1.1 representing square numbers · 5 9 perfect square (or square number) the square of a whole...

Copyright © 2009 Nelson Education Ltd.8 Lesson 1.1: Representing Square Numbers

1.1Student book page 4

Representing Square Numbers

You will need

• counters

• a calculator

A. Calculate the number of counters in this square array.

� �

number of counters in a row

number of counters in a column

number of counters in

the array

25 is called a square number because you can arrange 25 counters into a 5-by-5 square.

B. Use counters and the grid below to make square arrays. Complete the table.

Number of counters in:

Each row or column Square array

5 25

4

9

4

1

Is the number of counters in each square array a square number?

How do you know?

Use materials to represent square numbers.

5 5 25

Yes

3

16

1

2

You can arrange those numbers of

counters into a square.

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Lesson 1.1: Representing Square Numbers 9Copyright © 2009 Nelson Education Ltd.

C. What is the area of the shaded square on the grid?

Area � s � s

� units � units

� square units

When you multiply a whole number by itself, the result is a square number.

Is 6 a whole number?

So, is 36 a square number?

D. Determine whether 49 is a square number.

Sketch a square with a side length of 7 units.

Area � units � units

� square units

Is 49 the product of a whole number multiplied by itself?

So, is 49 a square number?

E. Square 9 and 10.

9 � 9 � or 92 �

10 � 10 � or 102 �

Are both of these products square numbers?

How do you know?

F. Identify two square numbers greater than 100.

( )2 �

( )2 �

whole numbersthe counting numbers that begin at 0 and continue forever (0, 1, 2, 3, …)

square numberthe product of a whole number multiplied by itself

The “square” of a number is that number times itself.

For example, the square of 8 is 8 � 8 � .

8 � 8 can be written as 82 (read as “eight squared”).

64 is a square number.terms

s

s

They are each the result of

multiplying a whole number by itself.

7 7

36

Yes

Yes

Yes

Yes

64

81 81

100 100

49

11

12 144

121

6 6

Yes

Note: Answers to Part F may vary.


Copyright © 2009 Nelson Education Ltd.10 Lesson 1.2: Recognizing Perfect Squares

Use materials to represent square numbers.

Method 1: Using diagrams

The area of a square with a whole-number side length is a perfect square.

This 9-by-9 square has an area of square units, so is a perfect square.

Method 2: Using factors

PROBLEM A perfect square can be written as the product of 2 equal factors. Is 225 a perfect square?

Draw a tree diagram to identify the prime factors of 225.

Continue factoring until the end of each branch is a prime number.

The ones digit of 225 is , so 5 is a factor of 225.

The factor partner is 225 ÷ 5 � .

225 � 5 �

45 is not a prime number, because 9 � � 45.

45 � 9 �

9 is not a prime number, because 9 � 3 � .

9 � 3 �

The ends of the branches are now all prime numbers: 5, 5, 3, and 3. Write 225 as the product of these prime factors.

9 units

9 units

Use a variety of strategies to identify perfect squares.

225

5 �

9 �

�

perfect square (or square number)

the square of a whole number

prime factora factor that is a prime number

A prime number has only itself and 1 as factors.

The fi rst few prime numbers are 2, 3, 5, 7, 11, 13, 17, ….

1.2Student book pages 5–9

Recognizing PerfectSquares

terms

You will need

• a calculator

81 81

5

5

5

3

33 3

45

4545

5

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Lesson 1.2: Recognizing Perfect Squares 11Copyright © 2009 Nelson Education Ltd.

225 � 5 � � �

Group the prime factors to create a pair of equal factors.

225 � 5 � 5 � 3 � 3

� (5 � 3) � ( � )

� 15 � or ( )2

Is 225 the square of a whole number?

So, is 225 a perfect square?

PROBLEM Is 170 a perfect square?

Complete the tree diagram.

Write 170 as a product of prime factors.

170 � 17 � �

Can you group the prime factors to create a pair of equal factors?


Method 3: Look at the ones digit

The table shows the fi rst 10 perfect squares.

Circle the possible ones digits for a perfect square.

0 1 2 3 4 5 6 7 8 9

Look at the ones digit of 187. Could 187 be a perfect square?

A number with ones digit 0, 1, 4, 5, 6, or 9 may or may not be a perfect square.

Look at the table of the fi rst 10 perfect squares. Is 6 a perfect square? Is 36 a perfect square?

Refl ecting

� Show that 400 is a perfect square without using a drawing or tree diagram.

4 � (2)2, so 400 � ( )2

170

17 �

2 �

Whole number

Perfect square

0 0

1 1

2 4

3 9

4 16

5 25

6 36

7 49

8 64

9 81

10 100

35 3

35

15 15

Yes

5210

5

No

No

No

No Yes

20

Yes


Copyright © 2009 Nelson Education Ltd.12 Lesson 1.2: Recognizing Perfect Squares

Practising

3. The area of this square is 289 square units.

Is the side length a whole number?

So, is 289 the square of a whole number?


4. Show that each number is a perfect square.

a) 16

Sketch a square with an area of 16 square units.

Side length of the square � units

Is the side length a whole number?


b) 1764

Represent the factors of 1764 in a tree diagram.

Use divisibility rules to help you identify factors.

1764

2 �

2 �

9 �

� 7 �

Divisibility rules

• If the number is even, 2 is a factor.

• If the sum of the digits is divisible by 3, then 3 is a factor.


Write 1764 as a product of prime factors.

Group the factors to create a pair of equal factors.

� � or ( )2

Is 1764 a perfect square?

1764 � � � � � �

1764 � ( � � ) � ( � � )

17 units

17 units

Yes

Yes

Yes

Yes

Yes

4

882

441

49

3 37

2

2 2

2 3

3 3

3 7 7

7 7

42 42 42

Yes


Lesson 1.2: Recognizing Perfect Squares 13Copyright © 2009 Nelson Education Ltd.

7. Maddy started to draw a tree diagram to determine whether 2025 is a perfect square.

How can Maddy use what she has done so far to determine that 2025 is a perfect square?

Solution:

Write 2025 as the product of the factors at the ends of the branches in Maddy’s tree diagram.

2025 � � � �

These factors are not all prime numbers, but you can rearrange them to create a pair of equal factors.

2025 � ( � ) � ( � )

� � or ( )2

Is 2025 the square of a whole number?


8. Guy says: “169 is a perfect square when you read the digits forward or backward.”

Is Guy correct? Explain.

Solution:

Use the strategy of guess and test.

102 � , so 169 is than 102.

Try some squares greater than 102.

112 � 122 � 132 �


169 written backward is .

302 � , so 961 is than 302.

Try 312.

312 �


Explain why 169 and 961 are perfect squares.

2025

5 405

5 81

9 9

Hint

Use 32 � 9 to solve 302 � ■.

5 5

5 5

9

9 9

9

45 45 45

100

121 144 169

961

961

900 more

Yes

Yes

Yes

Yes

Each is the square of a whole number.

more


Square Roots of Perfect Squares

Copyright © 2009 Nelson Education Ltd.14 Lesson 1.3: Square Roots of Perfect Squares

A square has an area of 16 m2.

Determine the side length, s.

Area of a square � s2

Solve s2 � 16 m.

Which whole number multiplied by

itself equals 16?

So, s � m.

4 is called the square root of 16, because 42 � 16.

Using the square root symbol, 4 � �___

16 .

Determine √ − 144 by guess and test.

PROBLEM A square has an area of 144 m2. Determine the

side length, s � �____

144 .

Solve the related equation s2 � 144.

Use the strategy of guess and test.

102 � 202 �

144 is between 100 and 400.

So, s2 is between 102 and ( )2.

Is 144 closer to 100 or 400?

So, is s2 closer to 102 or 202?

Square 11. 112 �

Square 12. 122 �

s2 � 144, so s � �____

144 � m.

Use a variety of strategies to identify perfect squares.

square root ( √ − )

one of 2 equal factors of a number

For example, the square root of 25 is 5, because 52 � 25.

Using the symbol, �

___ 25 � 5.

Notice that

�___

25 � �___

25 � 25.

Area = 16 m2

s metres

A = 144 m2

s metres


term

You will need

• a calculator

4

4

100 400

121

144

12

20

closer to 100

closer to 102


Lesson 1.3: Square Roots of Perfect Squares 15Copyright © 2009 Nelson Education Ltd.

Determine √ − 225 by factoring.

This factor rainbow shows all the factors of 225.

Complete the table to show the factor partners.

The factor with an equal partner is the square root.

15 � 15 � 225

So, �____

225 � .

A perfect square is the square of a whole number.

Is 225 the square of a whole

number?


Determine √ − 256 using factors.

Complete the tree diagram of the factors of 256.

Then, write 256 as a product of prime numbers.

256 � � � � � � � �

Group these factors to create a pair of equal factors.

256 � ( � � � ) � ( � � � )

� �

� ( )2

So, �____

256 � .

Refl ecting

� How can you check your answer when you calculate the square root of a number?

Use �___

81 � 9 and 92 � 81 to explain.

Factors of 225

0 225

3

5

9

15

256

2 �

4 �

� � 2

� 2

4 �

�

1 3 5 9 15 15 25 45 75 225

225√

15

2

2 2 2 2 2 2 2 2

2 2 2 2 2 2 2128

32

2 2

8

2

2 2

16

16

16

Calculate 92. If it equals 81, then �___

81 � 9.

16

Yes

Yes

75

45

25

15

16


Copyright © 2009 Nelson Education Ltd.16 Lesson 1.3: Square Roots of Perfect Squares

Checking

2. Calculate.

a) �__

4

If the area of a square is 4 square units, then the side length of the square is units.

�__

4 �

b) �___

16


�___

16 �

c) �___

81


�___

81 �

Practising

3. a) Complete the factor rainbow.

441 ÷ 7 � , so 7 � � 441.

The factor partner for 7 is .

441 ÷ 9 � , so 9 � � 441.


441 ÷ 21 � , so � 21 � 441.


A = 4 m2

A = 16 m2

A = 81 m2

b) Is 9 the square root of 441?

Why or why not?

Which factor of 441 is the square root?

�____

441 �

c) Square the square root to check your answer.

( )2 �

1 3 7 9 21 147 441

2

2

4

4

9

9

63 63

63

49 49

49

21 21

21

21

21

21 144

No

It is not one of 2 equal factors of 441.

21 49 63


Lesson 1.3: Square Roots of Perfect Squares 17Copyright © 2009 Nelson Education Ltd.

15. Describe 2 strategies to calculate �____

324 .

Guess and test

102 � � �

202 � � �

Is 324 closer to 100 or 400?

So, �____

324 is closer to ( )2 than to ( )

2.

Guess the number whose square is 324.

Square the number.

( )2 �

If the number you guessed is not the square root, continue guessing until you identify �

____ 324 .

( )2 �

( )2 �

So, �____

324 � .

Factoring

Represent the factors of 324 in a tree diagram.

Use divisibility rules to identify factors of 324.

Write 324 as a product of prime numbers.

324 �

Group the factors to create a pair of equal factors.

324 � ( ) � ( )

� � or ( )2

So, �____

324 � .

324

�

Divisibility rules

• If the number is even, 2 is a factor.



10 10

20

20 10

17

17

18

18

324

18

289

20

100

400

closer to 400

3 � 3 � 3 � 3 � 2 � 2

3 � 3 � 2 3 � 3 � 2

18

9

3 3 9 4

2233

36

18

18

Note: A number of tree diagrams are possible for 324.

�

� �

�



Estimating Square Roots

Copyright © 2009 Nelson Education Ltd.18 Lesson 1.4: Estimating Square Roots

If a number is not a perfect square, you can estimate its square root.

Estimate √ − 10 by comparing it to roots of perfect squares.

Estimate the side length of a square with an area of 10 square units.

Step 1: On the grid paper, draw a 2-by-2 square, a 3-by-3 square, and a 4-by-4 square.

Complete the table. ( )2

Estimate the square root of numbers that are not perfect squares.

You will need

• a calculator

Square Side length (s)

Area (s2)

Side length ( �

__ A )

2-by-2 2 4 �__

4

3-by-3

4-by-4

Step 2: Use the side lengths of the squares you drew to estimate �

___ 10 .

�__

4 � 2

�__

9 �

�___

10 � ■

�___

16 �

Step 3: Determine �___

10 to 2 decimal places.

Square 3.1. 3.12 � (too low)

Square 3.2. 3.22 � (too high)

The square of 3.2 is close to 10.

So, �___

10 is approximately .

�___

10 is between and , and closer to than .

So, �___

10 is not a whole number.

�____

�_____

4 16

9 9

16

3

3

4

9.61

3.2

10.24

3 4

43

17


Lesson 1.4: Estimating Square Roots 19Copyright © 2009 Nelson Education Ltd.

Determine square roots using a calculator.

Calculators have a square root button, �__

.

Different calculators use different key sequences.

PROBLEM Calculate �___

10 . Round the result to 3 decimal places.

Try each sequence below.

�__

10 � or 10 �__

�

Circle the sequence above that works with your calculator.

�___

10 ��

PROBLEM Calculate 0.5 �____

300 .

0.5 �____

300 means the same as 0.5 � �____

300 .

First, estimate 0.5 �____

300 . Use mental math.

Now, calculate 0.5 �____

300 . Use a calculator.

Round the result to 4 decimal places.

0.5 �____

300 ��

Refl ecting� 8.6603 and 8.6602 are both the same distance from

8.66025. Why is it more likely that you chose 8.6603 when rounding 8.66025 to 4 decimal places?

� �___

10 �� 3.162, but 3.1622 � 10. Why is this?

0.5 � �__

300 � or 0.5 � 300 �__

�

The symbol “�� ” means “approximately equal to.”

When you round a number, the answer is an approximation.

Use “�� ” instead of “�” when you write your answer.

Communication Tip

Step 1: Use �____

100 � 10 and �____

400 � 20 to estimate �____

300 .

�____

300 ��

Step 2: 0.5 �____

300 is half of �____

300 .

Halve your estimate in step 1.

0.5 �____

300 ��

Hint

The symbol � means “is not equal to.”

3.162

17

8.5

8.6603

The convention—how it’s usually done—is to round up.

3.162 is an approximation of �___

10 , so the

square of 3.162 is close to but not equal to 10.

Note: Keystroke sequence circled may vary.


Copyright © 2009 Nelson Education Ltd.20 Lesson 1.4: Estimating Square Roots

Practising

4. Estimate to determine whether each answer is reasonable.

Correct any unreasonable answers using the square root key on your calculator.

a) �___

10 �� 3.2

The area of a square with side length 3 units is square units.


Is 3.2 a reasonable estimate for the square root of 10?

Use your calculator to check.

�___

10 ��

b) �___

15 �� 4.8



Is 4.8 a reasonable estimate for the square root of 15?

Use your calculator to check.

�___

15 ��

5. Calculate each square root to 1 decimal place.

Choose one of your answers and explain why it is reasonable.

a) �___

18 �� c) �___

38 ��

b) �___

75 �� d) �____

150 ��

�� is reasonable because

.

Hint

Use the correct key sequence for your calculator.

For example, to calculate �

___ 10 , use

either 10 or

10 .

�____

�

9

16

Yes

3.2

16

25

No

3.9

4.2

38

6.2

8.7

6.2

between 62 and 72

38 is

12.2

Note: Students may choose to justify any one of 5 a), b), c) or d).

�

�__

�__


Lesson 1.4: Estimating Square Roots 21Copyright © 2009 Nelson Education Ltd.

8. Tiananmen Square in Beijing, China, is the largest open “square” in any city in the world. It is actually a rectangle of 880 m by 500 m.

a) What is the approximate side length of a square with the same area as Tiananmen Square?

Solution:

What is the area of Tiananmen Square?

Area � length � width

� m � m

� m2

What is the side length of a square with this area?

�� m

b) 6002 �

7002 �

Explain how you know your answer to part a) is reasonable.

�� m is reasonable because

.

10. Estimate the time an object takes to fall from each height using this formula:

time (s) �� 0.45 �______

height (m)

Record each answer to 1 decimal place.

a) 100 m

time �� 0.45 �

�� s

b) 200 m

time ��

�� s

c) 400 m

time ��

�� s

�__________

�__________

�_____

�_____

�_____

880

440 000

440 000

440 000

440 000 is between 6002 and 7002

663

663

4.5

0.45

6.4

9

0.45

360 000

490 000

500

400

200

100


1.5Student book page 24

Exploring Problems Involving Squares and Square Roots

Copyright © 2009 Nelson Education Ltd.22 Lesson 1.5: Exploring Problems Involving Squares and Square Roots

You will need

• square tiles

• a calculator

Create and solve problems involving a perfect square.

How many tiles are in each diagram?

3 � 3 � 9

(3)2 � 9 tiles

� � �

( )2 � � tiles

� � �

( )2 � � tiles

PROBLEM Joseph had 12 tiles. He made a square with some tiles and had 3 tiles left over. What is the side length of the square?

Solve s 2 � 3 � 12.

What number added to 3 makes 12?

What is the square root of that number?

So, ( )2 � 3 � 12. s � tiles

PROBLEM There are 104 tiles. What is the side length of the square?

Let the variable s represent the unknown side length.

s2 � 4 � 104

s 2 � 4 � � 104 �

s 2 �

So, s � �____

100 � .

?

?

Write an equation.

Subtract 4 from each side of the equation to isolate the variable.

Side length � tiles

3 72 2

2 3 7

3 3 1

1

10

103

9

3

33

4 4

100

10 10


Lesson 1.5: Exploring Problems Involving Squares and Square Roots 23Copyright © 2009 Nelson Education Ltd.

PROBLEM A game is played with a deck of 52 square cards.

You deal the cards in equal rows and equal columns to form a square. Three cards are left over and not used.

What is the side length of the square of cards?

Solution:

Draw a diagram similar to the ones on the previous page to represent the problem.

Choose a variable to represent the side length.

Write an equation to represent the situation.

( )2 � �

Hint

Use one of these problem-solving strategies:

• Make a model

• Work backward

The side length of the square of cards is cards.

PROBLEM Create a problem that uses a square number and another whole number.

Solve the problem.

Solve the equation.

s

s 3 52

7

Answers may vary. For example: Mark has 40 tiles. He

makes a square with tiles and has 4 left over. What is the

length of the square?

s2 � 3 � 52

s2 � 52 � 3 � 49

s � �___

49 � 7

s2 � 4 � 40

s2 � 40 � 4 � 36

s � �___

36 � 6

The side length of the square is 6 tiles.

?

?



The Pythagorean Theorem

Copyright © 2009 Nelson Education Ltd.24 Lesson 1.6: The Pythagorean Theorem

You will need

• counters

• cutout 1.6

Model, explain, and apply the Pythagorean theorem.

On each right triangle

• label the hypotenuse c

• label the smallest leg a

• label the other leg b

right triangle

a triangle with 1 right angle (90�)

The hypotenuse is the longest side of a right triangle, the side opposite the right angle.

The 2 shorter sides are called the legs.

Hint

To calculate 92 using a calculator:

9 x2 �

Pythagorean Theorem

In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the 2 legs.

c2 � a2 � b2 or a2 � b2 � c2

Use the Pythagorean theorem to determine if the triangle below is a right triangle.

length of hypotenuse: c �

length of shortest leg: a �

length of other leg: b �

Check if a2 � b2 � c2.

� �

� �

�

Is a2 � b2 � c2 true for this triangle?

So, is the triangle a right triangle?

15 m12 m

9 m

ca

b

leg

leg

hypotenuse

terms

9

12

92 122 152

22514481

225 225

Yes

Yes

15

b

a

cba

c


Lesson 1.6: The Pythagorean Theorem 25Copyright © 2009 Nelson Education Ltd.

Is the Pythagorean theorem true for all types of triangles?

Use Cutout 1.6, which shows 2 acute triangles, 1 right triangle, and 2 obtuse triangles. Each triangle has one side 60 mm long and another side 80 mm long.

A. Measure the third side of each triangle to the nearest millimetre. Record the length on the cutout page.

B. Write the missing side lengths (a, b, or c) in the table.

Calculate the missing squares (a2, b2, or c2).

Hint

In the table above, �A and �B are acute, �C is a right triangle, and �D and �E are obtuse.

Triangle a b a2 b2 a2 � b2 c c2 Comparison

A 60 80 3600 6400 10 000 a2 � b2 c2

B 60 3600 80 6400 a2 � b2 c2

C 60 80 3600 6400 10 000 a2 � b2 c2

D 60 80 3600 6400 10 000 a2 � b2 c2

E 60 3600 80 6400 a2 � b2 c2

C. For each triangle, calculate a2 � b2 and c2.

Compare the 2 values.

Record each comparison in the table. Use <, �, or >.

D. Is the Pythagorean theorem true for all types of

triangles? Explain.

Refl ecting

� Match the type of triangle with the equation or inequality.

Acute triangle a2 � b2 < c2

Right triangle a2 � b2 > c2

Obtuse triangle a2 � b2 � c2

Hint

� greater than � less than

32

60 3600

1024 4624

No

7200

87

100

118 13 924

10 000

7569

The theorem is only true for right triangles.

>

=

<

<

<


Copyright © 2009 Nelson Education Ltd.26 Lesson 1.6: The Pythagorean Theorem

Practising

3. Herman formed a triangle with grid-paper squares.How can you tell that he formed a right triangle?

Solution:

The side lengths of the 3 squares and the 3 side lengths of the triangle are the same.

If c2 � a2 � b2, then the triangle is a right triangle.

c is the length of the longest side: units

a and b are the other 2 side lengths: and units

c2 � a2 � b2

( )2 � ( )

2 � ( )

2

� �

�

Is the triangle a right triangle?

5. A Pythagorean triple is any set of 3 whole numbers, a, b, and c, for which a2 � b2 � c2.

Show that each set of numbers is a Pythagorean triple.

a) a � 5, b � 12, and c �13

a2 � b2 � c2

( )2 � ( )

2 � ( )

2

� �

�

b) a � 7, b � 24, and c � 25

( )2 � ( )

2 � ( )

2

� �

�

c) a � 9, b � 40, and c � 41

( )2 � ( )

2 � ( )

2

� �

�

5

3

4

5

5

25

7

49

9

81 1600

1681

1681

1681

25

25 25

3

12

144

169

24

576

625

40

16

4

13

169

169

25

625

625

41

9

Yes


Lesson 1.6: The Pythagorean Theorem 27Copyright © 2009 Nelson Education Ltd.

8.0 cm

9.0 cma

7. About how far would a hockey puck travel when shot from one corner of the rink (at the goal line) to the opposite corner (at the goal line)?

Think of the rink as a rectangle divided into 2 right triangles.

Label the sides of the shaded right triangle a, b, and c in the diagram above.

a � m b � m

Use the Pythagorean theorem to calculate the distance, c, travelled by the puck.

Round your answer to the nearest whole number.

Step 1: Step 2:

c2 � a2 � b2 c � �__

c2

� ( )2 � ( )

2 �

� � �� m

�

The puck would travel approximately m.

9. Calculate the unknown side to 1 decimal place.

a2 � b2 � c2

c � cm b � cm

Step 1: Step 2:

a2 � b2 � c2 a � �___

a2

a2 � ( )2 � ( )

2 �

a2 � � �� cm

a2 � �

�

Hint

The original measurements are precise to a tenth of a centimetre, so round your answer the nearest tenth of a centimetre.

Hint

Check a square root by multiplying it by itself. �

__ n �

__ n should

be close to n.

26 m

54 m

path of puck

�______

�___

26 54

26 54

2916676

3592

60

60

9.0 8.0

8

64

81

17

64

9

81 4.1

3592

17

26 m

54 m

path of puck

a

b

c



Solve Problems Using Diagrams

Copyright © 2009 Nelson Education Ltd.28 Lesson 1.7: Solve Problems Using Diagrams

You will need

• a calculator

• a ruler

Use diagrams to solve problems about squares and square roots.

Joseph is building a model of the front of a Haida longhouse.

He wants the model to have the measurements shown on the illustration.

How can Joseph calculate length c (at the top of the model)?

Solve a problem by identifying a right triangle

1. Understand the Problem

Draw a diagram that includes all you know about the model.

c represents the length you want to know.

Complete the diagram.

cc

cm

cm

cm

30 cm30 cm

9 cm

21

60

30


Lesson 1.7: Solve Problems Using Diagrams 29Copyright © 2009 Nelson Education Ltd.

2. Make a Plan

Draw a line on your diagram to connect the 2 dots at the tops of the sides of the model. This will make 2 right triangles at the top of the model.

The base of each triangle is half of 60 cm, or cm.

The height of the triangles is the height of the whole model minus the height of the side:

30 cm � cm � cm

Write these lengths on your diagram.

Now you know 2 sides of each triangle. Which theorem can you use to calculate the length of the unknown side

of the triangle, c?

3. Carry Out the Plan

Write the equation that relates the sides of a right triangle. c2 � �

Side c in the right triangle is unknown.

The lengths of the other 2 sides are known.

Use one of these lengths for a and one for b.

a � cm b � cm

Calculate c.

Step 1: Step 2:

c2 � a2 � b2 c � �__

c2

� ( )2 � ( )

2 �

� � �� m

�

c is approximately cm long.

Refl ecting

� How did drawing a diagram help solve the problem?

Hint

The hypotenuse (the longest side) in a right triangle is always labelled c.

�_____

30

21

a2

9

9

81

981

30

31.3900

31

It helped me see where a right triangle could be drawn

and it made it easy to keep track of the lengths.

30

b2

9

Pythagorean theorem

981


Copyright © 2009 Nelson Education Ltd.30 Lesson 1.7: Solve Problems Using Diagrams

Practising

5. The diagonal of a rectangle is 25 cm.

The shortest side is 15 cm.

What is the length of the other side?

Solution:

Draw a rectangle.

Write cm beside the shortest side.

Draw a diagonal on the rectangle.

Write cm beside the diagonal.

What does the problem ask you to determine?

Is the unknown length a side of a right triangle?

Shade one of the right triangles formed by the diagonal. The hypotenuse, c, � cm. Call the shortest side of the triangle a, so a � cm. The unknown side is b.

Use the Pythagorean theorem to calculate the other side length.

Step 1: Step 2:

a2 � b2 � c2 b � �___

b2

( )2 � b2 � ( )

2 �

� b2 � �

b2 � �

�

The length of the other side is cm.

diagonal

In a 2-D shape, a diagonal can join any 2 vertices that are not next to each other.

term

diagonal

diagonals

�_____

15

25

25

15

15

225

25

625

625 225

400

20

20

Yes

the length of the other side of the rectangle

400

15 cm25 cm

c


Lesson 1.7: Solve Problems Using Diagrams 31Copyright © 2009 Nelson Education Ltd.

6. Fran cycles 6.0 km north along a straight path.

She then rides 10.0 km east.

Then she rides 3.0 km south.

Then she turns and rides in a straight line back to her starting point.

What is the total distance of her ride?

Solution:

The fi rst 3 legs of Fran’s ride have been drawn.

Draw the path that takes Fran back to her starting point.

Draw a line on the diagram to divide the shape into a rectangle and a right triangle.

Label the hypotenuse of the triangle c.

b � other side of Δ

� long side of rectangle

� km.

Let a � short side of Δ

� 6 km � km

� km.

Use the Pythagorean theorem to calculate c.

c2 � a2 � b2 c � �___

c2

� ( )2 � ( )

2 �

� � ��

�

The total distance of Fran’s ride is

6.0 km � 10.0 km � 3.0 km � km � km.

Hint

The original measurements are precise to a tenth of a centimetre, so round the value of c to the nearest tenth of a centimetre.

N

S

W E

6.0 km

10.0 km

3.0 km

START

�_____

3

3 10

3

109

10.4 29.4

9

10

100 10.4

109

c


Acute triangles

(all 3 angles less than 90°)

Right triangle

(1 angle is 90°)

Obtuse triangles

(1 angle is greater than 90°)

Cutout 1.6

B

a = 60 mm

c = 80 mm

a = 60 mm

b = 80 mmA

c = mm

b = mm

a = 60 mm

b = 80 mm

C

c = mm

b = 60 mm

c = 80 mm

a =

E60 mm

80 mm

D

mm

mm

60

87

100

32

118


1.1 representing square numbers · 5 9 perfect square (or square number) the square of a whole...

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