MATHEMATICS 9 - mrtroanca.weebly.com

NAME: _________________________________

Date: ___________________________

MATHEMATICS 9 CHAPTER 4

SCALE FACTORS & SIMILARITY

Miller High School Mathematics Page 2

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……………………………..

What do you notice about shapes A, B, C & D below? How do they compare to the original?

Enlargement is when the dimensions are multiplied by . Which shapes above are enlargements? Why? . Which shapes above are not enlargements? Why? . What do you notice about shapes E, F, G & H below? How do they compare to the original?

Reduction is when dimensions are multiplied by . Which shapes above are reductions? Why? . Which shapes above are not reductions? Why? by . Similar Shapes are shapes created by .

(original)

A) B) D)

C)

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Scale Factor is the number multiplied to the dimensions;

a scale factor greater than 1 makes .

a scale factor less than 1 makes .

Example 1 – Enlargement: create a shape which is twice as big

Original Method 1:

use a grid twice as big

Method 2:

Use same grid, but double the dimensions (scale factor

2)

Question: what percentage is the second shape of the first? .

Practice: create a similar shape which is 150% times as big

Original Method 1:

use a grid 1.5 times as big

Method 2:

multiply dimensions by scale factor 1.5


Example 2 – Reduction: create a similar shape using scale factor of 0.50

Original Method 1:

use grid 0.50 times as big

Method 2:


Practice – Reduction: create a shape which is 85% of the original

Original Method 1:

use grid times as big

Method 2:


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.


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 .

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 transforms original measurements to copy measurements.

Practice:

a) What is the scale factor from the original head (24cm tall) to the above picture?

b) What is the scale factor from the original bacterium (0.000000009m) to the above

picture


Review Ratios

Notice that the scale factor is a – which are being compared.

This is called a , and can be written as a .

Let’s review two kinds:

Example - Ratios

Consider these shapes…

a) Part‐to‐part

‐ the ratio of circles to triangles: .

‐ the ratio of triangles to circles is: .

b) Part‐to‐whole

‐ the ratio of circles to all shapes: .

‐ the ratio of triangles to all shapes is: .

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. . ii. .

c) Write each ratio as a decimal and a percent.

i. . ii. .


Finding Missing Measurements

Method 1: Using a scale factor

Method 2: equal ratios (called proportions)

The same scale factor is applied to all dimensions. We’ll make two and set them equal.

Therefore…

Solve by cross multiplication…

Then divide…

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 on the

map? .

b) What measurement is this? Copy or Original?

c) What is the scale factor of the map?

Recall: scale factor transforms from the original dimensions to the

copy’s. Now we can see how:

d) How far away is Port Moody from Coquitlam?

Given Ratio:

Desired Ratio:

(answer from first page)


Practice:

1. The scale for the enlarged image of a housefly is 1: 0.3. What is the actual?

Ask yourself: what should we use as the original measurement?

Method 1: scale factor

Method 2: proportion

2. Calculate the missing value in each proportion.

a)

b) .

c)

.

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?


4. Determine the Scale Factor

An original laptop has a width of 39.5 cm.

Calculate the scale factor going from the original laptop to

the copied 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.

5. A scale of 2:5 means

a) There are 2 units of the copied image for every unit of original size

b) There are 2 units of the copied image for every 5 units of original size

c) There are 5 units of the copied image for every unit of original size

d) There are 5 units of the copied image for every 2 units of original size

Answer:


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: .

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

quadrilateral is .

Compare corresponding angles: Compare the corresponding sides:

130° 130° 50° 50°

C= 130°

D=

E=

and

and

and

or

or

Response:


Determining the interior angles of a polygon:

Polygons can be divided into triangles. The sum of the interior angles

in one triangle is . You can determine the sum of the interior angles in a polygon

by the number of triangles by 180°.

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.

Quadrilateral has is triangles, therefore is .

Pentagon has triangles, therefore is .

Example 2: What is the sum of the interior angle of a decagon?


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…

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:

Scale factor from diagram to actual is

b) How long is the fence surrounding the pasture? Show your work.

This side is the

shortest side. a

b

c

actual farmland


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 similar polygons, similar triangles have been multiplied by a

_________________ 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:


Example 2: Determine if ΔDEF is similar to ΔPQR.

Compare corresponding angles:

D= and = .

E= and = .

F= and = .

The corresponding angles are .

Compare corresponding sides:

The corresponding sides are proportional with a scale factor of .

Therefore:

Example 3: Determine if ΔCBD is similar to ΔSRT.

80°

60°

60°

40°

If angles in a triangle add to

180°, how can you find the

missing angle?

D

E F

R

Q

P 6 10

7.5

3

4

2.4

4.0


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.

Second, compare corresponding sides to determine the scale factor.

The scale factor is . 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 (see day 2)

Method 2: Proportions (see day 2)


StudyFocusThe following sections can be used to study.

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)

To study most effectively, identify your strengths and weaknesses. To identify strengths, choose a focus

from the chart below and ensure you can do the related problems. For weaknesses, choose the focus

below and work on the related problems in order to improve.

Focus Within the Chapter Chapter Review (pp. 160, 161)

Practice Test (pp. 162, 163)

What are similar shapes?

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

1, 2, 3, 4, 13, 16 1, 3, 4, 6, 11, 13, 14

Can you find the scale factor?

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:

9, 12 8, 9,

Can you use scale factor for drawing?

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

5, 6, 7, 8 7

Can you use scale factor for finding a length?

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

10, 11, 14, 15, 17, 18

2, 5, 10, 12, 15

Hand‐InDAY SECTION ASSIGNMENT

1 4.1 pp. 136‐138 #4, 6, 7,8, 9, 12, 14

2 4.2 Pg. 142‐145 # 4,5,7‐10, 11,15, 17, 19, 20; pp. 158 #9

3 4.4 Pg. 157‐159 # 3, 5, 10, 12, 13, 15; pp.151 #10

4 4.3 Pg. 150‐153 #4, 6, 7, 9, 11, 12, 14; pp. 158 #11

5 Review / Quiz

6 Test Date: ____________________________

4.1 Enlargements and Reductions MathLinks 9, pages 130–138

2. Draw an enlargement of each fi gure using a scale factor of 2.

a)

b)

3. Draw a reduction of each letter using a scale factor of 0.5.

a)

Nb)

K

Key Ideas ReviewChoose from the following terms to complete #1.

constant enlargement larger reduction scale factor smaller

1. a) A scale factor greater than 1 indicates a(n) , which results in an image that is the same shape but proportionally

than the original.

b) A scale factor less than 1 indicates a(n) , which results in

an image that is the same shape but proportionally than the original.

c) The is the amount by which all dimensions of an object are enlarged or reduced in a scale drawing.

Check Your Understanding

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4. For each image in column A, state whether the image in column B has a scale factor• greater than 1• less than 1• equal to 1

A B

a)

b)

c)

5. a) Draw an enlargement of the butterfl y using a scale factor of 4.

b) Explain how you know that your drawing is correct.

6. Draw an image of the fl ag of Greece using a scale factor of 1 __ 4 .

7. Alicia copied a headline from the school newspaper and included it on her election poster.

Alicia Wows Students WithCampaign Promises

Alicia received a huge round of applause after presenting her

policies for the upcoming election for student president



Vote for Alicia!

a) Is the headline on the poster an enlargement or a reduction of the headline in the newspaper?

b) What is the scale factor? How do you know?


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4.2 Scale Diagrams MathLinks 9, pages 139–145

Key Ideas Review

Match the term in column A with the correct description in column B.

A B

1. scale diagram a) 1 ______ 120000 = 4 __ ?

2. scale b) 3 cm × 1 200 000 = ?

3. solve using a scale c) 1:1 200 000

4. solve using a proportion representation

d) a proportionally smaller or larger object

Check Your Understanding5. State which numbers you multiply or

divide to fi nd the missing value.

a) 1 __ 5 = ? __ 85

b) 1 __ ? = 6 ___ 132

6. Determine the missing value. Show your thinking.

a) 1 __ 9 = 13.5 ____

b) 1 ____ = 12.5 ____ 50

7. Use the scale factor to calculate the actual length of each object.

a) The scale factor for the image of this hockey stick is 1:42.

3.1 cm

b) The Euvira Micmac beetle below is enlarged using a scale factor of 1:0.05.

40 mm


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8. Determine the scale factor for each question below. Show your thinking.

a) ____ = 30 ___ 225

b) ____ = 3.8 ____ 15.2

9. What scale factor is used to create each image below?

a) The actual size of this award is 34.3 cm.

2.1 cm

b) The average height of a male giraffe is 6 m.

0.045 mm

10. A blueprint is used to show all the measurements needed to build rooms in a house and the house itself.

Master Bedroom3.1 m × 4.2 m

BathroomKitchen

4.1 m × 7.4 m

sink

fridge wash/drystove

sink

Living Room3.1 m × 4.5 m

Bedroom3.1 m × 3.3 m

Porch2.4 m × 7.2 m

fireplace

7.2 m

closet2.1 m × 1.4 m

12 m

tub&

shower

closet

a) What is the scale factor used to draw the blueprint? Express the denominator of the scale factor to the nearest whole number.

b) Draw the master bedroom using a scale factor of 1:290. Express the calculations for the width and length of your drawing to the nearest tenth.

c) What is the area of your drawing in part b)? Express your answer to the nearest hundredth.


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4.3 Similar Triangles MathLinks 9, pages 146–153

Key Ideas Review

Choose from the following terms to complete #1 to 2.

angles both not proportion scale factor sides similar

1. Triangles are similar if one of the following conditions is true:

a) Corresponding are equal in measure.

b) Corresponding are proportional in length.

2. You can solve problems for similar triangles using a

or a .

3.

E F

D

B C

AP

Q R

45°

3 2.6

2.2

1.5 1.3

1.1

22.8

260°

60° 75°

45°

75°

a) Is ΔDEF similar to ΔABC? YES NO Explain.

b) Is ΔDEF similar to ΔPQR? YES NO Explain.


4. What are the corresponding angles and the corresponding sides for the following pairs of similar triangles?

a)

B

A

C K

J

L

b)

Q

P

R N

M

L


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5. Determine which pair of triangles is similar. Explain how you know.

Triangle Angles Sides

�PQR ∠P = 90°

∠Q = 45°

∠R = 45°

PQ = 3 QR = 4.2 PR = 3

�STU ∠S = 90°

∠T = 60°

∠U = 30°

ST = 9.2 TU = 18.4 SV = 15.9

�VWX ∠V = 45°

∠W = 90°

∠X = 45°

VW = 11.3WX = 11.3 VX = 16

6. Are these triangles similar? Explain how you know.

17 8.5

9.23.68.9

19.2

7. Determine the missing side lengths of the triangles below. Show your calculations.

a)

8.4

6.1

S

T U G H

F

7

25.2

18.3X

b) A

6.910

7.4

20

14.8

x

BD

E

C

8. Draw a triangle that is similar to the one shown. Label the measurements for the angles and sides on your triangle.

A B

C

9. Kaylee is 100 cm tall and is standing so that her mother’s shadow covers her shadow. She is 90 cm from her mother and her mother’s shadow is 225 cm long. How tall is her mother? Express your answer to the nearest centimetre.

100 cm

90 cm

225 cm


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4.4 Similar Polygons MathLinks 9, pages 154–159

Key Ideas Review

Decide whether each of the following statements is true or false. Circle the word True or False. If the statement is false, rewrite it to make it true.

1. True/False Polygons that are similar have some angles that are equal in measure.

2. True/False You can use polygons that are not similar to determine unknown side lengths.

3. True/False A polygon is a three-dimensional closed fi gure made of more than three line segments.


4. Is each pair of polygons similar? How do you know?

a)

3

3

3

3

1.5

1.5

1.5

1.5

b) 2.4 2.4

1.4 1.4

11.6

12.72

1.7 1.7

5. a) Draw lines to connect all sets of similar polygons found in the space below.

b) Draw any polygons that do not have a pair.

c) Sketch a similar polygon for the ones found in b).


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6. Use each pair of similar polygons to determine each unknown side length.

a) 3.4

3.4

x

3.4

1

2 2

2

b) 13.5

8.77.5

y21.25 1.45

x

7. As part of an art project, Jamal made an outline of a shape with string. He wanted to create another shape inside the fi rst one.

3.4 m

1.7 m

3.4 m

2.5 m

2.5 m

2.5 m

2.5 m

a) Calculate the unknown side lengths of the inside shape if it is similar to the outside shape.

b) What is the total length of string Jamal used for his art project?

8. Determine the value of the missing values to the nearest tenth. Show your thinking.

A

D

E

x

y

z

G C

B

10 14

12

16

15

9. A pattern is cut showing the dimensions of a pair of similar trays. How much trim will you need to cover the outside edge of the larger tray? Justify your response.

48 cm

30 cm

20 cm


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Check Your Understanding Communicate the Ideas 1. Jesse thinks many of the photographs in this book are reductions. Do you agree with him?

Circle YES or NO. Give 1 reason for your answer. ________________________________________________________________________________ ________________________________________________________________________________ 2. Mary used a scale factor of 3 to enlarge the rectangle.

5 cm

3 cm

3 X 3 = 9 The length of each side for the enlargement is 9 cm. Is she correct? Circle YES or NO. If she is wrong, explain her mistake. ________________________________________________________________________________ ________________________________________________________________________________ ________________________________________________________________________________ 3. Jacob is editing a photo on the computer.

a) If he enters 0.5 as his scale factor, what happens to the photo? _____________________________________________________________________________ b) What happens if he uses 8 as his scale factor? _____________________________________________________________________________ c) What scale factor should he use if he wants the photo to be the same size as the original? _____________________________________________________________________________

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Practise 4. Draw an enlargement of the flag using a scale factor of 4.

France

Measurement of sides × 4:

5. Draw a reduction of the letter using a scale factor of 0.5.

Measurement of sides × 0.5:

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6. Draw a line from the picture to the correct scale factor.

a)

b) c)

• greater than 1

• less than 1

• equal to 1

7. Draw a reduction of the flag of Poland using a scale factor of 0.25.

Measurement of sides × 0.25: Apply 8. Anna received a postcard of the Great Pyramid of Giza.

a) Is this picture an enlargement or a reduction? b) What scale factor could have been used?

0.25 = 14

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8 cm

9. A Canadian flag is 8 cm long. The ratio of the length to the width is 2 : 1. a) How wide is the flag?

×

21

8

× Sentence: _____________________________________________________________________ b) Enlarge the flag by a scale factor of 3. What are the new dimensions? Sentence: _____________________________________________________________________ c) Reduce the flag by a scale factor of 0.25. What are the new dimensions? Sentence: _____________________________________________________________________

10. Draw an enlargement of the letter on grid paper. Use a scale factor of 2.

Use your answer from part a) to answer part b) and c).

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4. Warm Up 1. Find equivalent fractions.

×

a) 1 56

× ÷

c) 1 1442

÷

b) 1 8100

d) 1 1272

2. Write each ratio as a fraction.

a) 1 : 15 b) 1 : 1000

3. Round to the nearest tenth (1 decimal place).

a) 24.88 b) 159.249 4. Round to the nearest hundredth (2 decimal places).

a) 1124.828 b) 58.2496 5. Find the missing measurements.

a) 300 000 cm = 300 000 ÷ 100 = m

b) 3000 m = 3000 ÷ 1000 = km

c) 1 400 000 cm = m d) 40 000 m = km

1 m = 100 cm 1000 m = 1 km

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2. How could you check that the larger image of the plane is proportional to the original image? _______________________________________________________________________________ _______________________________________________________________________________

Practise

3. Complete the proportions.

×

a) 13 36

×

c) 19 90

e) 17 1400

÷

b) 1 728

÷

d) 1 4

48

f ) 1 121200

4. Find the scale factor.

a) 2412

Scale factor = 24 ÷ 12 =

c) 0.525

b) 30200

Scale factor = 30 ÷ 200 =

d) 1.63.2

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5. Find the actual length of each object. a) The image of the school bus is 5 cm long. The scale is 1 : 300.

5 cm ×

1 5300

×

b) The diagram of the mosquito is 32 mm long. The scale is 1 : 0.5.

32 mm

× 32

10.5

× 32

6. A flying distance of 800 km is 5 cm on a map. What is the scale factor?

1 km = 100 000 cm So, 800 km = cm

diagram measurementscaleactual measurement

÷

5 cm 1

cm

÷

The scale factor is .

Use the same units in your proportion.

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Apply 7. The actual footprint of an adult male polar bear measures 30 cm across.

a) What is the scale factor?

diagram measurementactual measurement


b) What is the length of the footprint in the diagram?

c) Use the scale factor from part a) to find the actual length of the footprint.

8. An eagle’s wingspan is 4 cm long on a drawing. The scale is 1:50. Find the actual length of the eagle’s wingspan.

Sentence: ____________________________________________________________ 9. Find the scale factor for each enlargement or reduction.

a) from A to B

b) from A to C

c) from B to A d) from C to A

2 cm

4 cm

A

C

B

Write the fraction in lowest terms.

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10. Julie wants to build a scale model of a volcano for the science fair. The volcano is actually 2500 m tall. If she uses a scale of 1 : 100, how tall will her volcano be? Will it fit into the classroom?

Sentence: _______________________________________________________________________

Math Link a) Look at your design in Math Link 4.1 on page 187. Did you use an enlargement or a reduction?

Circle ENLARGEMENT or REDUCTION. Give 1 reason for your answer. _______________________________________________________________________________

b) Why is it important to use the same scale factor for each part of your diagram?

______________________________________________________________________________

c) What scale factor did you use?

Show your work.

diagram measurementactual measurement

d) Choose a new image to add to your design. Draw it on your scale diagram on page 187.

e) Calculate the actual dimensions of the new image. Show your work.

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4.3 Warm Up 1. Measure the angles.

a) b) 2. Find the missing angle.

3. Divide.

a) 64 =8 b) 62.4 =0.2

4. Find the missing numbers.

a) 5

= 6 b) 28 = 7

5. Use equivalent fractions to complete each proportion.

×

a) 182 6

×

b) 3514 2

Example: The sum of the angles in a triangle = 180°. To find the missing angle, subtract the 2 given angles from 180°. x = 180 – 90 – 30 x = 60°

30º

6 = 61

27º

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Check Your Understanding Communicate the Ideas 1. If 2 triangles are similar, a) what do you know about the angles of the triangles? _____________________________________________________________________________ b) what do you know about the sides of the triangles? _____________________________________________________________________________ 2. Amanda says these triangles are similar. Is she correct?

Circle YES or NO. Give 1 reason for your answer. ________________________________________________________________________________ ________________________________________________________________________________ Practise 3. List the corresponding angles and the corresponding sides for each pair of triangles.

a) PQR and TUV b) ABC and WXY

Q R

U V

TP

AW

X

Y

B C

P corresponds to . corresponds to . corresponds to . PQ corresponds to . corresponds to . corresponds to .

Y

X

2

4.6

5 3.1

2.4

2

L

Z M N

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4. Use a scale factor to determine if these triangles are similar.

S

1.22.5

2.2

12.5

11

6

R

T

V

W

U

RSUV

2.512.5

STVW

RTUW

Are the triangles similar? Circle YES or NO. Give 1 reason for your answer. ________________________________________________________________________________ 5. Use a scale factor to find the missing side length. Show how you know.

UVSR


8x =

x = The missing side length is units.

10.5

73.5

73.5

810.5

R

T

S

V

U

W

x

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Find the scale factor for each pair of corresponding sides.

6. Determine if the triangles are similar.

A

X Y

W

4 6

8 12

8.5 10

B C

ABWX

BC AC

ABWX BC AC

Are the triangles similar? Circle YES or NO. Give 1 reason for your answer. ________________________________________________________________________________ Apply 7. Sam is building 2 bike ramps. They are different sizes, but they are similar. How high is the larger ramp? Sentence: ________________________________________________________________________

1.2 m

3 m 6 m

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The sum of the angles in a triangle = 180°. Find the measure of the third angle.

8. Two ladders are leaning against a wall. They each create a similar right triangle with the wall.

a) Label the dimensions on the similar right triangles.

x

b) How far up the wall does the longer ladder reach? Use a scale factor to solve.

Sentence: _____________________________________________________________________

c) How much farther up the wall does the longer ladder reach than the shorter ladder?

Sentence: _____________________________________________________________________ 9. Are the triangles similar?

a) The angles of the first triangle are 45° and 75°. The angles of the second triangle are 45° and 60°. ____________________________________________________________________________ b) The angles of the first triangle are 60° and 70°. The angles of the second triangle are 50° and 80°. ____________________________________________________________________________

2.4 m 3 m

8 m

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Math Link Draw a logo for your drum company. Include a triangle similar to the one shown here. a) Measure the angles and side lengths of the triangle. 1 = ° 2 = ° 3 = ° Side 1 = cm Side 2 = cm Side 3 = cm b) Design your logo on a sheet of 8.5 × 11 paper. Include a triangle that is similar to the one shown. c) Draw a smaller scale diagram of the logo to fit your design project. Calculate the scale factor.

The angles must have the same measurement but the sides can

be different lengths.

1

2 3

1

2

3

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4.4 Warm Up 1. Measure the angles in each polygon.

a)

A = ° B = ° C = ° D = °

b)

J = ° K = ° L = ° M = ° N = °

2. Measure the angles and the sides of the trapezoid.

P

S R

Q

P = ° Q = °

R = ° S = °

PQ = cm QR = cm RS = cm PS = cm

A

B

C

D

J K

L

M

N

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Check Your Understanding Communicate the Ideas

1. Draw a parallelogram similar to this one. Explain how you know they are similar.

F

G

E

H _______________________________________________________________________________

Practise 2. Use a proportion to find the missing side length of the similar polygons.

A B

x

D C H G

FE

32.4

4.8

BCFG

ABEF

×

2.43

×

The missing side length is units. 3. Find the missing angles of the similar polygons.

x = y =

90°

115°

110°110° 110°

115°115°

90°

x

y

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4. Is each pair of polygons similar?

a)

Compare the angles. A = ° and J = ° B = ° and K = ° C = ° and L = ° D = ° and M = ° E = ° and N = °

AEJN

EDNM

CDLM

Are the polygons similar? Circle YES or NO. Give 1 reason for your answer. _____________________________________________________________________________ b)

R = ° and V = ° S = ° and W = ° T = ° and X = ° U = ° and Y = °

SRWV RU

VY TU

XY ST

Are the polygons similar? Circle YES or NO. Give 1 reason for your answer. _____________________________________________________________________________

C D

L M

K NJB E

A

12

12

12 4 4

4

2.5 2.5

7.57.5

W

S R

T U

3.2 3.2

2

3

15

13 13

10

V

X Y

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Apply 5. If 2 polygons have the same side lengths, are they similar? Circle YES or NO.

Give 1 reason for your answer.

______________________________________________________________________________ 6. What scale factor should you use to reduce the side length of the square to 2.4 cm? Show how you know.

7. Find the dimensions of a similar parallelogram using a scale factor of 0.8.

The dimensions of the similar parallelogram are . 8. Tracy is building a dog house.

The dimensions on the blueprints are too small. She wants to enlarge the house by a scale factor of 3.5. Find the new dimensions of the dog house.

_______________________________________________________________________________

Math Link Add a polygon similar to the ones shown here to your drum design on page 187. What scale factor did you use?

6 cm

5 cm

3 cm

T S

Q R

30 cm

45 cm45 cm

50 cm

30 cm

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Chapter 4 Practice Test

For #1 to #4, choose the best answer.

1. What is the value of x if 1x = 8

32 ?

A 2 C 4

B 3 D 7

2. GHI is similar to KLM. Find the missing side length.

A 4

B 7

C 10 D 14 3. On a scale diagram, what does 1 in the scale of 1 : 5 represent?

A how many times larger the object is

B how many times smaller the object is

C the actual size D the diagram size 4. Which 2 shapes are similar?

A Figures 1 and 2

B Figures 1 and 3

C Figures 1 and 4 D Figures 2 and 3 Complete the statements in #5 and #6. 5. The amount by which an object is enlarged or reduced is called the (2 words). 6. An umbrella is 75 cm long. An image of the umbrella is 15 cm long.

The image is a of the original.

Figure 1 Figure 4Figure 3Figure 2

21

39 481613

x

G

H I

L M

K

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Short Answer 7. Draw a reduction that is half the size of this figure.

8. A pencil is 18 cm long. The drawing of the pencil is 4 cm long.

What is the scale factor?

Sentence: ______________________________________________________________________ 9. A western spruce budworm larva can grow to 32 mm long.

A drawing of the larva uses a scale of 1 : 1.5. How long is the drawing?

Sentence: ______________________________________________________________________

4 cm

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10. a) Measure the angles and sides of the 2 quadrilaterals.

original image

B

X W

ZY

A

DC

Quadrilateral ABCD: A = °, B = ° C = °, D = ° AB = cm, BC = cm CD = cm, DA = cm

Quadrilateral WXYZ: W = °, X = ° Y = °, Z = °

WX = cm, XY = cm YZ = cm, ZW = cm

b) Are the quadrilaterals similar? Circle YES or NO. Give 1 reason for your answer.

_____________________________________________________________________________

c) If they are similar, what is the scale factor?

Math Link: Wrap It Up! Finish your design project. a) Decide on a layout. Include the following 4 parts: an enlargement or reduction of your design from Math Link 4.2, page 195 your logo using a similar triangle from Math Link 4.3, page 207 the title of your design project surrounded by a similar polygon from Math Link 4.4, page 214 a scale diagram of your design b) Present your design using a poster or a multimedia presentation. Your presentation must include: your design and the scale you used an actual sample of your finished design project what you learned about scale diagrams and similarity

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