Chapter 10
Traditional woodcarvers make Russian stacking dolls hollow and similar in shape so
the dolls can be placed inside one another to form a nested set.
293
10
Chapter
Focus on Skills
ARITHMETIC
Express each fraction in simplest form.
6
1·
10
8
2· 30
9
6
8
7. 12
21
10. 24
9. 16
75
12
13. 60
4. 16
10
8. 12
45
70
14. 100
15. 100
Complete.
±ft
18
12
3 · 10
16. 100
19. 12 em = • mm
20. 2
22. 3.2m = • em
23. 1~ 1b = • oz
= • in.
6.
5. 30
11.
16
10
12. 25
20
30
60
17. 360
21.
15
45
18. 360
1± h
= • min
24. 12 min =
li h
GEOMETRY
The polygons shown are congruent. Complt:te.
25. a. h.JKL =: •
b. h.JLK =: 0
26. a. ABCD =:
JdLZ~y
b. DCBA =: .
H
~BE~
~G
A
~
F
The polygons shown are congruent.
a. Name all pairs of corresponding angles.
b. Name all pairs of corresponding sides.
c. Write a congruence statement for the polygons.
27.
c
8
A~BRVT
2 •
~P
o 0~
M
s
=
29. If h.GHI
h. KLM, then LH and • are corresponding angles and
MK and • are corresponding sides.
294
Chapter 10 Focus on Skills
D
N
F
~
10.1
Ratios
Objective: To express ratios in simplest form.
A ratio is a comparison of two numbers by division. The ratio of a to b may
be written as ~ or a: b. Ratios are usually expressed in simplest form.
Example 1: Express each ratio in simplest for m.
a. ACto CB
b. CB toAC
Solution: a. AC = _l_
CB
b. CB = 12
12
1
AC
c.
3
4
c
B
AC = ]_
AB
15
1
5
1
4
12
3
A
c. AC toAB
EXPLORING
•
Express each r atio in simplest form.
1. Use the table at the right to determine the ratio of the
number of free throws made (FfM) to the number of free
throws attempted (FfA) for each basketball player.
Player
FTM
FTA
Beth
15
50
14
23
60
21
Maria
2. The team has won 12 games and lost 8 games.
a. Write the ratio of games won to games lost.
b. Writ~ the ratio of games lost to games won.
c. Write the ratio of games won to total games.
Kim
Example 2: Express each ratio in simplest form.
b. 1 dime to 1 dollar
a. 3 em to 15 mm
Solution: a.
3 em = 30 mm = ~
15mm
15mm
1
1dime
b. 1 dollar
10¢
1
= -- = 100¢
10
Thinking Critically
1. Jon's paycheck is $180. He saves~ qf it, or $72. What comparison is
being made by the ratio~?
2. a. Name three ratios equal to 1 : 3.
b. Name three ratios equal to 12: 16.
10.1 Ratios
295
Class Exercises
Make a drawing to represent each ratio.
1. shaded parts to all parts is 5 : 12
2. shaded parts to unshaded parts is 5 : 12
Express each ratio in simplest form.
3. _!_
4. 24
36
36
5. 45: 60
7. 8 mrn to 2 em
6. 16: 20
8. 2 quarters to 3 dimes
Exercises
Express each ratio in simplest form.
1. shaded squares to unshaded squares
2. unshaded squares to shaded squares
3. shaded squat:es to all squares
Make a drawing to represent each ratio.
4. shaded parts to all parts is 3 to 7
5. shaded parts to unshaded parts is 3 to 7
6. A room is 10 ft wide and 13 ft long. Find the ratio of length to width.
7. Rick has 3 hits in 5 times at bat. Find the ratio of hits to times at bat.
8. A geometry class has 17 girls and 13 boys. Find each ratio.
a. girls to boys
b. boys to girls
.I'\
c. girls to all students
d. boys to all students
1\;
e. all students to boys
Express each ratio in simplest form.
10.
9. _2._
27
11
45
11. ~
12. 24
60
30
13. 6 m to 18m
14. 10 em to 15 em
15. 15 em to 10 em
16. 10 ini to 4 mi
17. 8: 20
18. 9: 36
19. 24: 32
20. 28: 24
21. RS: ST
22. ST: RS
23. RS: RT
2
IL
R
S
3 b
T
A
24. AX
25. XB
26. AX
XB
AX
AB
27. BC
28. AY
29. AY
XY
YC
AC
296
Chapter 10
Similarity
X~Y
y
B
~
8
20
C
l
30.
33.
DE
RS
31. FD
RS
DE
34.
32.
F
TS
TR
FE
TR
FD
35. FE
2:14T 2::16
R
TS
6
S
D
9
E
36. 2 ft to 6 in.
39. 15 min to 1 h
37. 1 dime to 1 quarter
38. 6 em to 50 mm
40. 7 oz to 1 lb
41. 3 nickels to 2 quarters
42. the measure of a right angle to the measure of a straight angle
43. the length of a side of a square to the perimeter of the square
44. If EF
= ~ • XY, find EF : XY.
45. If RS
= 3 • AB, find AB : RS.
46. The ratio of the measures of two consecutive angles of a parallelogram
is 2 ; 3. Find the measure of each angle. (Hint: Represent the measures
by 2x and 3x.)
I
47. The ratio of the measures of the three interior angles of a triangle is
2 : 3 : 4. Find the measure of each angle.
APPLICATION
48: Perimeter and Area
D
Find the indicated ratios for rectangles A and B.
a. base length of B to base length of A
b. height of B to height of A
c. perimeter of B to perimeter of A
6m
4m
B
0
3
2m
m
A
d. area of B to area of A
J
'
Everyday Geometry
The musical sound of a guitar or violin is produced by the
vibration of the string. The number of vibrations each second
is the frequency measured in Hertz (Hz). When the ratio of the
frequencies of two sounds is 2 : 1, the sounds are one octave
apart. The higher sound has twice the frequency of the sound
an octave lower.
1. The lowest string on a guitar vibrates with a frequency of
82.5 Hz. Find the frequency of a sound:
a. one octave higher
b. two octaves higher
2. The highest note on a piano is produced by a string vibrating
with a frequency of 4,186 Hz. Find the frequency of a sound:
a. one octave lower
b. four octaves lower
.,
~
10.1 Ratios
297
10~2
Proportions
Objective: To solve proportions.
Since both~ and 192 are equal to ~' they are equal to each other. A statement
that two ratios are equal is called a proportion. A proportion can be written
in either of the following ways.
6
=
8
9
12
6: 8 = 9: 12
or
------~-___,
EXPLORING .
Choose the ratios that are equal to the given ratio.
1. ~
a. _i_
b. 10
5
10
4
20
50
a. 9 : 12
b. 18 : 24
c. 16: 12
2. 12 : 16
c.
The following terms are used to describe proportions.
1st term 3rd term
~
1st term 3rd term
~
~
a
c
- -b
d
1'
1'
~
a:b = c:d
1'
2nd term 4th term
1'
2nd term 4th term
The first and fourth terms of a proportion are called the extremes. The
second and third terms of a proportion are called the means. You can use
the following property to solve a proportion for one term when the other
three terms are known.
Means-Extremes Property
In a proportion, the product of the means equals the product of the
extremes.
= be.
If a : b =e : d, then ad = be.
If
298
Chapter 10
!!:._
b
Similarity
= .£ , then ad
d
d.
g
30
d. 15: 20
Example:
Solution:
Solve each proportion.
- l60
l
a• i
X -
b.
a. 12 • x = 4 • 60
b. 42 • n = 6 • 7
12x = 240
X
=
!!:_ =
6
20
1
X
3•
C.
X
= 5•5
3x = 25
= 42
n =
=
5
42
42n
240
12
c.l
2_
42
=
42
= 25 = 8 _!_
3
3
X
1. In the proportion a : b = c : d , are there any restrictions on b and d?
Explain.
2
:
2. A proportion like ; 6 = % =
is called an extended proportion.
a. Describe how you would solve for x andy.
b. Solve for x andy.
Class Exercises
1. For the proportion ~ = %, name each of the following.
a. the first term
b. the second term
d. the fourth term
c. the third term
f. the extremes
e. the means
Solve each proportion.
'
~
I
2.
!!:_ -
15 -
2
5
6
9
3.=16
X
!!:_ =
l
5.
4. 3: X = 12 : 16
l
5
= £.
8
Exercises
Solve each propor tion.
1. l = ~
2.
2
8
5.= -
6.-1 = -d
2
7
4
l.
9. 9: X = 3: 2
13. 7.5 : 15 = b : 20
12
2
4
9
10. 4:7 = 8: y
14. 1 : 2.5 = 4 : X
4
2
3.= -
4. £
5
3
l
8. _!_
y
7. £
10
=
11. n: 2
15. 4 : d
4
= 5: 3
= 5 : 7.5
4
=
=
10
5
!!:_
5
12. 5 : a =, 2 : 5
16. 1.5 : 4.5 = 2 : z
17. The numbers 2, 3, and 6 are the first three terms (listed in order) of a
proportion. Find the fourth term.
10.2 Proportions
299
If possible, use the given numbers to complete the proportion.
18. 8 16· ~
'
' 4
.
22. 6, 8,
= •
15 225 -
19. 2 8· l_
•
'
' 12
=•
. 27
•
ii
20. 6 2· 2_
•
'
= •
' 15
•
. 18
•
24. 20, 8, 45 = •
•
23. 16, 18, 24 = •
21. 9 12· ~ - •
'
25. 16 21·
'
Solve each proportion.
26.
l =
5
7
27 . 2x - 5
2x + 5
_x_
4 + x
5
28. 1._ =
12
2._ =
20
21
y
APPLICATIONS
29. Elections A candidate won an election by a 3 to 2 margin. Her
opponent received 1,110 votes. How many votes did the winner
receive?
30. Calculator Use a calculator to solve the proportion
70 : 175 = 98 : X.
31. Property Taxes The property tax on the Johnsons' house, which is
valued at $100,000, is $1,500. Next door, the Lees' house is valued at
$110,000. What is the property tax on the Lees' house?
Seeing in Geometry
Complete.
1.
D
isto
2.
D
is to
3.
3
is to
is to
300
0
Chapter 10
as
DD
D
4.
D
D
as
0
is to • .
0
0
isto • .
E
as
~
isto • .
(]
as
~
isto • .
Similarity
'6 -
•
.!2 - •
'20 -
.
10.3
Similar Figures
Objective: To recognize figures that appear to be similar and identify
corresponding parts of similar polygons.
Figures that are the same shape but not necessarily the same size are called
similar (-). A photograph and its enlargement are a familiar example of
similar figures. In each case below, the rectangles are similar.
-
[QJ
c=J
The red rectangle is a reduction
of the black rectangle.
The red rectangle is an enlargement
of the black rectangle.
EXPLORING
Part A
1. Use the Geometric Supposer: Triangles disk to create an acute triangle.
Sketch the triangle and record its measurements.
2. Use the Scale change option. Sketch the new figure. What do you
notice about the two figures?
I
~
3. Repeat Steps 1 and 2 for a right triangle and an obtuse triangle.
4. Insert the Quadrilaterals disk and repeat Steps 1 and 2 for several
different quadrilaterals.
PartB
1. The two triangles shown at the right are similar. Using "H"
for "corresponds to," write correspondences between six
pairs of corresponding parts.
2. A similarity statement that indicates the corresponding
order of the vertices is MBC- LPQR, where"-" is read
as "is similar to." Write five other similarity statements for
the two triangles.
A~
c
P~R
/
10.3 Similar Figures
301
Example: Assume that GHIJ- RSTW. Complete.
a. LH ~ •
b. HI~ •
c.IJGH- .
d. IHGJ- .
Solution: a.LH ~ LS
b. HI~ST
c.IJGH- TWRS
a.o
d. IHGJ- TSRW
1. Tell whether each pair of figures appear to be similar.
0
b.
d.OD
D
L--/_____..
e.~//\
c.D
f.~~
2. Are congruent figures also similar?
3. What appears to be true about corresponding angles of similar
polygons?
Class Exercises
The polygons shown are similar. Name the side or angle that
corresponds to each indicated part.
~~
l.a. ST
b. XY
c. LR
d. LX
X
2. a. HK
y
b.BP
c. LD
d. LP
c:JE
p
Q~
A
B
l
H
3. Complete each similarity statement for the triangles in Exercise 1.
a. 6. XYZ - •
b. 6. STR - •
c. 6. ZYX - 11
d. 6. TSR - •
4. Assume that 6.PTO- 6.RMB. Complete.
a. LO ~ •
302
b. OT ~ •
Chapter 10 Similarity
c. MR~ ·
d.
LBMR~
I
'l
'"
Exercises
True or false?
1. Similar figures must be the same size.
2. Congruent figures must be similar.
3. If MBC - fs.. WXY, then BC corresponds to WX.
4. If"~' is true, then "6CAB- 61"Wx' is also true.
5. Any two triangles 'are similar.
6. Any two rectangles are similar.
Select the figure that appears to be similar to the given figure.
a. ~ b.
~
~
9.
DI
·,
j\
7.
LJ
\7 c. 1\
~
v
b.~
c.~
a.
8.
b.
I. I
lO.d CJb.d
c.D
I
c.
d
Complete each similarity statement for the similar polygons.
,
11. a: BQNST- •
b. CDFGH- •
c. NQBTS- •
d. GFDCH- •
Q.
B~N
\
~
'T
_\\
8
.
12. a. TRPS- •
b. AEBD- .
c. PRTS- •
d. EADB- .
VG
f
.It
_.·
D
c
, ~
:osA~=1,
a
D
r
,)....
,
E
<:..:
"" -·- _//
13. Use the similar polygons in Exercise 11.
a. Name five pairs of corresponding angles.
t
b. Name five pairs of corresponding sides.
y
14. a. Write a similarity statement for the two
similar triangles shown.
b. Name three pairs of corresponding angles.
c. Name three pairs of corresponding sides.
X
I
'f
CL
z
.A
~
A~c -- \:)
15. Assume that 6. RST - 6. JKL.
a. Name three pairs of corresponding angles.
b. Name three pairs of corresponding sides.
10.3 SimilarFigures
303
For each pair of similar triangles:
a. Name three pairs of corresponding angles.
b. Name three pairs of corresponding sides.
c. Write two other similarity statements for the two triangles.
17. 6AMN- 6AEF
16. 6ABC- 6XYC
d2E
c
A
18. 6ACD- 6BCD
7~ z B
A
N
c
A~B
F
Draw and label similar polygons to fit each description.
19. 6 CDE- 6PLG
20. rectangle ABCD- rectangle WXYZ
21. 6RST-6RWY
22. 6ABC-6ABD
APPLICATION
23. Patterns
a. The length of each side of the large rectangle
is twice the length of the corresponding side
of the other rectangle. How do the areas of the
rectangles compare? (Hint: How many of the
small rectangles can you place inside the large
one?)
D
b. The length of each side of the large rectangle
is three times the length of the corresponding
side of the other rectangle. How do the areas
of the rectangles compare?
CJ
c. The length of each side of a large rectangle is 10 times the length of
the corresponding side of another rectangle. How do the areas of the
rectangles compare?
Test Yourself
Express each ratio in simplest form.
1. -24
2. 32:48
3. ~
Solve each proportion.
5. ~ = 6
6. 5 : 6
7.
40
5
304
y
Chapter 10
Similarity
= X : 24
4. 4 in. to 1 ft
15
l
c
=
11
35
8. a: 8 = 8: 1
10.4
Similar Polygons
Objective: To apply the definition of similar polygons.
In the previous lesson we defined similar figures as figures having the same
shape but not necessarily the same size. We can develop a more precise
definition of similarity for polygons.
When the ratios of the lengths of corresponding sides of two polygons are
equal, we say that the lengths are proportional. The scale factor of the
similarity is the ratio of the lengths of the corresponding sides.
EXPLORING
J~
-------------,
1. Use LOGO to enter and run the procedure at the right.
To similar.pentagons
pu lt 90 fd 70 lt 90 pd
repeat 5 [rt 72 fd 30]
pu rt 90 fd 80 lt 90 pd
repeat 5 [rt 72 fd 60]
end
2. The pentagons shown are similar. What measures are the
same? How do their other measures compare? What is the
scale factor of the first pentagon to the second pentagon?
~
3. Change the procedure so the sides of the second pentagon are three
times as long as the sides of the first pentagon. Are the pentagons
similar? What is the scale factor of the first pentagon to the second
pentagon?
4. Write a procedure that draws four similar regular hexagons. Make
the sides of the first hexagon 12 units. The scale factors of the first hexagon
to each of the other three hexagons should be 1 : 3, 1 : 4, and 1 : 5. Find the
scale factor of the second hexagon to the third hexagon.
The EXPLORING activity suggests the following definition of similarity for
two polygons.
Similar polygons are polygons for which
1. corresponding angles are congruent, and
2. corresponding side lengths are proportional.
10.4 Similar Polygons
305
Example:
~RST- ~DEF
Given:
T
a. Find the scale factor of ~RST to ~DEF.
b. Find the scale factor of ~DEF to ~RST.
Solution:
c.mLE= •
d.mLR= •
e. mLT =
g. RS = •
f. RS
h. FE
•
R
a
= • • DE
=•
F
~
s
D
5
18
E
a. Since ~~ = 1~ = ~, the scale factor of ~RST to ~DEF is~·
b. Since ~~ =
c. mLE
1
:
=
%, the scale factor of ~DEF to ~RSTis ~·
= mLS = 41°
d. mLR = mLD = 57°
e. mLT = 180°- (41 o + 57°) = 82°
f. RS = ~ • DE, since the scale factor of ~RSTto ~DEF is ~·
g. Two methods of solution are shown.
Using the scale factor:
RS = 13 • DE
RS = 13 • 18
= 336 = 12
Using proportions:
RS = RT
DE
DF
RS = ~
18
12
12 • RS = 8 • 18
= 144
RS = 12
h. FE = 12 . TS
FE = 1.
10 = 15
2
1. Are similar polygons always congruent? Why or why not?
2. Are congruent polygons always similar? Why or why not?
3. Are all rectangles similar? Why or why not?
4. Are all rhombuses similar? Why or why not?
5. Are all squares similar? Why or why not?
306
Chapter 10
Similarity
'1
Class Exercises
z
Given: 6ABC- 6XYZ
1. What is the scale factor of 6ABC to 6XYZ?
2. What is the scale factor of 6XYZ to 6ABC?
3. BC = •
4. BC = •
• YZ
5. BC = •
6.xz= •
7.XZ= •
·AC
8.XZ= •
YZ
AC
9. If mLC
c~
A
y
10
B
6
= 29°, then mLZ = J.
1
Exercises
The polygons shown are similar. Find each length or angle measure.
1. a. a
b. mL 1
ab::h
3.f
20
2
6
2. a. c
c. e
b. d
d. mL2
d
~
6
~o
4. a.
.r
c. mL3
b. y
d. mL4
5
The triangles shown are similar.
=
.
=.
5. a. What is the scale factor of 6DEF to 6XYZ?
b. EF
YZ
d. EF
f. DE= •
c. EF = •
e.
• XY
c. BC =
ST
• ST
~
9
E
X
XY
g. XY = •
b. BC
RS
D
DE = •
h. If mLX = 48°, then mLD =
6. a. AB = •
• YZ
z
F
=.
d. BC = •
e. AC = •
f. If mLT = 53°, then mLA = • .
~
A~c~ Lj
12
B
R
~
10.4 Similar Polygons
y
T
6-
s
307
The triangles shown are similar.
A
7. a. What is the scale factor of ~ABC to
b. If AD = 5, thenAB = • .
c. IfmLADE = 35°, then mLABC
~ADE?
=• .
B
8. a. OQ = •
b. What is the scale factor of ~NOP to
c. If NP = 12, thenMQ = • .
d. If mLM = 38°, then mLONP
9. L.ABC -
6. PQR and AB
PQ
c. PQ
=.
=.
d. If PQ
e. IfAB
f. CA
2~
Q
10.
b. PQ = •
a. AB = •
12
0
~MOQ?
=• .
= 3 • PQ
A:
6. WLC- L.XYZ and LC
a. LC
YZ
AB
=•
c.YZ= •
•AB
= 6 em, thenAB = • .
= 15 em, thenPQ = • .
g. CA = •
RP
h. If mLP = 45°, then mLA = • .
i. What is the scale factor of ~ABC
to ~PQR?
b. YZ
LC
= 12 em, then LC = • .
e. If LC = 30 em, then YZ = • .
•XY
g.WL= •
XY
h. If mLY = 30°, then mLL = • .
i. What is the scale factor of 6. WLC
to L.XYZ?
= 5 em, and PQ = 10 em
12. L.ACE- L.MOT, AC = 6 em, and MO = 2 em
13. ABCD - JKLM, KL = 5 em, and BC = 2 em
14. The lengths of the sides of the smaller of two similar triangles are
3 em, 4 em, and 5 em. The shortest side of the larger triangle is 9 em.
Find the lengths of the remaining sides of the larger triangle.
15. The smaller of two similar rectangles has dimensions of 6 ft and 8 ft. If
the ratio of a pair of corresponding sides is 2 to 5, find the dimensions
of the larger rectangle.
16. The lengths of the sides of the larger of two similar triangles are 8 em,
14 em, and 18 em. The longest side of the smaller triangle is 9 em.
Find the lengths of the remaining sides of the smaller triangle.
308
Chapter 10
Similarity
=
•LC
Find the scale factor of the first polygon to the second polygon.
11. L.ABC- L.PQR, AB
= ~ • YZ
d. If YZ
f. WL = •
•RP
M
Draw and label a pair of polygons to fit each description. If the
polygons described cannot be drawn, write "not possible."
17. rectangles that are similar
18. rectangles that are not similar
19. squares that are not similar
20. isosceles triangles that are similar
21. isosceles triangles that are not similar
22. parallelograms that are not similar
23. regular polygons that are similar
24. regular polygons that are not similar
APPLICATIONS
25. Photography A rectangular photo
is 5 in. by 7 in. It is enlarged so
that the longer dimension is 21 in.
What is the shorter dimension of
the enlargement?
26. Perimeter and Area
The rectangles shown are similar.
a. AB =
b. BC =
EF
FG
c. Perimeter of ABCD =
d. Area of ABCD =
Perimeter of EFGH
Area of EFGH
27. Computer Examine the two LOGO procedures.
shown at the right. Are the triangles similar? Why
or why not? If the triangles are not similar,
change the procedures so that MBC- LXYZ.
What is the scale factor?
Dl
A
c
E
l2cm
6cm
D3cm
B
F
To ABC
fd 30 rt 120
fd 30 rt 120
fd 30
end
Jl/i I em
G
ToXYZ
fd 75 rt 90
fd 75 rt 135
fd 75*sqrt 2
end
Computer
Packing boxes are often nested inside one another, as
shown at the right. Write a LOGO procedure that will
draw similar nested rectangles.
lgl
10.4 Similar Polygons
309
Thinking About Proof
Using Definitions
The exercises in Lesson 1a.4 applied the definition of similar
polygons only one way-to show what is true of corresponding sides
and angles if the polygons are known to be similar. To use the
definition to show that two polygons are similar, you must show that
there is a correspondence between the polygons such that corresponding
angles are congruent and corresponding side lengths are proportional.
Exam pie: Are the triangles similar? If so, write a
R
similarity statement.
Solution: LA =: LT, LB =: LS, LC =: LR
AB
_
TS
-
15
10
_
-
3
2'
BC
_
SR
-
12
_
B
3
S - 2'
CA
18
3
AB
BC
CA
=-=-so-=-=RT
12
2 '
TS
SR
RT
A
Therefore 6ABC - 6TSR.
D
8
~:
T
2
c
18
Exercises
1. Given: 6ABC - 6EDC
D
a. Write congruence statements for
three pairs of congruent angles.
AB
b. Complete: -
= -BC = -AC
B
c. If AB = 8, BC = 1S, AC = 12,
and ED = 6, find DC and CE.
Are the polygons described similar? Why or why not?
2. a rectangle 1S em by 1a em and a rectangle 12 em by 8 em
sao and a
3. a rhombus with sides 12 em long and one angle of measure
rhombus with sides S em long and one angle of measure 13ao
4. a. Explain why the triangles are similar.
R
v nz
b. Write a similarity statement for the triangles.
5. In 6ABC, mLA =
and mLB = 7a 0 • In 6DEF,
mLE =
As part of showing that the triangles
are similar, what must you show about LD and LF?
sao.
310
Chapter 10
sao
Thinking About Proof
T
50
32
30
40
S
24
X
40
y
10.5
Scale Drawings
Objective: To find actual lengths represented on scale drawings.
Scale drawings are drawings or plans of objects that are either too large or
too small to draw actual size on a sheet of paper. The relationship between
the size of a scale drawing and the actual size of the object represented is
indicated by the scale. The scale may be expressed in several ways, such as
3~, ±in.= 1 ft, or 1:48.
A scale such as 3 ~ means that for every unit of length on the scale
drawing, there are 36 units of length on the actual object. For example, a
length of 1 em on the drawing would represent 36 em on the actual object.
A scale such as ±in. = 1 ft means that every ±in. on the drawing represents
a length of 1 ft on the actual object. This scale represents a ratio of 1:48
between lengths on the drawing and actual lengths.
EXPLORING .
Measure the appropriate part of the scale
drawing and use the given scale to find the
following actual measurements.
Patio
- -- - - ~---~--~---
1. the width of Bedroom 1
2. the length and width of Bedroom 2
Bedroom
1
Kitchen
Living Room
J
3. the length and width of the patio
T J,cale
4. the length and width of the living room
~
5. the length and width of the garage
Bedroom
2
lcm=2m
Garage
10.5 Scale Drawings
311
Think'ln9~' t:rliicall:y
·
1. Using the scale 1 em= 2m, what would be the length on a drawing of
a room that is 7 m long?
2. A scale drawing has a scale of~ in.= 6ft. What would be the length
on the drawing corresponding to each of the following?
a. an actual length of 18 ft
b. an actual length of 9 ft
'I
Class Exercises
Suppose a scale drawing has a scale of 1 : 4. Find the actual length
represented by each length on the drawing.
1. 2 em
2. 2.5 em
3. 6 em
4. 0.5 em
Suppose a scale drawing has a scale of~ in.= 6ft. Find the actual
length represented by each length on the drawing.
.
.
6. -1 In.
7. -5 m.
.
5. -1 m.
8• 2'm.
4
2
8
Exercises
Suppose a scale drawing has a scale of ~ in. = 8 ft. Find the actual
length represented by each length on the drawing.
1. 1 in.
2. 3 in.
3. 4
±in.
4 1 .
. 4m.
1
Suppose a scale drawing has a scale of 16 in. = 1 in. Find the actual
length represented by each length on the drawing.
1 .
5. _!_ in.
6. 1 in.
7. 2 in.
8. -m.
4
2
On a certain map, a distance of 50 mi is represented by 2 ~ in. Find the
number of miles represented by each length on the map.
9. 5 in.
10. 2 in.
11. 1 in.
12. 10 in.
13. On a scale drawing of a room, the dimensions are 2 ~ in. by 2 ±in.
Suppose the scale is ±in. = 2 ft. Find the following information.
a. the actual dimensions of the room
b. the actual area of the room in square feet
312
Chapter 10
Similarity
Minimum regulation sizes for various athletic fields are given. Using
the indicated scale, make a scale drawing of each.
14. Soccer field: 91 m by 46 m
15. Baseball diamond: 90ft by 90ft
Scale: 1 mm = 1 m
Scale: ±in. = 10 ft
16. Basketball court: 84 ft by 50 ft
Scale: 1~ in. = 2 ft
17. Football field: 100 yd by 160 ft
Scale: ± in. = 20 ft
Each scale drawing is labeled with the actual dimensions being
represented. Measure each drawing and find an appropriate scale.
18.
I
lis
ft
19.
45ft
D Sft
14ft
20.
I
120m
21.
10m
5m f 1
_jl5rn
60m
35m
APPLICATIONS
22. Technical Drawing Choose an appropriate scale and make a scale
drawing of your classroom.
./
23. Research Collect some examples of scale drawings from newspapers,
magazines, and other sources. List the different scales used.
Calculator
To make using scales easier, you can store the scale in your calculator's
memory. Find out how to use your calculator's memory.
A map has a scale of 1 em= 2.5 km. Use a calculator to find the actual
distance represented by each distance on the map.
1. 2.2 em
2. 4.6 em
3. 7 em
4. 11 em
5. 8.4 em
6. 14.6 em
7. 15 em
8. 21 em
9. 8.3 em
10. 10.5 em
Use the scale 1 em= 2.5 km to find the map distance that represents
each actual distance.
11. 12.5 km
12. 32.5 km
13. 15 km
14. 12 km
15. 29 km
16. 17 km
17. 8.75 km
18. 49 km
19. 55.5 km
20. 93.5 km
10.5 Scale Drawings
313
10.6
Problem Solving Application:
Using Proportions to Estimate
Distances
Objective: To solve problems by using proportions to estimate distances.
You can use proportions based on similar polygons to measure distances
indirectly. For example, astronomers cannot measure directly the heights of
mountains on the moon or other planets. However, they can estimate those
heights by using a simple proportion based on similar triangles. One of the
terms in this proportion is the distance of the mountain from the terminator,
the line that separates the sunlit side of the moon or planet from the side
facing away from the sun. The proportion is written:
height of mountain _
length of shadow
distance to terminator
radius of moon
Example: A photograph of the moon shows a mountain casting a shadow
that astronomers have calculated to be about 4.5 mi long. The
distance of the mountain from the terminator is about 720 mi.
The radius of the moon is 1,080 mi. How high is the mountain?
Solution: Substitute the known values in the proportion. Let h represent
the unknown height.
h
720
1,080
Moon
--=--
4. 5
~
1,080 h = 720.4.5
1,080 h = 3,240
h = 3, 240
1, 080
h= 3
The mountain is about 3 mi high.
Sun
•
•
Jt:
Astronomers used this method to discover that the borders of some of the
huge craters on the moon are over 15,000 ft high. They also learned that
some mountains exceed 30,000 ft, making them higher than Mount Everest,
the highest point on Earth.
Class Exercises
1. Measurements from a photo of the moon show a mountain that casts a
shadow 22.5 mi long. The mountain appears to be 204 mi from the
terminator. How high is the mountain?
2. A mountain on the moon is 500 mi from the terminator. The mountain
casts a shadow 2.6 mi long. How high is the mountain?
Exercises
1. Mars has a radius of 2,100 mi. Suppose a photograph of Mars shows a
mountain 1,575 mi from the terminator. The shadow cast by the
mountain is 5 mi long. How high is the mountain?
2. Astronomers have determined that the highest mountain on a certain
planet is 24 mi high. A photograph of the planet shows that the
mountain casts a shadow 32 mi long. The distance from the mountain
to the terminator is 42,000 mi. What is the radius of the planet?
3. A botanist used a photograph of a redwood
tree to measure the height of the tree. On the
photo, the tree was 1~ in. tall and cast a
shadow ~ in. long. When the photograph
was taken, the botanist measured the actual
length of the tree's shadow as 72 ft.
a. Using the fact that the right triangle
formed by the tree and its shadow is
similar to the right triangle formed by
the picture of the tree and its shadow on
the photograph, write a proportion
relating the length of the shadow and the
height of the tree on the photograph to
the actual length and height.
b. Solve the proportion to find the height of
the tree.
I
Photo
~
shad
I
(notto
actual
scale)
I
I
I
I
I
shadow.
tree
I
I
I
I
10.6 Problem Solving Application: Using Proportions to Estimate Distances
315
·Chapter 10 Review
'.
'
·; ·•
~
Vocabulary and Symbols
You should be able to write a brief description, draw a picture, or give an
example to illustrate the meaning of each of the following terms.
scale of a scale drawing (p. 311)
similar figures (p. 301)
similar polygons (p. 305)
similarity statement (p. 301)
Vocabulary
extremes (p. 298)
means (p. 298)
proportion (p. 298)
proportional (p. 305)
ratio (p. 295)
scale drawing (p. 311)
scale factor (p. 305)
Symbols
-(is similar to) (p. 301)
~ or a : b (the ratio of a to b) (p. 295)
Summary
The following list indicates the major skills, facts, and results you should
have mastered in this chapter.
10. 1 Express ratios in simplest form. (pp. 295-297)
1 0.2 Solve proportions. (pp. 298-300)
10.3 Recognize figures that appear to be similar and identify
corresponding parts of similar figures. (pp. 301-304)
10.4 Apply the definition of similar polygons. (pp. 305-309)
10.5 Find actual lengths represented on scale drawings. (pp. 311-313)
10.6 Solve problems by using proportions to estimate distances. (pp. 314-315)
Exercises
Express each ratio in simplest form.
1. the ratio of shaded triangles to unshaded triangles
2. the ratio of unshaded triangles to all triangles
DDDDDD
Express each ratio in simplest form.
3. Q
4. 12
39
18
7. 18 : 24
316
Chapter 10
8. 12 m to 60 m
Review
5.
12
9.
1 em to 25 mm
35
6. 25: 10
10. 4 in. to 2ft
_\
Solve each proportion.
12.
11. l = _±_
24
y
15. 9 : 24
=x :4
g
18
13. ~ =
= ~
6
16. 12 : 4
12
=5 :n
12
14. 1_ =
!!:_
6
12
l
18. 5 : 7 =
17. c : 8 = 15 : 24
D DD
z : 21
Select the figure that appears to be similar to the given figure.
19.
L
a.~ b.L] c.~
20.
b.
a.
The polygons shown are similar.
c.
D
A
21. Name three pairs of corresponding sides.
22. Name three pairs of corresponding angles.
23. Write three similarity statements for the triangles.
E
LA.
~
'"7
c
lA
24. What is the scale factor of the smaller triangle to the
larger triangle?
25. Name the side that corresponds to GH.
F
26. N arne the angle that corresponds to LH.
27. What is the scale factor of FGHJ to KLMN?
]~
4
28. Find mLN.
K
G
79AH
N
29. a. JH = •
b. JH =
NM
• NM
L
LJ
12
M
c. JH = •
30. Find KL.
Suppose a scale drawing has a scale of 1 em = 3.5 m. Find the actual
length represented by each length on the drawing.
31. 2 em
32. 3.5 em
33. 5 em
34. 0.5 em
35. 4.6 em
Suppose a scale drawing has a scale of -± in. = 3 ft. Find the actual
length represented by each length on the drawing.
36. 1 in.
37. _!_ in.
2
38.
l
4
in.
39. 1 _!_ in.
2
40. 3 in.
PROBLEM SOLVING
41. The moon has a radius of 1,080 mi. A photograph of the moon shows a
mountain 195 mi from the terminator. The shadow cast by the mountain is
9.6 mi long. How high is the mountain?
Chapter 10 Review
317
Chapter 10 Test
Express each ratio in simplest form.
1. 20
2. 26
35
10
5. 18 : 27
6. 16 em to 4 em
Solve each proportion.
= 2
10 36
9• ~
8
2
• 42
13. 3 : 4
= 5 :X
14. 7 : Z
3. 16: 20
4.12:2
7. 8 in. to 3ft
8. 2 dimes to 5 nickels
6
=
11.
2
6
d. GHEF
~
Given: L. UVW
~
5
n
12. 6 = 2
r4
15. m : 12 = 4.5 : 8
12 : 18
17. The polygons shown are similar. Complete.
a. ABCD ~ II
b. EFGH ~
c. BCDA
=
16. 6:4 = y: 3
A\
~
( HI
D
L. XYZ
C
y
19. a. What is the scale factor of L. UVW to L.XYZ?
b. What is the scale factor of L. XYZ to L. UVW?
w
12
c. mL 1
b. y
6
3°
~~
y
15
37°
8
12
Suppose a scale drawing has a scale of ~in. = 5 ft. Find the actual
length represented by each length on the drawing.
22. 1 in.
23.
l
4
in.
24. 1 _!_ in.
2
25. 4 in.
PROBLEM SOLVING
27. The moon has a radius of 1,080 mi. A photograph of the moon shows a
mountain 354 mi from the terminator. The shadow cast by the
mountain is 17.6 mi long. How high is the mountain?
318
Chapter 10 Test
z
c. Find YZ.
21. Find each of the following for the similar triangles shown.
a. x
F
~
b. N arne three pairs of corresponding angles.
b. Find UV.
E
X
u
18. a. N arne three pairs of corresponding sides.
20. a. Find mL Y.
G
\
'.
26.
1 .
4
m.