d. n. a
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D. N. A.1) Find the ratio of BC to DG.
7
56 a)
x
0 10 20 30
A B C D E F G
2) Solve each proportion.
3
4
5
32 b)
x
x
Lesson 2 MI/Vocab
• similar polygons
• scale factor
• Identify similar figures.
• Solve problems involving scale factors.
Standard 11.0 Students determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids.
Similar Polygons• Have congruent corresponding angles.
• Have proportional corresponding sides.
• “~” means “is similar to”
A
BD
C
F
G
H
E
HE
DA
GH
CD
FG
BC
EF
AB
ABCD ~ EFGH
A E
B F
C G
D H
Writing Similarity Statements• Decide if the polygons are similar. If they
are, write a similarity statement.A B
C
D
6
12
915
W
X
Z
Y
10
8
6
4
2
3
4
6
WY
AB
2
3
6
9
YZ
BC
2
3
8
12
ZX
CD
2
3
10
15
XW
DA
A W
B Y
C Z
D X All corr. sides are
proportionate and all corr. angles are
ABCD ~ WYZX
Scale Factor• The ratio of the lengths of two corresponding
sides.
• In the previous example the scale factor is 3:2.
Lesson 2 Ex1
Similar Polygons
A. Determine whether each pair of figures is similar. Justify your answer.
The vertex angles are marked as 40º and 50º, so they are not congruent.
Similar Polygons
Answer: None of the corresponding angles are congruent, so the triangles are not similar.
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
Similar Polygons
B. Determine whether each pair of figures is similar. Justify your answer.
All the corresponding angles are congruent.
Similar Polygons
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 ΔABC ~ ΔRST.
A. Yes, ΔAXE ~ ΔWRT.
B. Yes, ΔAXE ~ ΔRWT.
C. No, the Δ's are not ~.
D. not enough information
A. Determine whether the pair of figures is similar.
A. Yes, ΔTRS ~ ΔNGA.
B. Yes, ΔTRS ~ ΔGNA.
C. No, the Δ's are not ~.
D. not enough information
B. Determine whether the pair of figures is similar.
ARCHITECTURE 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.
1 foot = 12 inches
Answer: The ratio comparing the two heights is
or 1:1100. The scale factor is , which
means that the model is the height of the
real skyscraper.
Animation:Similar Polygons
Each pair of polygons is similar. Find x and y.
1)
x
8
10
y9
12 2)5.4
10
y
5.2
5
x3)
x 36
40 18 y
30
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?
A.
B.
C.
D.
Proportional Parts and Scale Factor
A. 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.
polygon ABCDE ~ polygon RSTUV
Proportional Parts and Scale Factor
Now write proportions to find x and y.
To find x:
Similarity proportion
Cross products
Multiply.
Divide each side by 4.
Proportional Parts and Scale Factor
To find y:
Similarity proportion
Cross products
Multiply.
Subtract 6 from each side.
Divide each side by 6 and simplify.
AB = 6, RS = 4, DE = 8, UV = y + 1
Proportional Parts and Scale Factor
B. 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:
A. TRAP ~ OZDL
B. TRAP ~ OLDZ
C. TRAP ~ ZDLO
D. TRAP ~ ZOLD
A. The two polygons are similar. Write a similarity statement.
A. a = 1.4
B. a = 3.75
C. a = 2.4
D. a = 2
B. The two polygons are similar. Solve for a.
C. The two polygons are similar. Solve for b.
A. b = 7.2
B. b = 1.2
C.
D. b = 7.2
D. The two polygons are similar. Solve for ZO.
A. 7.2
B. 1.2
C. 2.4
D.
1. A
2. B
3. C
4. D
E. The two polygons are similar. What is the scale factor of polygon TRAP to polygon ZOLD?
A.
B.
C.
D.
Rectangle WXYZ is similar to rectangle PQRS with a scale factor of 1.5. If the length and width of PQRS are 10 meters and 4 meters, respectively, what are the length and width of rectangle WXYZ?
Enlargement or Reduction of a Figure
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
A. 9.8 in, 19.6 in
B. 7 in, 14 in
C. 6 in, 12 in
D. 5 in, 10 in
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?
Scales on Maps
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
Scales on Maps
Solve
Cross products
The distance across the city is 30 miles.
Plan Create a proportion relating the measurements to the scale to find the distance in miles. Then use the formula to find the time.
Divide each side by 0.25.
Answer: 3 hours
Scales on Maps
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.
A. 3.75 hr
B. 1.25 hr
C. 5 hr
D. 2.5 hr
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?
Forced Perspective
Using Ratios Example #1• The Perimeter of a rectangle is 60 cm. The ratio of
AB:BC is 3:2. Find the length and width of the rectangle. A
D C
B
3:2 is in lowest terms.
AB:BC could be 3:2, 6:4, 9:6, 12:8,
etc.
AB = 3x
BC = 2x
Perimeter = l + w+ l + w
60 = 3x + 2x + 3x + 2x
60 = 10x
x = 6
L = 3(6) = 18
W = 2(6) = 12
Find the measures of the sides of each triangle.
12. The ratio of the measures of the sides of a triangle is 3:5:7, and its perimeter is 450 centimeters.
13. The ratio of the measures of the sides of a triangle is 5:6:9, and its perimeter is 220 meters.
14. The ratio of the measures of the sides of a triangle is 4:6:8, and its perimeter is 126 feet.
Find the measures of the angles in each triangle.
15) The ratio of the measures of the angles is 4:5:6.
mA+ mB+ mC = 180o Triangle Sum Thm.
2x + 3x + 4x = 180o
9x = 180o
x = 20o
mA = 40o
mB = 60o
mC = 80o
• The angle measures in ABC are in the extended ratio of 2:3:4. Find the measure of the three angles.
Using Ratios Example #2
A
C
B
2x3x
4x
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