you
can see the top of the object on the dot, as shown.
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Step 4: Ask your partner to sketch a picture of you, the mirror, and the object.
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Step 5: Review the sketch with your partner. Decide where to place right
angles, and where to locate the sides of the two triangles.
1. How can similar triangles be used to calculate the height of the object?
2. Use your sketch to write a proportion to calculate the height of the object and
solve the proportion.
3. Compare your answer with others measuring the same object. How do the
answers compare?
4. What are some possible sources of error that could result when using this
method?
5
5. Switch places with your partner and identify a second object to measure.
Repeat this method of indirect measurement to solve for the height of the
new object.
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Step 6: Determine which segments in your sketch can easily be measured
using the tape measure. Describe their locations and record the
measurements on your sketch.
PROBLEM 2
How Tall is That Oak Tree?
© 2010 Carnegie Learning, Inc.
1. You go to the park and use the mirror method to gather enough information to
calculate the height of one of the trees. The figure shows your measurements.
Calculate the height of the tree.
2.
Take Note
Remember,
whenever you are
solving a problem
that involves
measurements like
length (or weight),
you may have to
rewrite units so
they are the same.
For instance, if a
problem involves
weight, all of the
weights should be
measured in the
same unit of
measure.
Stacey wants to try the mirror method to measure the height of one
of her trees. She calculates that the distance between her and the
mirror is 3 feet and the distance between the mirror and the tree is
18 feet. Stacey’s eye height is 60 inches. Draw a diagram of this
situation. Then calculate the height of this tree.
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295
© 2010 Carnegie Learning, Inc.
3. Stacey notices that another tree casts a shadow and suggests that you could
also use shadows to calculate the height of the tree. She lines herself up with
the tree’s shadow so that the tip of her shadow and the tip of the tree’s shadow
meet. She then asks you to measure the distance from the tip of the shadows
to her, and then measure the distance from her to the tree. Finally, you draw
a diagram of this situation as shown below. Calculate the height of the tree.
Explain your reasoning.
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PROBLEM 3
How Wide is That Creek?
It is not reasonable for you to directly measure the width of a creek, but you can use
indirect measurement to measure the width. You stand on one side of the creek and
your friend stands directly across the creek from you on the other side as shown in
the figure.
1. Your friend is standing 5 feet from the creek and you are standing 5 feet from
the creek. Mark these measurements on the diagram shown.
2. You and your friend walk away from each other in opposite parallel directions.
Your friend walks 50 feet and you walk 12 feet. Mark these measurements
on the diagram shown. Draw a line segment that connects your starting point
and ending point and draw a line segment that connects your friend’s starting
point and ending point.
© 2010 Carnegie Learning, Inc.
3. Draw a line segment that connects you and your friend’s starting points and
draw a line segment that connects you and your friend’s ending points. Label
any angle measures and any angle relationships that you know on the diagram.
Explain how you know these angle measures.
4. How do you know that the triangles formed by the lines are similar?
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297
5. Calculate the distance from your friend’s starting point to your side of the
creek. Round your answer to the nearest tenth, if necessary.
6. What is the width of the creek? Explain your reasoning.
© 2010 Carnegie Learning, Inc.
7. There is also a ravine (a deep hollow in the earth) on another edge of the park.
You and your friend take measurements like those in Problem 3 to indirectly
calculate the width of the ravine. The figure shows your measurements.
Calculate the width of the ravine.
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8. There is a large pond in the park. A diagram of the pond is shown below. You
___
want to calculate the distance across the widest part of the pond, labeled as DE .
To indirectly calculate this distance, you first place a stake at point A. You chose
point A so that you can see the edge of the pond on both sides at points D and
E, where you also place stakes. Then you tie string from point A to point D and
from point A to point E. At a narrow portion
of the pond,___
you place stakes at
___
points B and C along the string so that BC is parallel to DE . The measurements
you make are shown on the diagram. Calculate the distance across the widest
part of the pond.
Be prepared to share your solutions and methods.
5
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299
Chapter 5 Checklist
KEY TERMS
●
●
●
●
ratio (5.1)
probability (5.1)
proportion (5.1)
means (5.1)
●
●
●
●
extremes (5.1)
similar (5.2)
scale model (5.2)
scale (5.2)
●
●
●
●
included angle (5.3)
included side (5.3)
geometric mean (5.5)
indirect measurement (5.6)
POSTULATES
●
●
●
Angle-Angle Similarity Postulate (5.3)
Side-Side-Side Similarity Postulate (5.3)
Side-Angle-Side Similarity Postulate (5.3)
THEOREMS
●
●
Angle Bisector/Proportional Side Theorem (5.4)
Right Triangle/Altitude Similarity Theorem (5.5)
●
●
Right Triangle Altitude Theorem 1 (5.5)
Right Triangle Altitude Theorem 2 (5.5)
CONSTRUCTIONS
●
triangles (5.3)
altitudes (5.5)
5.1
Writing Ratios
To write a ratio of one quantity to another quantity, write the ratio as a fraction or use
a colon.
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●
Example:
The table shows the results of a survey in which 55 students were asked to name
their favorite type of movie.
What is Your Favorite Type of Movie?
Comedy
21
Drama
16
Horror
7
Action
6
Science Fiction
2
Other
3
The ratios of the number of students who chose comedy as their favorite type of
movie to the number of students surveyed is:
21 students
As a fraction: ___________
55 students
Using a colon: 21 students : 55 students
The ratio of the number of students who chose drama as their favorite type of movie
to the number of students who chose horror as their favorite type of movie is:
16 students
As a fraction: ___________
7 students
Using a colon: 16 students : 7 students
5.1
Solving Proportions to Make Predictions
© 2010 Carnegie Learning, Inc.
To solve a proportion, set the product of the means equal to the product of the
extremes. Then, solve for the variable in the proportion.
Example:
Suppose that there are 495 students in your school. You can use the results from the
survey in the previous example to predict the number of students that would choose
each type of movie.
To predict the number of students in your school who would choose comedy as their
favorite type of movie, solve the following proportion.
21 students ____________
x students
___________
Write a proportion.
55x 21 495
Set the product of the means equal to the product
of the extremes.
55x 10,395
Multiply.
55 students
x 189
495 students
5
Divide each side by 55.
You can predict that 189 students from your school would choose comedy as their
favorite type of movie.
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5.2
Identifying Similar Polygons
Two polygons are similar when the corresponding angles are congruent and the
ratios of the measures of the corresponding sides are equal.
Example:
B
80°
S
18 ft
80°
12 ft
12 ft
8 ft
34°
66°
R
T
34°
14 ft
66°
A
C
21 ft
Corresponding angles are congruent: ⬔R ⬵ ⬔A, ⬔S ⬵ ⬔B, and ⬔T ⬵ ⬔C.
RS ___
ST ___
8 __
2, ___
12 __
2 , and
Ratios of corresponding sides are equal: ___
12
3 BC
18
3
AB
2.
RT ___
14 __
___
21
AC
3
So, triangle RST is similar to triangle ABC. You can write 䉭RST ⬃ 䉭ABC.
5.2
Using a Scale Model to Calculate a Measure
A scale model is a replication of an actual object that is similar to the object, but is
either larger or smaller. To calculate a measure using a scale model, write and solve
a proportion.
A museum gift shop sells plastic alligators that are scale models of an actual
alligator.
The scale of the model is 22 : 1. The tail length of the model alligator is 3 inches.
To determine the tail length of the actual alligator, write and solve a proportion.
x inches _________
22 inches
________
3 inches
1 inch
x 1 22 3
x 66
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Write a proportion.
Set the product of the means equal to the product
of the extremes.
Multiply.
The tail length of the actual alligator is 66 inches, or 5.5 feet.
Chapter 5
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Example:
5.3
Using the Angle-Angle Similarity Postulate
The Angle-Angle Similarity Postulate states: “If two angles of one triangle are
congruent to two angles of another triangle, then the triangles are similar.”
Example:
D
Z
Y
28°
32°
32°
X
28°
F
E
m⬔D m⬔X and m⬔E m⬔Y, so 䉭DEF ⬃ 䉭XYZ
5.3
Using the Side-Side-Side Similarity Postulate
The Side-Side-Side Similarity Postulate states: “If three pairs of corresponding sides
of two triangles are proportional, then the triangles are similar.”
Example:
Q
B
32 m
20 m
48 m
30 m
A
25 m
P
C
40 m
R
5, ____
BC ___
AC ___
20 __
30 __
5 , and ___
25 __
5 , so 䉭ABC ⬃ 䉭PQR
AB ___
___
© 2010 Carnegie Learning, Inc.
PQ
5.3
32
8 QR
48
8
PR
40
8
Using the Side-Angle-Side Similarity Postulate
The Side-Angle-Side Similarity Postulate states: “If two of the corresponding sides
of two triangles are proportional and the included angles are congruent, then the
triangles are similar.”
Example:
G
100°
J
16 ft
5
28 ft
H
14 ft
L
100°
8 ft
F
K
FG ___
GH ___
28 2, m⬔G m⬔K, and ____
16 2, so 䉭FGH ⬃ 䉭JKL
___
JK
14
KL
8
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5.4
Using the Angle Bisector/Proportional Side Theorem
The Angle Bisector/Proportional Side Theorem states: “A bisector of an angle in a
triangle divides the opposite side into two segments whose lengths are in the same
ratio as the lengths of the sides adjacent to the angle.”
Example:
R
18 cm
Q
15 cm
12 cm
S
x
T
QR ___
SR
____
QT
ST
18 ___
12
___
x
15
18x 180
x 10 cm
5.5
Using the Right Triangle/Altitude Similarity Theorem
The Right Triangle/Altitude Similarity Theorem states: “If an 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.”
Example:
B
A
C
䉭ABC ⬃ 䉭ACD, 䉭ABC ⬃ 䉭CBD, and 䉭ACD ⬃ 䉭CBD.
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D
5.5
Using the Right Triangle Altitude Theorem 1
The Right Triangle Altitude Theorem 1 states: “The measure of the altitude drawn
from the vertex of the right angle of a right triangle to its hypotenuse is the geometric
mean between the measures of the two segments of the hypotenuse.”
A geometric mean of two positive numbers a and b is the positive number x
a __
x.
such that __
x
b
Example:
x
20 ft
8 ft
8 ___
x
__
x
20
x 160
2
____
x √160
x 艐 12.65 ft
5.5
Using the Right Triangle Altitude Theorem 2
The Right Triangle Altitude Theorem 2 states: “If the altitude is drawn to the
hypotenuse of a right triangle, each leg of the right triangle is the geometric mean of
the hypotenuse and the segment of the hypotenuse adjacent to the leg.”
© 2010 Carnegie Learning, Inc.
Example:
15 in.
32 in.
x
32 ___
x
___
x
15
x 480
2
____
5
x √480
x 21.91 in.
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5.6
Using Indirect Measurement to Solve a Problem
Indirect measurement is a technique that uses proportions to determine a
measurement when direct measurement is not possible. To calculate a length by
using indirect measurement, create two similar triangles and then write and solve a
proportion that uses the ratios of the corresponding lengths of the similar triangles.
Example:
You want to determine the height of a sign in front of a store. You notice that the
sign is casting a shadow on the ground. So, you ask your friend to line herself up
with the sign’s shadow so that the tip of her shadow and the tip of the sign’s shadow
meet. Then, you measure the distance between your friend and the sign, and you
also measure the length of your friend’s shadow. You know that your friend is
5.5 feet tall. You draw and label a diagram of this situation as shown.
You can use a proportion to solve for the height x of the sign.
11
x 19
___
________
11
5.5
11
11x 165
x 15
The sign is 15 feet tall.
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5.5
30
x ___
___