Post on 01-Jan-2016
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FeatureLesson
Course 3Course 3
LessonMain
How many miles are in 21,120 yd? (Hint: 1 mi = 5,280 ft)
12 mi
LESSON 4-1LESSON 4-1
Ratios and RatesRatios and Rates
Problem of the Day
4-1
FeatureLesson
Course 3Course 3
LessonMain
LESSON 4-1LESSON 4-1
Ratios and RatesRatios and Rates
(For help, go to Lesson 2-3.)
1. Vocabulary Review What is the least common denominator of two rational numbers?
Determine which rational number is greater.
2. , 3. , 4. , 5. ,39
16
1525
45
4554
23
47
712
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4-1
FeatureLesson
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Ratios and RatesRatios and RatesLESSON 4-1LESSON 4-1
Solutions
1. The least common denominator is the smallest multiple the denominators have in common.
2. 3. 4. 5. 39
45
4554
712
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4-1
FeatureLesson
Course 3Course 3
LessonMain
Write the ratio 36 seconds to 12 minutes in
simplest form.
Ratios and RatesRatios and RatesLESSON 4-1LESSON 4-1
36 s12 min
36 s720 s=
Convert minutes to seconds so that both measures are in the same units. Divide the common units.
36720
= 36 ÷ 36720 ÷ 36
Divide the numerator and denominator by the GCF, 36.
=1
20 Simplify.
The ratio of 36 seconds: 12 minutes is .1
20 Quick Check
Additional Examples
4-1
FeatureLesson
Course 3Course 3
LessonMain
Computer time costs $4.50 for 30 min. What is the
unit rate?
Ratios and RatesRatios and RatesLESSON 4-1LESSON 4-1
costnumber of minutes
$4.5030 min= Write a rate comparing cost to minutes.
= $.15/min Divide.
The unit rate is $.15 per minute.
Quick Check
Additional Examples
4-1
FeatureLesson
Course 3Course 3
LessonMain
Keneesha drove her car 267 mi using 11 gal of gas. Vanessa
drove her car 210 mi using 9 gal. Give the unit rate for each. Which car
got more miles per gallon of gas?
Ratios and RatesRatios and RatesLESSON 4-1LESSON 4-1
Keneesha’s car got more miles per gallon.
milesgallons
267 mi11 gal=
Write the rates comparing
miles to gallons.
milesgallons
210 mi9 gal=
Keneesha Vanessa
Divide. 24.27272727 mi/gal 23.33333333 mi/gal
24.3 mi/gal 23.3 mi/galRound to the nearest tenth.
Additional Examples
4-1
FeatureLesson
Course 3Course 3
LessonMain
(continued)
Ratios and RatesRatios and RatesLESSON 4-1LESSON 4-1
Check for Reasonableness 24.3 • 11 = 267.3 and 267.3 267. Also, 23.3 • 9 = 209.7 and 209.7 210. The answers are reasonable.
Quick Check
Additional Examples
4-1
FeatureLesson
Course 3Course 3
LessonMain
Ratios and RatesRatios and Rates
Express each ratio in simplest form.
1. 27 laps : 81 minutes
2. 12 minutes : 3 hours
3. Carli walked 16 miles in 5 hours. Find the unit rate.
4. A 21-oz bottle of shampoo costs $2.80. A 12-oz bottle costs $1.35. Which has the better unit rate?
LESSON 4-1LESSON 4-1
13
115
3.2 mi/h
12-oz bottle
Lesson Quiz
4-1
FeatureLesson
Course 3Course 3
LessonMain
Write a fraction in lowest terms, with a single digit numerator, that is about the same as each decimal: 0.52, 0.13, 0.74, 0.88.
LESSON 4-2LESSON 4-2
12 , , ,
18
34
78
Converting UnitsConverting Units
Problem of the Day
4-2
FeatureLesson
Course 3Course 3
LessonMain
LESSON 4-2LESSON 4-2
Converting UnitsConverting Units
(For help, go to Lesson 2-5.)
1. Vocabulary Review What is the product of a number and its reciprocal?
Find each product. Write the answer in simplest form.
2. 3.
4. 5.
103
14
46 •
56
•
49
32•
67
83•
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4-2
FeatureLesson
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LESSON 4-2LESSON 4-2
Solutions
1. 1 2.
3. 4.
5.
10 • 13 • 4 = =
56
10 • 13 • 4
5
2
4 • 56 • 6 =
4 • 56 • 6 =
1018
59=
2
3= =
4 • 39 • 2
4 • 39 • 2
23
2
1
1
3
26 • 87 • 3 =
6 • 87 • 3 =
1671
= 227
Converting UnitsConverting Units
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4-2
FeatureLesson
Course 3Course 3
LessonMain
Convert 0.7 mi to ft.
LESSON 4-2LESSON 4-2
There are 3,696 feet in 0.7 miles.
= (0.7)(5,280) ft
1 Simplify.
= 3,696 ft Divide.
Converting UnitsConverting Units
Since 5,280 ft = 1 mi, use the conversion factor 5,280 ft.1 mi
0.7 = •0.7 mi
15,280 ft
1 miMultiply by a conversion factor .5,280 ft
1 mi
Quick Check
Additional Examples
4-2
FeatureLesson
Course 3Course 3
LessonMain
A rowing team completed a 2000-m course at a
rate of 6.84 m/s. Convert this rate to kilometers per minute.
LESSON 4-2LESSON 4-2
The team rowed at a rate of 0.4104 km/min.
Estimate 6.84 7. Then, 7 • 60 ÷ 1000 = 0.42.
= • •6.84 m
1 s1 km
1000 m
Multiply by two ratios thateach equal one.
60 s1 min
6.84 m1 s
= Simplify.
= 0.4104 Use a calculator.
(6.48)(1)(60) km(1)(1,000)(1) min
Check for Reasonableness The answer 0.4104 km/min is close to the estimate 0.42. The answer is reasonable.
Divide by the common units.
Converting UnitsConverting Units
Quick Check
Additional Examples
4-2
FeatureLesson
Course 3Course 3
LessonMain
Use compatible numbers to estimate the number
of gallons in 33 quarts.
LESSON 4-2LESSON 4-2
There are about 8 gallons in 33 quarts.
The conversion factor for changing gallons to quarts is .
= •32 qt
1 1 gal4 qt
Multiply by the conversion factor.Divide by the common units.
= 8 gallons Divide.
= gallons Simplify.324
1 gal4 qt
33 qt 32 qtRound to the nearest numberdivisible by 4.
Converting UnitsConverting Units
Quick Check
Additional Examples
4-2
FeatureLesson
Course 3Course 3
LessonMain
Convert 650 g to ounces.
LESSON 4-2LESSON 4-2
Converting UnitsConverting Units
Quick Check
There are about 22.9 oz in 650 g.
650 g =650 g
1 •1 oz
28.4 gMultiply by the conversion factor .
1 oz28.4 g
(650)(1) oz
28.4 22.9 oz
a calculator.Simplify. Divide using=
Additional Examples
4-2
FeatureLesson
Course 3Course 3
LessonMain
1. Convert 0.75 hours to seconds.
2. $150 per hour is how much per minute?
3. 69.2 cm is about how many meters?
4. Convert 12 qt to liters.
LESSON 4-2LESSON 4-2
2,700 seconds
$2.50 per min
0.7 m
Converting UnitsConverting Units
about 11.3L
Lesson Quiz
4-2
FeatureLesson
Course 3Course 3
LessonMain
Write each word phrase as an algebraic expression.
a. 12 times a number
b. 8 less than a number
c. twice the sum of 5 and a number
LESSON 4-3LESSON 4-3
12n
n – 8
2(5 + n)
Solving ProportionsSolving Proportions
Problem of the Day
4-3
FeatureLesson
Course 3Course 3
LessonMain
LESSON 4-3LESSON 4-3
Solving ProportionsSolving Proportions
(For help, go to Lesson 2-2.)
1. Vocabulary Review Is the fraction in simplest form? Explain.
a + 2b + 2
Write each fraction in simplest form.
2. 3. 4. 5.3099
4212
132602
7025
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4-3
FeatureLesson
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Solving ProportionsSolving ProportionsLESSON 4-3LESSON 4-3
2. 3.
4. 5.
3099
3 • 103 • 33= =
1033
4212
6 • 76 • 2= =
72
1
1= 3
1
1
12
132602
2 • 662 • 301= =
1
1
66301
7025
5 • 145 • 5= =
145 = 2
1
1
45
Solutions
1. Yes; there is no common factor between the numerator and denominator.
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4-3
FeatureLesson
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Solving ProportionsSolving ProportionsLESSON 4-3LESSON 4-3
Do and form a proportion? Explain. 49
818
49
818
gallons Write as a proportion.
gallons Use number sense to find acommon multiplier.
Quick Check
49
818
Since = ,they form a proportion.
Additional Examples
4-3
FeatureLesson
Course 3Course 3
LessonMain
The fixed rate of conversion is 1 euro = 0.7876 Irish pounds.
How many euros would you receive for 125 Irish pounds?
Solving ProportionsSolving ProportionsLESSON 4-3LESSON 4-3
You would receive 158.71 euros.
Let p = the number of euros.
=0.7876
1125
pWrite theproportion .
Irish poundseuros
0.7876 • p = 1 • 125 Write the cross products.
=0.7876 • p
0.7876125
0.7876Divide each side by 0.7876.
Quick Check
125 0.7876 Use a calculator.
Additional Examples
4-3
FeatureLesson
Course 3Course 3
LessonMain
Solve each proportion.
2. =
3. =
4. Suppose the exchange rate for dollars to Indian rupees is 0.02. How many rupees should you receive for $100?
w12
34
20r
45
Solving ProportionsSolving ProportionsLESSON 4-3LESSON 4-3
9
25
5,000 rupees
1. Is proportional to ? Explain.58
1024
No; the fractions are not equal.
Lesson Quiz
4-3
FeatureLesson
Course 3Course 3
LessonMain
A football team scored 38 points in a game. They scored 3 points for a field goal and 7 points for each touchdown with an extra point. How many field goals did they make? How many touchdowns?
1 field goal and 5 touchdowns or 2 touchdowns and 8 field goals
LESSON 4-4LESSON 4-4
Similar Figures and ProportionsSimilar Figures and Proportions
Problem of the Day
4-4
FeatureLesson
Course 3Course 3
LessonMain
LESSON 4-4LESSON 4-4
Similar Figures and ProportionsSimilar Figures and Proportions
(For help, go to Lesson 4-3.)
1. Vocabulary Review What are the cross products for 1015
23
Solve each proportion.
2. = 3. = 4. = 713
21 t
k50
2210
1625
324 m
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= ?
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4-4
FeatureLesson
Course 3Course 3
LessonMain
Similar Figures and ProportionsSimilar Figures and ProportionsLESSON 4-4LESSON 4-4
Solutions
7t = 273
=
t = 39
7t7
2737
14
10k = 1,100
=
k = 110
10k10
1,10010
4. 16m = 8,100; m = 506
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4-4
FeatureLesson
Course 3Course 3
LessonMain
Is rectangle ABCD similar to rectangle RSTU? Explain why or
why not.
Similar Figures and ProportionsSimilar Figures and ProportionsLESSON 4-4LESSON 4-4
Additional Examples
4-4
First, check to see if corresponding angles are congruent. A R B S All right angles are 90°.
C T D U
FeatureLesson
Course 3Course 3
LessonMain
(continued)
Similar Figures and ProportionsSimilar Figures and ProportionsLESSON 4-4LESSON 4-4
Next, check to see if corresponding sides are in proportion.
Quick Check
Additional Examples
4-4
The corresponding sides are in proportion, so rectangle ABCD is similar to rectangle RSTU.
ABRS
DAUR
AB corresponds to RS. DA corresponds to UR.
648
324
Substitute.
6 • 24 48 • 3 Write the cross products.
144 = 144 Simplify.
FeatureLesson
Course 3Course 3
LessonMain
A stonemason’s sketch of a carving to be madeon a building includes the letter “E” shown below. If thewidth of the actual letter in the arrangement is 22 in.,what is the height?
Similar Figures and ProportionsSimilar Figures and ProportionsLESSON 4-4LESSON 4-4
2.75 • x = 5 • 22 Write the cross products.
Simplify.2.75 x = 110
Divide each side by 2.75.2.75x2.75
110 2.75
=
x = 40 Simplify.
Set up a proportion.2.75 in.5 in.
22 in.x=
Quick CheckThe height of the letter is 40 inches.
Additional Examples
4-4
FeatureLesson
Course 3Course 3
LessonMain
RST ~ PSU. Find the value of d.
Similar Figures and ProportionsSimilar Figures and ProportionsLESSON 4-4LESSON 4-4
12 • d = 21 •14 Write the cross products.
Simplify.12d = 294
Divide each side by 12.12d12
29412
=
d = 24.5 Simplify.
Write a proportion.14d
1221
=
Quick Check
The value of d is 24.5.
Additional Examples
4-4
FeatureLesson
Course 3Course 3
LessonMain
Similar Figures and ProportionsSimilar Figures and Proportions
1. Are the triangles similar? Explain.
2. A model of a building is 18 in. tall and 24 in. wide. The building is 30 ft tall. How wide is the building?
LESSON 4-4LESSON 4-4
No; their sides are not proportional.
40 ft
Lesson Quiz
4-4
FeatureLesson
Course 3Course 3
LessonMain
Similar Figures and ProportionsSimilar Figures and ProportionsLESSON 4-4LESSON 4-4
3. In the figure at the right, MNO ~ LNP. Find the value of a.
18
4. If all the lengths in Exercise 3 are doubled, are the triangles still similar? Explain why or why not.
Yes; corresponding values are multiplied by the same factor.
Lesson Quiz
4-4
FeatureLesson
Course 3Course 3
LessonMain
There are three different 1-digit numbers greater than zero and all odd. Their sum is 15. What are the numbers?
3, 5, 7 or 1, 5, 9
LESSON 4-5LESSON 4-5
Similarity TransformationsSimilarity Transformations
Problem of the Day
4-5
FeatureLesson
Course 3Course 3
LessonMain
LESSON 4-5LESSON 4-5
Similarity TransformationsSimilarity Transformations
(For help, go to Lesson 3-4.)
1. Vocabulary Review The first coordinate in an ordered pair is the ? -coordinate.
Graph each point on a coordinate plane.
2. A(3, 6) 3. B(–2, 7)
4. C(5, –1) 5. D(–3, 0)
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4-5
FeatureLesson
Course 3Course 3
LessonMain
Similarity TransformationsSimilarity TransformationsLESSON 4-5LESSON 4-5
Solutions
1. 2-5.x
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4-5
FeatureLesson
Course 3Course 3
LessonMain
Find the image of ABC after a dilation with center A and a
scale factor of 3.
Similarity TransformationsSimilarity TransformationsLESSON 4-5LESSON 4-5
A B is 3 times AB.
A C is 3 times AC.
Since A is the center of dilation
A = A .
A = A
Quick Check
A B C is the image of ABC after a dilation with a scale factor of 3. ABC ~ A B C
Additional Examples
4-5
FeatureLesson
Course 3Course 3
LessonMain
Find the coordinates of the image of quadrilateral KLMN after
a dilation with a scale factor of . Quadrilateral KLMN has vertices
K (–2, –1), L (0, 2), M (4, 2), and N (4, –1).
Similarity TransformationsSimilarity TransformationsLESSON 4-5LESSON 4-5
12
Step 1 Multiply the x- and
y-coordinates of each point by . 12
Step 2 Graph the image.
K (–2, –1) K (–1, – )
L (0, 2) L (0, 1)
M (4, 2) M (2, 1)
N (4, –1) N (2, – )
12
12
Quick Check
Additional Examples
4-5
FeatureLesson
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LessonMain
Similarity TransformationsSimilarity TransformationsLESSON 4-5LESSON 4-5
The figure below PQR shows the outline of aplaying field. A city planner dilates the design to showthe area available for community youth to play sports.Find the scale factor. Is it an enlargement or a reduction?
Quick Check
The scale factor is 1.5.
The dilation is an enlargement.
= = = 1.5imageoriginal
P Q PQ
64
32
Additional Examples
4-5
FeatureLesson
Course 3Course 3
LessonMain
ABC has coordinates A(0, 0), B(10, 0), and C(5, 5). Find the coordinates of the image of ABC after a dilation with each scale factor.
1.
2. 4
3.
Similarity TransformationsSimilarity TransformationsLESSON 4-5LESSON 4-5
15
A (0, 0), B (2, 0), C (1, 1)
A (0, 0), B (40, 0), C (20, 20)
Figure ABCD shows the outline of a porch. The figureA′B′C′D′ is the outline of a table formed by dilatingABCD. Find the scale factor. Is it an enlargementor a reduction?
, reduction13
Lesson Quiz
4-5
FeatureLesson
Course 3Course 3
LessonMain
Mirror primes are pairs of prime numbers in which the digits are reversed, such as 13 and 31. Find all the mirror primes less than 100.
13 and 31, 17 and 71, 37 and 73, and 79 and 97; 11 is its own mirror image.
LESSON 4-6LESSON 4-6
Scale Models and MapsScale Models and Maps
Problem of the Day
4-6
FeatureLesson
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LessonMain
LESSON 4-6LESSON 4-6
Scale Models and MapsScale Models and Maps
(For help, go to the Skills Handbook page 632.)
1. Vocabulary Review A product is the result of which operation?
Multiply.
2. 4 3.2 3. 7.6 5.9
4. 1.8 22 5. 13 6.5
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4-6
FeatureLesson
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Scale Models and MapsScale Models and MapsLESSON 4-6LESSON 4-6
Solutions
1. multiplication 2. 12.8
3. 44.84 4. 39.6
5. 84.5
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4-6
FeatureLesson
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On a blueprint, the cellar is 4 in. by 3 in. The scale is
in. = 8 ft. What are the length and width of the actual cellar?
Scale Models and MapsScale Models and MapsLESSON 4-6LESSON 4-6
12
First, find the actual length of the cellar.
Let = the actual length of the cellar.
blueprint measure (in.)actual measure (ft)
128
=4 blueprint length (in.)
actual length (ft)
Additional Examples
4-6
FeatureLesson
Course 3Course 3
LessonMain
(continued)
Scale Models and MapsScale Models and MapsLESSON 4-6LESSON 4-6
12
= 32 Simplify.
= 64 Simplify.
12
12
=12
32 Divide each side
by .12
12
• = 8 • 4 Write the cross products.
Additional Examples
4-6
FeatureLesson
Course 3Course 3
LessonMain
(continued)
Scale Models and MapsScale Models and MapsLESSON 4-6LESSON 4-6
The length of the actual room is 64 ft.
Next, find the actual width of the cellar.
Let w = the actual width of the cellar.
blueprint measure (in.)actual measure (ft)
128
=3 blueprint length (in.)
actual length (ft)w
Additional Examples
4-6
FeatureLesson
Course 3Course 3
LessonMain
(continued)
Scale Models and MapsScale Models and MapsLESSON 4-6LESSON 4-6
w = 48 Simplify.
12
w = 24 Simplify.
12
• w = 8 • 3 Write the cross products.
12
12
=12
24 Divide each side
by .12
w
The width of the actual room is 48 ft. Quick Check
Additional Examples
4-6
FeatureLesson
Course 3Course 3
LessonMain
The map distance from El Paso, Texas, to Chihuahua, Mexico, measures about 7.5 cm. The scale is 1 cm = 50 km. What is the actual distance?
Scale Models and MapsScale Models and MapsLESSON 4-6LESSON 4-6
1 • d = 50 • 7.5 Write the cross products.
d = 375 Simplify.
The actual distance from El Paso, Texas to Chihuahua, Mexicois 375 kilometers.
Let d be the actual distance from El Paso, Texas to Chihuahua, Mexico.
map (cm)actual (km)
150
=7.5 map (cm)
actual (km)dSet up a proportion.
Quick Check
Additional Examples
4-6
FeatureLesson
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1. A 6-ft man is designing a new chair that would make him feel like a 2.5-ft child. The seat of a normal chair is 1.5 ft high. How high should he make the seat in his new chair?
2. A map scale shows 4 cm to represent 6 km. Two intersections measure 1 cm apart on the map. What is the actual distance?
Scale Models and MapsScale Models and MapsLESSON 4-6LESSON 4-6
3.6 ft
1.5 km
Lesson Quiz
4-6
FeatureLesson
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Scale Models and MapsScale Models and MapsLESSON 4-6LESSON 4-6
3. A tennis court is 36 ft wide. A drawing of the court is 2 in. long and 1 in. wide. Find the scale used.1
4
4. Find the actual length of the court.
1 in. = 36 ft
81 ft
Lesson Quiz
4-6
For Exercises 3–4, use the diagram.
FeatureLesson
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LessonMain
A rectangular field is 120 yd long and 53 yd 1 ft wide. How much longer is the field than it is wide?
66 yd 2 ft
LESSON 4-7LESSON 4-7
Similarity and Indirect MeasurementSimilarity and Indirect Measurement
Problem of the Day
4-7
FeatureLesson
Course 3Course 3
LessonMain
LESSON 4-7LESSON 4-7
Similarity and Indirect MeasurementSimilarity and Indirect Measurement
(For help, go to Lesson 4-4.)
1. Vocabulary Review Similar figures have the same ? but not necessarily the same size.
2. If ABC ~ XYZ, which angle is congruent to B?
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Check Skills You’ll Need
4-7
FeatureLesson
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Solutions
1. shape
2. Y
Similarity and Indirect MeasurementSimilarity and Indirect MeasurementLESSON 4-7LESSON 4-7
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4-7
FeatureLesson
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When a 6-ft student casts a 17-ft shadow, a flagpole casts a shadow that is 51 ft long. Find the height of the flagpole.
Similarity and Indirect MeasurementSimilarity and Indirect MeasurementLESSON 4-7LESSON 4-7
Set up a proportion for the similar triangles.
17h = 6 • 51 Write the cross products.
h = 18 Simplify.
Divide each side by 17.17h17
6 • 5117=
The height of the flagpole is 18 ft.
Words
Let h = the flagpole’s height.
Proportion
flagpole’s heightstudent’s height
length of flagpole’s shadowlength of student’s shadow=
h6
5117=
Quick Check
Additional Examples
4-7
FeatureLesson
Course 3Course 3
LessonMain
In the figure below, ABC ~ EDC. Find d.
Similarity and Indirect MeasurementSimilarity and Indirect MeasurementLESSON 4-7LESSON 4-7
=
Use similar triangles to set up a proportion involving the lengths of corresponding sides.
EDAB
CDCB
ED corresponds to AB. CD corresponds to CB.
=d
416141312
Substitute.
312 • d = 416 • 141 Write the cross products.
Additional Examples
4-7
FeatureLesson
Course 3Course 3
LessonMain
(continued)
Similarity and Indirect MeasurementSimilarity and Indirect MeasurementLESSON 4-7LESSON 4-7
The length d is 188 m.
312d = 58,656 Simplify.
312d 58,656 312 312= Divide each side by 312.
58,656 312 188 Use a calculator.
Quick Check
Additional Examples
4-7
FeatureLesson
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1. A 5-ft tall student casts a 12-ft shadow. A tree casts a 27-ft shadow. How tall is the tree?
2. A 6-ft man casts a 9-ft shadow. A sculpture casts a 45-ft shadow. How tall is the sculpture?
Similarity and Indirect MeasurementSimilarity and Indirect MeasurementLESSON 4-7LESSON 4-7
11.25 ft tall
30 ft
Lesson Quiz
4-7
FeatureLesson
Course 3Course 3
LessonMain
Similarity and Indirect MeasurementSimilarity and Indirect MeasurementLESSON 4-7LESSON 4-7
3. The diagram shows an outline of a village green EFG next to a small park JHG. The length of JH is 47.4 m, FG is 31 m, and HG is 15.8 m. Find the length of EF.
93 m
Use the diagram for Exercise 3. EFG ~ JHG
Lesson Quiz
4-7