MATHEMATICS 9
CHAPTER 4
SCALE FACTORS & SIMILARITY
NAME: _________________________________
DATE: __________________________
BLOCK: ___________
TEACHER: ___________________________
Day 1 (Lesson 4.1) What are Similar Shapes?
– Enlargements and Reductions –
Lesson Focus: After this lesson, you will be able to…
 Identify enlargements and reductions, and interpret the scale factor
 Draw enlargement and reductions to scale
Consider this moon shape……………………………..
(original)
What do you notice about shapes A, B, C & D below? How do they compare to the original?
A)
B)
C)
D)
Enlargement is when the dimensions in all directions are multiplied by greater than 1.
Which shapes above are enlargements? Why? A and D (enlarged in all directions equally)
Which shapes above are not enlargements? Why? B and C (enlarged in only one direction)
What do you notice about shapes E, F, G & H below? How do they compare to the original?
E)
F)
G)
H)
Reduction is when dimensions in all directions are multiplied by less than 1.
Which shapes above are reductions? Why? A and D (reduced in all directions equally)
Which shapes above are not reductions? Why? B and C (reduced in only one direction)
Similar Shapes are shapes created by enlargement or reduction.
Scale Factor is the number multiplied to the dimensions;
 a scale factor greater than 1 makes enlargement
 a scale factor less than 1 makes reduction
Example 1 – Enlargement: create a shape which is twice as big
Method 2:
Original Method 1:
use a grid twice as big
Terry Fox Secondary Mathematics
multiply dimensions by scale factor 2
Page 1
Question: what percentage is the second shape of the first? Scale factor of 2=2x100=200%
Practice: create a similar shape which is 150% times as big
Method 2:
Original Method 1:
use a grid 1.5 times as big
Czech Republic
multiply dimensions by scale factor 1.5
𝑛𝑒𝑤 𝑤𝑖𝑑𝑡ℎ = 1.5 ∗ 𝑜𝑙𝑑 𝑤𝑖𝑑𝑡ℎ
=1.5*2=3
𝑛𝑒𝑤 ℎ𝑒𝑖𝑔ℎ𝑡 = 1.5 ∗ 𝑜𝑙𝑑 ℎ𝑒𝑖𝑔ℎ𝑡
=1.5*1=1.5
Czech Republic
Example 2 – Reduction: create a similar shape using scale factor of 0.50
Method 1:
Method 2:
Original
use grid 0.50 times as big
multiply dimensions by scale factor 0.5
New Top width=0.5*old top width
=0.5*3=1.5
New height=0.5*old height
=0.5*5=2.5
Practice – Reduction: create a shape which is 85% of the original
Method 1:
Method 2: multiply dimensions by scale
Original
use grid 0.85 times as big
factor 0.5
𝑛𝑒𝑤 𝑤𝑖𝑑𝑡ℎ = 0.85 ∗ 𝑜𝑙𝑑 𝑤𝑖𝑑𝑡ℎ
=0.85*9=7.65
New height =0.85*old height
=0.85*6=5.1
Russia
Example 3 – Reasoning with Similar Shapes
For the second image, is the scale factor
equal to 1? greater than 1? less than 1?
Explain how you know.
The second image is larger, meaning a scale factor>1
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Day 2 (Lesson 4.2) Scale Diagrams
Lesson Focus:



After this lesson, you will be able to…
Identify scale diagrams and interpret the scale factor
Determine the scale factor for scale diagrams
Determine if a given diagram is proportional to the original shape
Yesterday we looked at similar shapes, created through enlargement and reduction
Scale Diagrams are diagrams made smaller or larger with enlargement or reduction.
Examples of reduction: photo of yourself on your smartphone
Examples of enlargement: looking through a microscope at a bacterium
𝑆𝑐𝑎𝑙𝑒 𝐹𝑎𝑐𝑡𝑜𝑟 =
𝑛𝑒𝑤 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡
𝑠𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡
Scale factor transforms starting measurements to new measurements. How?
(we’ll find out next page)
Practice:
a) What is the scale factor from my actual head (24cm tall) to the above picture?
measure picture: 1.3cm
Scale factor = 1.3cm (new) ÷ 24cm (starting) = 0.054 (reduction)
b) What is the scale factor from the actual bacterium (0.000000009m) to the above picture?
Measure picture: 1.18cm=0.0118m
Scale factor = 0.0118m (new) / 0.000000009m (starting) = 2000000 (enlargement)
Ratios
Notice that the scale factor is a division of two numbers – both which are being compared.
This is called a ratio
, and can be written as a fraction, decimal, or percent.
Let’s review two kinds:
Example 1-Ratios
Consider these shapes…
a) Part-to-part
a. the ratio of circles to triangles: 4:3 or 4/3 or 1. 3̅ or 133%
b. the ratio of triangles to circles is: 3:4 or 3/4 or 0.75 or 75%
b) Part-to-whole
a. the ratio of circles to all shapes: 4:7 or 4/7 or 0.571… or 57%
b. the ratio of triangles to all shapes is: 3:7 or 3/7 or 0.428… or 43%
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Practice: For each regular polygon, what is the ratio of one side length to the perimeter?
a)
i.
ii.
b)
Write each ratio as an equivalent ratio in lowest terms:
i. 5:20 => 1:4
ii.
9:27 => 1:3
Write each ratio as a decimal and a percent.
i. 1/4=0.25=25%
ii.
1/3=0.33=33%
c)
Finding Missing Measurements
Method 1: Using a scale factor
Consider the map below. If the map has a ratio of 0.32cm:1km…
a) How many cm is it from Coquitlam to Port Moody? 1.2cm
b) What measurement is this? new or starting
c) What is the scale factor from the diagram to the actual?
𝑛𝑒𝑤 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡
1𝑘𝑚
𝑘𝑚
𝑆𝐹 =
=
= 3.125
𝑠𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡 0.32𝑐𝑚
𝑐𝑚
Recall: scale factor transforms from starting to new
measurement. Now we can see how.
d) How far away is Port Moody from Coquitlam?
𝑘𝑚
𝑠𝑡𝑎𝑟𝑡𝑖𝑛𝑔 × 𝑆𝐹 = 1.2𝑐𝑚 × 3.125
= 3.75𝑘𝑚
𝑐𝑚
𝑠𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝑚𝑒𝑎𝑠𝑢𝑟𝑚𝑒𝑛𝑡 × 𝑆𝐹 = 𝑛𝑒𝑤 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡 (answer from first page)
Method 2: equal ratios (called proportions)
The same scale factor is applied to all dimensions. We’ll make two and set them equal.
Given Ratio
1𝑘𝑚
0.32𝑐𝑚
Desired Ratio
𝑑
1.2𝑐𝑚
Therefore…
1𝑘𝑚
0.32𝑐𝑚
=
𝑑
1.2𝑐𝑚
Solve by cross multiplication…
1𝑘𝑚 × 1.2𝑐𝑚 = 𝑑 × 0.32𝑐𝑚
Then divide…
1𝑘𝑚 × 1.2𝑐𝑚
𝑑=
= 3.75𝑘𝑚
0.32𝑐𝑚
Practice:
1. The scale for the enlarged image of a housefly is 1: 0.3. What is the actual?
Method 1: scale factor
𝑆𝐹 =
𝑛𝑒𝑤
0.3
=
= 0.3
𝑠𝑡𝑎𝑟𝑡𝑖𝑛𝑔
1
Method 2: proportion
0.3
𝑑
=
1
30
𝑎𝑐𝑡𝑢𝑎𝑙 𝑓𝑙𝑦 = 0.3 × 30𝑚𝑚 = 10𝑚𝑚
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2. Calculate the missing value in each proportion.
a)
𝟏
𝟕𝟖
= 𝟔𝟐𝟒
𝟖
b)
𝟏
𝟐𝟓.𝟐
= 𝟏𝟐𝟔𝟎
𝟓𝟎
c)
𝟏
𝟓𝟖
= 𝟑𝟒.𝟖
𝟎.𝟔
3. A telephone pole that is 12 m tall casts a shadow that is 2 m long. What is the length of
the shadow cast by a student who is 1.5 m tall?
𝒓𝒂𝒕𝒊𝒐 𝒐𝒇𝒑𝒐𝒍𝒆 = 𝒓𝒂𝒕𝒊𝒐 𝒐𝒇 𝒔𝒕𝒖𝒅𝒆𝒏𝒕
12𝑚 1.5𝑚
=
2𝑚
𝑥
12𝑚 × 𝑥 = 2𝑚 × 1.5𝑚
2𝑚 × 1.5𝑚
𝑥=
= 0.25𝑚
12𝑚
x
4. Determine the Scale Factor
An actual laptop has a width of 39.5 cm.
Calculate the scale factor going from the actual laptop to the
image. Express the answer to the nearest tenth.
Solution:
The width of the computer diagram measures 4.2cm.
Set up a proportion for the scale and the measurements.
𝑆𝐹 =
𝑛𝑒𝑤 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡
4.2𝑐𝑚
=
= 0.11
𝑠𝑡𝑎𝑟𝑡𝑖𝑛𝑔 𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑚𝑒𝑛𝑡 39.5𝑐𝑚
5. A scale of 2:5 means
a) There are 2 units of the image for every unit of actual size
b) There are 2 units of the image for every 5 units of actual size
c) There are 5 units of the image for every unit of actual size
d) There are 5 units of the image for every 2 units of actual size
Answer: B
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Day 3 (Lesson 4.4) Similar Polygons
Lesson Focus: After this lesson, you will be able to…
 Identify similar polygons and explain why they are similar
 Solve problems using the properties of similar polygons
You have seen scale diagrams for odd shapes like flies and computers, now we will focus on
simpler mathematical shapes.
A Polygon is a two-dimensional closed shape made just with line segments: three or more
Examples:
Draw two other examples:
Draw two other non-examples:
Similar Polygons are polygons which have been multiplied by a scale factor with
enlargement or reduction. Consequently, similar polygons have:
 equal internal angles
 proportional side lengths (because of scale factor)
Example 1: Identify similar Polygons
The two quadrilaterals look similar. CDEF and RSTU are parallelograms. Is RSTU a true
enlargement of CDEF?
Note: The sum of the interior angles in a
50°
130°
130° 50°
quadrilateral is 360.
Compare corresponding angles:
Compare the corresponding sides:
𝐶𝐷
1.8
𝑅𝑆
2.7
C= 130° and
=
= 0. 6̅ or
=
= 1.5
𝑅𝑆
2.7
𝐶𝐷
1.8
R=130°
𝐷𝐸
3.6
D=50°
and S=50°
̅ or 𝑆𝑇 = 5.4 = 1.5
=
=
0.
6
𝑆𝑇
5.4
𝐷𝐸
3.6
E=130°
and T=130°
F=50° Yes,
andtheU=50°
Response:
quadrilaterals are similar: corresponding sides are proportional and
corresponding angles are equal.
Determining the interior angles of a polygon:
Polygons can be divided into Non-overlapping triangles. The sum of the interior angles in
one triangle is 180°. You can determine the sum of the interior angles in a polygon by
multiplying the number of triangles by 180°.
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To draw the triangles, start with any vertex of the polygon, and from there draw a line to
connect to each of the other vertices. The pentagon can be divided into three triangles.
Quadrilateral has 2 triangles, therefore _2x180°=360°__
Pentagon has 3 triangles, therefore __3x180°=540°____
Example 2: What is the sum of the interior angle of a decagon?
8 non-overlapping triangles
Therefore, internal angle=8x180°=1440°
Example 3: The trapezoid is a scale drawing of a cattle pasture. The actual length of the
shortest side of the pasture is 200 m.
2.5cm:200m
We need same units, giving…
2.5:20000
a) Use what you have learned about similar polygons to determine the actual length
of the other sides of the pasture. Show your work.
Hint: 200 m = 20 000 cm. Start with the values you know:
This side is the
shortest side.
Scale factor from diagram to
actual is 20000/2.5=8000
a
actual farmland
b
A=8000*4=32000cm=320m
B=8000*3=24000=240m
C=8000*5=40000=400m
c
b) How long is the fence surrounding the pasture? Show your work.
Total fencing = sum of all sides = 200m+320m+240m+400m=1160m
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Day 4 – (Lesson 4.3) Similar Triangles
Lesson Focus: After this lesson, you will be able to…
 Determine similar triangles & solve problems using properties of similar triangles
 Determine if diagrams are proportional
Similar Triangles are just a special case of what we studied last day.
They are a 3-sided polygon.
Just as with polygons, similar triangles have been multiplied by a scale factor with
enlargement or reduction. Consequently, similar triangles have:
 equal internal angles
 proportional side lengths (because of scale factor)
Unlike polygons in general, to check if triangles are similar, checking just one of the above
conditions suffices. If one is true, the other follows.
Example 1
Δ ABC is similar to Δ DEF if
A=D , B=E , C =F
Check ratio of sides:
𝐴𝐵
1.5 𝐴𝐶
1.3
=
=
𝐷𝐸
3 𝐷𝐹
2.6
=0.5
= 0.5
𝐵𝐶
1.1
=
𝐸𝐹
2.2
= 0.5
Example 2: Determine if ΔDEF is similar to ΔPQR.
If angles in a triangle add to
180°, how can you find the
missing angle?
D
2.4
80°
10
6
P
4
E
60°
7.5
Compare corresponding angles:
D=_80°_ and  P =_80°__
E=_60°_ and  Q =_60°__
F=_40°_ and  R =_40°__
Q
60° 3
40°
R
The corresponding angles are equal.
F
Compare corresponding sides:
𝐷𝐸
6
=
= 2.5
𝑃𝑄 2.4
𝐸𝐹 7.5
=
= 2.5
𝑄𝑅
3
𝐷𝐹 10
=
= 2.5
𝑃𝑅
4
The corresponding sides are proportional with a scale factor of equal
Therefore: ΔDEF ~ ΔPQR (ΔDEF is similar to ΔPQR)
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Example 3: Determine if ΔCBD is similar to ΔSRT.
𝐵𝐶 3.5
=
= 0. 3̅
𝑅𝑆 10
𝐵𝐷 1.4
=
= 0.35
𝑅𝑇 4.0
𝐶𝐷 2.4
=
= 0. 3̅
𝑆𝑇 7.2
4.0
These triangles are not similar because all sides are not proportional.
Example 4: Use Similar Triangles to Determine a Missing Side Length
Use Similar Triangles to Determine a Missing Side Length
The two vertical supports on a ramp
form two triangles. ABC is similar to
DEC. Find the height of the ramp by
calculating the missing length, y.
Show your work.
First, check that
ABC is similar to
CDE.
Since ΔABC and ΔDEC are right triangles, and since they share angles C, angle A and D must
be the same. Since all corresponding angles are the same, the triangles must be similar.
__________________________________________________________________________
________________________________________________________________
Second, compare corresponding sides to determine the scale factor.
5
= 1.666. ..
3
The scale factor is 1. 6̅. Since the triangles are similar, you can use the scale factor to
determine the missing length.
Third, determine the missing side.
Method 1: Scale Factor
Y=1.5 × 1. 6̅ = 2.5𝑚
Method 2: Proportions
5
𝑦
5 × 1.5
=
=> 5 × 1.5 = 3𝑦 => 𝑦 =
= 2.5𝑚
3 1.5
3
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Practice





Section 4.1 (pp. 136-138)
Section 4.2 (pp. 142-145)
Section 4.3 (pp. 150-153)
Section 4.4 (pp. 157-159)
Chapter Review/Practice Test (pp. 160-163)
Use the following to guide your studying, based on your strengths and weaknesses. If you struggle with one
section, then you can focus on that.
Focus
What are similar
shapes?
Can you find the scale
factor?
Can you use scale factor
for drawing?
Can you use scale factor
for finding a length?
Within the Chapter
















Section 4.1: 11
Section 4.2: 3, 20ab
Section 4.3: 1, 2, 3, 4, 5, 6,7,8, 17, 19
Section 4.4: 2, 3, 4, 7, 8, 11, 14a, 18
Section 4.1: 7,9,10
Section 4.2: 2, 8, 9, 10, 11, 12, 14a, 15,
18, 19a, 20 abcd, 22
Section 4.3: 11
Section 4.4:
Section 4.1: 2, 4, 5, 6, 8, 13, 14, 15,
16, 17
Section 4.2:
Section 4.3: 20
Section 4.4: 9, 11
Section 4.1: 12
Section 4.2: 1, 4, 5, 6, 7, 13, 14, 16, 17,
19, 21
Section 4.3: 9, 10, 11, 12, 13, 14, 15,
16, 18, 21, 22, 23
Section 4.4: 1, 5, 6, 9, 10, 12, 13, 14,
15, 16, 17
Chapter Review
(pp. 160, 161)
1, 2, 3, 4, 13, 16
Practice Test
(pp. 162, 163)
1, 3, 4, 6, 11,
13, 14
9, 12
8, 9,
5, 6, 7, 8
7
10, 11, 14, 15,
17, 18
2, 5, 10, 12, 15
Hand-In
DAY
SECTION
ASSIGNMENT
1
2
3
4
4.1
4.2
4.3
4.4
pp. 136-138 #4,6,7,8,9,12,14
Pg. 142-145 # 4,5,7-10, 11,15, 17, 19, 20; pp. 158 #9
Pg. 150-153 #4,6,7,9,10,12,14
Pg. 157-159 # 3, 5, 10, 12, 13, 15
5
Review / Quiz
6
Test
Date: ____________________________
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