Download Packet 1 for Unit 5 M2G

M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY
ASSIGNMENTS FOR PACKET 1 OF UNIT 5
This packet includes sections 7-1, 7-2, and 7-3 from our textbook.
Date Due
Number
Assignment
5A
p. 464 # 1 – 4 all
p. 465 # 34, 37, 40
5B
p. 465 # 23 – 26 all, 29 – 31 all
5C
p. 473-474
# 9, 11, 12, 14, 19, 21, 23, 28
5D
p. 483-484
# 1 – 4 all, 7, 8, 16, 17, 18, 22
HINT: In #2 & #18, use the
Pythagorean Theorem a2 + b2 = c2
to find the 3rd side of a right
triangle if you know the other two.
1
Topics
7-1:
Vocabulary: ratio, proportion, cross
products
Use ratios to solve problems
Use ratios and algebra to solve
problems.
7-1:
Vocabulary: proportion, cross products
Use algebra to solve proportions
involving linear or quadratic equations.
Use proportions to solve real world
problems.
7-2:
Vocabulary: similar polygons, scale
factor
Name and identify corresponding parts
of similar polygons
Use proportions to solve problems
involving similar polygons
7-3:
Use SSS, SAS, or AA Similarity to
determine whether two triangles are
similar
Use proportions to solve problems
involving similar triangles
M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY
7-1 Ratios and Proportions
Part 1: Ratios
A ratio compares two quantities using division. The ratio a to b can be written as
a
or a:b.
b
Ex. 1: In 2015 the Mets won 90 games out of 162 games played. Write a ratio for the number of
games won to the total number of games played. Give your answer as a reduced fraction
and as a percent rounded to the nearest tenth.
Solution:
To find the ratio, divide number of games won by number of games played. The result is
Using Math, Frac on a calculator to reduce, we get
90
.
162
5
.
9
As a decimal, 5/9 is 0.556, which means 55.6%.
Try these:
1. In 2016, Kris Bryant hit 39 home runs and was at bat 603 times. What is the ratio of home runs to the
number of times he was at bat? Give your answer as a reduced fraction and as a percent
rounded to the nearest tenth.
2. Suppose there are 182 girls in a sophomore class of 306 students. What is the ratio of girls to total
students? Give your answer as a reduced fraction and as a percent rounded to the nearest tenth.
2
M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY
Part 2: Extended Ratios
Ex. 2: The ratio of the measures of the angles in
JKH is 2:3:4. Find the measures of the angles.
Solution: This is called an extended ratio because it has more than two
parts. 2:3:4 can be rewritten as 2x:3x:4x. Sketch and label the angle
measures of the triangle as shown at right. Then write and solve an
equation to find the value of x.
2x + 3x + 4x = 180
9x = 180
x = 20
The angle measures are 2(20) = 40 , 3(20) = 60 , and 4(20) = 80 .
Try these:
3. The ratio of the sides of a triangle is 8:15:17. Its perimeter is 480 inches. Find the length of each
side.
4. The ratio of the measures of angles in a triangle is 7:9:20. Find the measure of each angle.
3
M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY
Part 3: Solving Basic Proportions
An equation stating that two ratios are equal is called a proportion.
In the proportion
a c
 , the cross products are equal. That is, ad  bc .
b d
9 27
.

16 x
Solution: 9 x  16  27
9 x  432
x  48
Ex. 3: Solve
Try these: Solve each proportion. Show all steps.
5.
9
3

y 18.2
x3
8
.

3
x2
Solution:  x  3 x  2   3  8
6.
x  22 30

x  2 10
8.
4
x5

x 3
5
Ex. 4: Solve
x2  2 x  3x  6  24
x2  x  6  24
x2  x  30  0
 x  6 x  5  0
x  6  0 or x  5  0
x  6 or x  5
Cross products
FOIL
Simplify
Get 0 on one side.
Factor
Set factors = to 0.
Solve each for x.
Try these: Solve each proportion. Show all steps.
7.
x5
6

5
x2
4
M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY
Part 4: Using Proportions to Solve Word Problems
Ex. 5: The mayor conducted a random survey of 200 voters and found that 135 approve of the
job he is doing. If there are 48,000 voters in this district, predict the total number of voters who
approve of the job he is doing.
Solution:
Surveyed voters who approve
All surveyed voters
135
x

200 48, 000
District voters who approve
All district voters
135  48,000  200x
6, 480,000  200x
32, 400  x
Based on the survey, about 32,400 registered voters approve of the job the mayor is doing.
Try these:
9. Mark is measuring plants in a field. Of the first 25 plants he measures, 15 are smaller than a foot in
height. If there are 4000 plants in the field, predict the total number of plants smaller than a foot in
height.
10. A storm produced a rainfall of 2 inches in one hour. At this rate, how many hours would it take to
get a rainfall amount of one foot?
5
M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY
7-2 Similar Polygons
Polygons are similar if:
(1) Corresponding angles are congruent, and
(2) Corresponding sides are in the same proportion.
The statement
XYZ means that triangle ABC is similar to triangle XYZ.
ABC
Ex. 1: If ABC
XYZ , list all pairs of congruent angles, and
write a proportion that relates the corresponding sides.
Solution: Use the order of the letters in the triangle names:
Congruent angles: A  X , B  Y , C  Z
Proportion:
AB BC CA


XY YZ ZX
1. Try this: List all pairs of congruent angles, and write a
proportion that relates the corresponding sides.
PQRS
TUWX
Congruent angles:  ____   ____ ,  ____   ____ ,  ____   ____ ,  ____   ____
Proportion:



Conversely, if two polygons have congruent corresponding angles and proportional corresponding
sides, then they are similar.
Ex. 2: Determine whether the figures are similar. If so, write a
similarity statement, and find the scale factor.
Solution: Compare corresponding angles.
W  P, X  Q, Y  R, Z  S
Corresponding angles are congruent.
Compare corresponding sides by writing ratios for each pair.
WX 12 3
 
PQ 8 2
XY 18 3
 
QR 12 2
YZ 15 3
 
RS 10 2
ZW 9 3
 
SP 6 2
Corresponding sides are proportional because all of the ratios are equal.
The scale factor is
3
, and WXYZ
2
PQRS .
6
M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY
Try these: Determine whether the figures are similar. If so, write a similarity statement, and find the
scale factor. If not, explain your reasoning.
2.
3.
Ex. 3: The two polygons are similar. Find x and y.
Solution:
Use the congruent angles to write corresponding
vertices in order: RST
MNP
Write and solve proportions:
RS
ST

MN NP
32 x

16 13
RS
RT

MN MP
32 38

16 y
16 x  32 13
32 y  38 16 
y  19
x  26
Try these: Each pair of polygons is similar. Find the value of x.
4.
5.
7
M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY
If two polygons are similar, the ratio of their perimeters will equal the ratio of their sides. We can use
this fact to find the perimeter of a figure without finding the lengths of all of its sides.
Ex. 4: If DEF
GHJ , find the scale factor of
and the perimeter of each triangle.
Solution: The scale factor is
The perimeter of
Then
DEF to GHJ
EF 8 2
  .
HJ 12 3
DEF is 10 + 8 + 12 = 30.
2 perimeter of DEF
.

3 perimeter of GHJ
2 30

3 x
2 x  3  30 
x  45
6. Try this: If ABCD
The perimeter of GHJ is 45.
PQRS , find the scale
factor of ABCD to PQRS and the
perimeter of each polygon.
8
M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY
7-3 Similar Triangles
Here are three ways to show that two triangles are similar.
AA Similarity
SSS Similarity
SAS Similarity
Two angles of one triangle are congruent to two angles of another triangle.
Lengths of all three pairs of corresponding sides of two triangles are
proportional.
Two side lengths of one triangle are proportional to the two corresponding
side lengths of another triangle, and the included angles are congruent.
Ex. 1 : Determine whether the triangles are similar.
a.
b.
Solution:
Solution:
AC 6 2 BC 8 2
AB 10 2
  ,
  , and
 
DF 9 3 EF 12 3
DE 15 3
ABC DEF by SSS Similarity.
MN NP
3 6

 , so
QR RS
4 8
Also, N  R .
NMP RQS by SAS Similarity.
Try these: Determine whether the triangles are similar. If so, write a similarity statement. Explain your
reasoning.
1.
2.
3.
9
M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY
Ex. 2 : Write a similarity statement, and explain why the triangles are
similar. Then find IU.
Solution:
G  G . Also,
GWH
Since
GW 26 2
GH 26 2

 and

 .
GU 39 3
GI 39 3
GUI by SAS Similarity.
GW HW
26 20
, then

 . So 26 x  39  20  , or x  30 .
GU
IU
39 x
Try these: Write a similarity statement, and explain why triangles are similar. Then find the requested
measure.
4. JL
5. BC
Ex. 3 : A person 6 feet tall casts a 1.5-foot-long shadow at the same
time that a flagpole casts a 7-foot-long shadow. How tall is the
flagpole?
Solution: The Sun’s rays form similar triangles.
Using x for the height of the pole,
1.5 6
 .
7
x
So 1.5 x  6  7  , and x  28 . The flagpole is 28 feet tall.
6. Try this: Two vertical posts are 2 meters and 0.45 meter tall. When the shorter post casts a shadow
that is 0.85 meter long, what is the length of the longer post’s shadow? Round your answer to the
nearest hundredth.
10
M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY
Practice Problems for 7-1 to 7-3
#1-3: Find the requested ratio. (See page 2 of this packet.)
1. Kieran scored 6 touchdowns in 14 games. Find the ratio of touchdowns per game. Express as a reduced fraction.
2. In a schedule of 6 classes, Katie has 2 elective classes. What is the ratio of elective to non–elective classes in Katie’s
schedule? Express as a reduced fraction.
3. Out of 274 listed species of birds in the United States, 78 species made the endangered list. Find the ratio of
endangered species of birds to listed species in the United States. Express as percent rounded to the nearest tenth.
#4-5: Write an equation, and solve. (See page 3 of this packet.)
4. The ratio of male students to female students in the Latin club at Campbell High School is 3:4. If there are 18 male
students in the club, how many female students are in the club?
5. The ratio of the measures of the sides of a triangle is 5:6:9, and its perimeter is 220 meters. What are the measures of
the sides of the triangle?
#6-8: Solve each proportion. (See page 4 of this packet.)
6.
20 4 x

5
6
7.
x 1 7

3
2
8.
11
15 x  3

3
5
M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY
#9-10: Solve each proportion. (See page 5 of this packet.)
9.
x3
11

4
x4
10.
x 5
4

8
x 1
#11-12: Determine whether the figures are similar. If so, write the similarity statement, and find the scale factor. If
not, explain your reasoning. (See pages 6-7 of this packet.)
11.
12.
#13-14: Each pair of polygons is similar. Write a proportion, and solve for x. (See pages 5-7 of this packet.)
13.
14.
+
12
M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY
#15-18: Determine whether each pair of triangles is similar. If so, write a similarity statement. Explain your
reasoning. (See page 9 of this packet.)
15.
16.
17.
18.
#19-21: Write a similarity statement, and explain why the triangles are similar. Then find the requested measure.
(See page 10 of this packet.)
19. AC
20. JL
21. VT
13
M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY
Review for 7-1 to 7-3
1. One ounce of cheddar cheese contains 9 grams of fat. Six of the grams of fat are saturated fats. Find the ratio of
saturated fats to total fat in an ounce of cheese. Write your answer as a reduced fraction and as a percent rounded to
the nearest tenth.
2. Write a proportion and solve: A worker at an automobile assembly plant checks new cars for defects. Of the first
280 cars he checks, 4 have defects. If 10,500 cars will be checked this month, predict the total number of cars that will
have defects.
3. Write an equation and solve: The ratio of goats to sheep at a university research farm is 4:7. The number of sheep at
the farm is 28. What is the number of goats?
4. Write an equation and solve: The ratio of the measures of the sides of a triangle is 3:4:6, and its perimeter is 104
feet. Find the measure of each side of the triangle.
5. Write an equation and solve: The ratio of the measures of the three angles is 4:5:6. Find the measure of each angle
of the triangle.
Solve each proportion.
6.
3x  5 5

4
7
7.
4
2

x2 x4
8.
2
x5

x4
3
9.
5
x4

x  1 10
14
M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY
Determine whether each pair of figures is similar. If so, write the similarity statement, and find the scale factor.
If not, explain your reasoning.
10.
11.
Each pair of polygons is similar. Find the value of x.
12.
13.
14. If ABCDE
PQRST , find the scale factor of ABCDE
to PQRST and the perimeter of each polygon.
Determine whether the triangles are similar. If so, write a similarity statement. If not, what would be sufficient to
prove the triangles similar? Explain your reasoning.
15.
16.
15
M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY
Write a similarity statement, and explain why the triangles are similar. Then find the requested measure.
17. LM, QP
18. NL, ML
19. A lighthouse casts a 128-foot shadow. A nearby lamppost that measures 5 feet 3 inches casts an 8-foot shadow.
a. Sketch a diagram, and label with the given information.
b. Write a proportion that can be used to determine the height of the lighthouse.
c. Find the height of the lighthouse.
Answers to Review Problems on p. 14-16 in this packet:
4
x
1. 2/3, 66.7%
2.
, 150 cars

280 10,500
3. 7x = 28, 16 goats
4. 3x + 4x + 6x = 104, 24 ft, 32 ft, 48 ft
5. 4x + 5x + 6x = 180, 48  , 60  , 72 
6. x = 5/7
7. x = 10
8. x = 7 or x = 2
9. x = 9 or x = 6
ABC UVT , scale factor 2/3
10. MLKJ PSQR, scale factor 5/3
11.
12. x = 1.5
13. x = 7
14. scale factor 5/7, perim of ABCDE = 65, perim of PQRST = 91
SWY by SAS Similarity
15. AJK
16. No. We need to know that the 3rd pair of sides has the same ratio or that LNM  Q .
17.
PQN by AA Similarity, x = 9
LMN
19. b.
x
5.25

128
8
18.
c. 84 ft
16
KLM
JLN by AA Similarity, x = 2