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Columbus, Ohio 43240
Lesson 6-1 Proportions
Lesson 6-2 Similar Polygons
Lesson 6-3 Similar Triangles
Lesson 6-4 Parallel Lines and Proportional Parts
Lesson 6-5 Parts of Similar Triangles
Lesson 6-6 Fractals and Self-Similarity
Example 1 Write a Ratio
Example 2 Extended Ratios in Triangles
Example 3 Solve Proportions by Using Cross Products
Example 4 Solve Problems Using Proportions
The total number of students who participate in
sports programs at Central High School is 520. The
total number of students in the school is 1850. Find
the athlete-to-student ratio to the nearest tenth.
To find this ratio, divide the number of athletes by the
total number of students.
0.3 can be written as
Answer: The athlete-to-student ratio is 0.3.
The country with the longest school year is China
with 251 days. Find the ratio of school days to total
days in a year for China to the nearest tenth. (Use
365 as the number of days in a year.)
Answer: 0.7
Multiple- Choice Test Item
In a triangle, the ratio of the measures of three sides
is 5:12:13, and the perimeter is 90 centimeters. Find
the measure of the shortest side of the triangle.
A 15 cm
B 18 cm
C 36 cm
D 39 cm
Read the Test Item
You are asked to apply the ratio to the three sides of the
triangle and the perimeter to find the shortest side.
Solve the Test Item
We can rewrite 5:12:13 as 5x:12x:13x and use those
measures for the sides of the triangle. Write an equation
to represent the perimeter of the triangle as the sum of
the measures of its sides.
Perimeter
Combine like terms.
Divide each side by 30.
Use this value of x to find the measures of the sides of
the triangle.
The shortest side is 15 centimeters. The answer is A.
Check Add the lengths of the sides to make sure that
the perimeter is 90.
Answer: A
Multiple- Choice Test Item
In a triangle, the ratio of the measures of three sides
is 3:4:5, and the perimeter is 42 feet. Find the
measure of the longest side of the triangle.
A 10.5 ft
Answer: C
B 14 ft
C 17.5 ft
D 37 ft
Solve
Original proportion
Cross products
Multiply.
Divide each side by 6.
Answer: 27.3
Solve
Original proportion
Cross products
Simplify.
Add 30 to each side.
Divide each side by 24.
Answer: –2
Solve each proportion.
a.
Answer: 4.5
b.
Answer: 9
A boxcar on a train has a length of 40 feet and a
width of 9 feet. A scale model is made with a length
of 16 inches. Find the width of the model.
Because the scale model of the boxcar and the boxcar
are in proportion, you can write a proportion to show the
relationship between their measures. Since both ratios
compare feet to inches, you need not convert all the
lengths to the same unit of measure.
Substitution
Cross products
Multiply.
Divide each side by 40.
Answer: The width of the model is 3.6 inches.
Two large cylindrical containers are in proportion.
The height of the larger container is 25 meters with
a diameter of 8 meters. The height of the smaller
container is 7 meters. Find the diameter of the
smaller container.
Answer: 2.24 m
Example 1 Similar Polygons
Example 2 Scale Factor
Example 3 Proportional Parts and Scale Factor
Example 4 Enlargement of a Figure
Example 5 Scale Factors on Maps
Determine whether the pair of figures is similar.
Justify your answer.
Q
The vertex angles are marked as 40º and 50º, so they
are not congruent.
Since both triangles are isosceles, the base angles in
each triangle are congruent. In the first triangle, the base
angles measure
and in the second
triangle, the base angles measure
Answer: None of the corresponding angles are
congruent, so the triangles are not similar.
Determine whether the pair of figures is similar.
Justify your answer.
T
Thus, all the corresponding angles are congruent.
Now determine whether corresponding sides are
proportional.
The ratios of the measures of the corresponding sides
are equal.
Answer: The ratio of the measures of the corresponding
sides are equal and the corresponding angles
are congruent, so
Determine whether the pair of figures is similar.
Justify your answer.
a.
Answer: Both triangles are isosceles with base angles
measuring 76º and vertex angles measuring 28º.
The ratio of the measures of the corresponding
sides are equal and the corresponding angles
are congruent,
Determine whether the pair of figures is similar.
Justify your answer.
b.
Answer: Only one pair of angles are congruent, so the
triangles are not similar.
An architect prepared a 12-inch model of a
skyscraper to look like a real 1100-foot building.
What is the scale factor of the model compared to
the real building?
Before finding the scale factor you must make sure that
both measurements use the same unit of measure.
1100(12) 13,200 inches
Answer: The ratio comparing the two heights is
The scale factor is
which means that the model is
of the real skyscraper.
,
the height
A space shuttle is about 122 feet in length. The
Science Club plans to make a model of the space
shuttle with a length of 24 inches. What is the scale
factor of the model compared to the real space
shuttle?
Answer:
The two polygons are similar. Write a similarity
statement. Then find x, y, and UV.
Use the congruent angles to write the corresponding
vertices in order.
Now write proportions to find x and y.
To find x:
Similarity proportion
Cross products
Multiply.
Divide each side by 4.
To find y:
Similarity proportion
Cross products
Multiply.
Subtract 6 from each side.
Divide each side by 6 and
simplify.
Answer:
The two polygons are similar. Find the scale factor
of polygon ABCDE to polygon RSTUV.
The scale factor is the ratio of the lengths of any two
corresponding sides.
Answer:
The two polygons are similar.
a. Write a similarity statement. Then find a, b, and ZO.
Answer:
;
b. Find the scale factor of polygon TRAP to polygon
Answer:
.
Rectangle WXYZ is similar to rectangle PQRS
with a scale factor of 1.5. If the length and width
of rectangle PQRS are 10 meters and 4 meters,
respectively, what are the length and width of
rectangle WXYZ?
Write proportions for finding side measures. Let one long
side of each WXYZ and PQRS be
and one
short side of each WXYZ and PQRS be
Answer:
Quadrilateral GCDE is similar to quadrilateral JKLM
with a scale factor of
If two of the sides of GCDE
measure 7 inches and 14 inches, what are the
lengths of the corresponding sides of JKLM?
Answer: 5 in., 10 in.
The scale on the map of a city is
inch equals 2
miles. On the map, the width of the city at its widest
point is
inches. The city hosts a bicycle race
across town at its widest point. Tashawna bikes at
10 miles per hour. How long will it take her to
complete the race?
Explore Every
equals 2 miles. The
distance across the city at its widest point is
Plan Create a proportion relating the measurements to
the scale to find the distance in miles. Then use
the formula
to find the time.
Solve
Cross products
Divide each side by 0.25.
The distance across the city is 30 miles.
Divide each side by 10.
It would take Tashawna 3 hours to bike
across town.
Examine To determine whether the answer is reasonable,
reexamine the scale. If 0.25 inches 2 miles,
then 4 inches 32 miles. The distance across the
city is approximately 32 miles. At 10 miles per
hour, the ride would take about 3 hours. The
answer is reasonable.
Answer: 3 hours
An historic train ride is planned between two
landmarks on the Lewis and Clark Trail. The scale
on a map that includes the two landmarks is 3
centimeters = 125 miles. The distance between the
two landmarks on the map is 1.5 centimeters. If the
train travels at an average rate of 50 miles per hour,
how long will the trip between the landmarks take?
Answer: 1.25 hours
Example 1 Determine Whether Triangles Are Similar
Example 2 Parts of Similar Triangles
Example 3 Find a Measurement
In the figure,
and
Determine which triangles in the figure
are similar.
by the Alternate Interior
Angles Theorem.
Vertical angles are congruent,
Answer: Therefore, by the AA Similarity Theorem,
In the figure, OW = 7, BW = 9, WT = 17.5, and WI = 22.5.
Determine which triangles in the figure are similar.
I
Answer:
ALGEBRA Given
QT 2x 10, UT 10, find RQ and QT.
Since
because they are alternate interior angles. By AA Similarity,
Using the definition of similar polygons,
Substitution
Cross products
Distributive Property
Subtract 8x and 30 from each
side.
Divide each side by 2.
Now find RQ and QT.
Answer:
ALGEBRA Given
and CE x + 2, find AC and CE.
Answer:
INDIRECT MEASUREMENT Josh wanted to measure
the height of the Sears Tower in Chicago. He used a
12-foot light pole and measured its shadow at 1 P.M. The
length of the shadow was 2 feet. Then he measured the
length of the Sears Tower’s shadow and it was 242 feet
at that time. What is the height of the Sears Tower?
Assuming that the sun’s rays form similar triangles, the
following proportion can be written.
Now substitute the known values and let x be the height
of the Sears Tower.
Substitution
Cross products
Simplify.
Divide each side by 2.
Answer: The Sears Tower is 1452 feet tall.
INDIRECT MEASUREMENT
On her trip along the East coast,
Jennie stops to look at the tallest
lighthouse in the U.S. located at
Cape Hatteras, North Carolina.
At that particular time of day,
Jennie measures her shadow
to be 1 feet 6 inches in length
and the length of the shadow
of the lighthouse to be 53 feet 6 inches. Jennie
knows that her height is 5 feet 6 inches. What is
the height of the Cape Hatteras lighthouse to the
nearest foot?
Answer: 196 ft
Example 1 Find the Length of a Side
Example 2 Determine Parallel Lines
Example 3 Midsegment of a Triangle
Example 4 Proportional Segments
Example 5 Congruent Segments
In
and
Find SU.
S
From the Triangle Proportionality Theorem,
Substitute the known measures.
Cross products
Multiply.
Divide each side by 8.
Simplify.
Answer:
In
and
B
Answer: 15.75
Find BY.
In
whether
and
Explain.
Determine
In order to show that
Since
we must show that
the sides have
proportional length.
Answer:
since the segments have proportional
lengths,
In
Determine whether
and AZ = 32.
Explain.
X
Answer: No; the segments are not in proportion since
Triangle ABC has vertices A(–2, 2), B(2, 4,) and C(4, –4).
is a midsegment of
Find the coordinates of
D and E.
(2, 4)
(-2, 2)
(4, –4)
Use the Midpoint Formula to find the midpoints of
Answer: D(0, 3), E(1, –1)
Triangle ABC has vertices A(–2, 2), B(2, 4) and C(4, –4).
is a midsegment of
Verify that
(2, 4)
(-2, 2)
(4, –4)
If the slopes of
slope of
slope of
Answer: Because the slopes of
Triangle ABC has vertices A(–2, 2), B(2, 4) and C(4, –4).
is a midsegment of
Verify that
(2, 4)
(-2, 2)
(4, –4)
First, use the Distance Formula to find BC and DE.
Answer:
Triangle UXY has vertices U(–3, 1), X(3, 3), and Y(5, –7).
is a midsegment of
a. Find the coordinates of W and Z.
Answer: W(0, 2), Z(1, –3)
b. Verify that
Answer: Since the slope of
and the slope of
c. Verify that
Answer:
Therefore,
In the figure, Larch, Maple, and Nuthatch Streets are all
parallel. The figure shows the distances in city blocks
that the streets are apart. Find x.
Notice that the streets form a triangle that is cut by parallel
lines. So you can use the Triangle Proportionality Theorem.
Triangle Proportionality Theorem
Cross products
Multiply.
Divide each side by 13.
Answer: 32
In the figure, Davis, Broad, and Main Streets are all
parallel. The figure shows the distances in city blocks
that the streets are apart. Find x.
Answer: 5
Find x and y.
To find x:
Given
Subtract 2x from each side.
Add 4 to each side.
To find y:
The segments with lengths
are congruent
since parallel lines that cut off congruent segments on one
transversal cut off congruent segments on every
transversal.
Equal lengths
Multiply each side by 3 to
eliminate the denominator.
Subtract 8y from each side.
Divide each side by 7.
Answer: x = 6; y = 3
Find a and b.
Answer: a = 11; b = 1.5
Example 1 Perimeters of Similar Triangles
Example 2 Write a Proof
Example 3 Medians of Similar Triangles
Example 4 Solve Problems with Similar Triangles
If
and
find the perimeter of
C
Let x represent the perimeter of
The perimeter of
Proportional
Perimeter Theorem
Substitution
Cross products
Multiply.
Divide each side
by 16.
Answer: The perimeter of
If
RX = 20, find the perimeter of
and
R
Answer:
and
length of an altitude of
Find the ratio of the
to the length of an
altitude of
are similar with a ratio of
According
to Theorem 6.8, if two triangles are similar, then the
measures of the corresponding altitudes are proportional
to the measures of the corresponding sides.
Answer: The ratio of the lengths of the altitudes is
and
length of a median of
of
Answer:
Find the ratio of the
to the length of a median
In the figure,
and is an altitude of
and
is an altitude of
Find x if
K
Write a proportion.
Cross products
Divide each side by 36.
Answer: Thus, JI = 28.
In the figure,
and
and
is an altitude of
is an altitude of
Find x if
N
Answer: 17.5
The drawing below illustrates two poles supported by
wires.
,
, and
Find
the height of the pole
.
are medians of
since
and
If two triangles are similar, then the measures of the
corresponding medians are proportional to the measures
of the corresponding sides. This leads to the
proportion
measures 40 ft. Also, since
both measure 20 ft.
Therefore,
Write a proportion.
Cross products
Simplify.
Divide each side by 80.
Answer: The height of the pole is 15 feet.
The drawing below illustrates the legs,
of a
table. The top of the legs are fastened so that AC
measures 12 inches while the bottom of the legs open
such that GE measures 36 inches. If BD measures 7
inches, what is the height h of the table?
Answer: 28 in.
Example 1 Self-Similarity
Example 2 Create a Fractal
Example 3 Evaluate a Recursive Formula
Example 4 Find a Recursive Formula
Example 5 Solve a Problem Using Iteration
below is found by connecting the midpoints of
the sides of
Prove that
Given: E, D, and F are midpoints of
respectively.
Prove:
C
Proof:
Statements
Reasons
1. E, D, and F are midpoints
of
respectively.
1. Given
2.
2. Triangle Midsegment
Theorem
3.
3. Alternate Interior Angles
Theorem
Proof:
Statements
Reasons
4.
4. Corresponding Angles
Postulate
5.
5. Transitive Property
6.
6. AA Similarity
is formed by connecting the midpoints of the
sides
and
of
Prove that
Given: Z and X are the midpoints of
respectively.
Prove:
w
Proof:
Statements
Reasons
1. Z and X are the midpoints
1. Given
of
respectively.
2.
2. Triangle Midsegment
Theorem
3.
3. Corresponding Angles
Postulate
4.
4. Reflexive Property
5.
5. AA Similarity
Draw an equilateral triangle. Create a fractal by drawing
another equilateral triangle within it and shading above
or beneath the triangle that shares the horizontal side
of the equilateral triangle. Stages 1 and 2 are shown.
Answer:
Draw a square. Create a fractal by bisecting the top
and left side of the square and drawing a smaller
square inside the larger square as shown.
Answer:
Find the value of
where x initially equals 1.
Then use that value as the next x in the expression.
Repeat the process three more times and describe
your observations.
The iterative process is to square the value of x, multiply
that value by 3, and then subtract 1. Begin with
The
value of
becomes the next value of x.
x
1
2
2
11
362
11 362 393,131
Answer: 2, 11, 362, 393,131; x values increase with each
iteration, approaching infinity.
Find the value of
where x initially equals 0.
Then use that value as the next x in the expression.
Repeat the process three more times and describe
your observations.
Answer: 3, –15, –447, –399,615; x values decrease with
each iteration, approaching negative infinity.
The diagram below represents the odd integers 1, 3, 5,
and 7. Find a formula in terms of the row number for
the sum of the values in the diagram.
Notice that each sum is the row number squared.
Answer:
What is the sum of the first 10 odd numbers?
The sum of the values in the tenth row will be 102 or 100.
Answer: 100
Examine the pattern shown below.
Row
Sum
1

2
 
3
  
4
   
a. Find a formula for the sum of the values in the diagram.
Answer: Sn = Sn–1
n
b. Find the number in row 7.
Answer: 28
BANKING Joaquin has $1500 in a savings account
that earns 4.1% interest. If the interest is compounded
annually, find the balance of his account after 4 years.
First, write an equation to find the balance after one year.
Answer: After 4 years, Joaquin will have $1761.55 in his
account.
BANKING Anna has $3500 in a savings account that
earns 3.8% interest. If the interest is compounded
annually, find the balance of her account after 5 years.
Answer: $4217.50
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