Page 1
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.
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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?
So, is 170 a perfect square?
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
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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?
So, is 289 a perfect square?
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?
So, is 16 a perfect square?
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.
• If the sum of the digits is divisible by 9, then 9 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
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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?
So, is 2025 a perfect square?
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 �
Is 169 a perfect square?
169 written backward is .
302 � , so 961 is than 302.
Try 312.
312 �
Is 961 a perfect square?
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
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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
1.3Student book pages 10–15
term
You will need
• a calculator
4
4
100 400
121
144
12
20
closer to 100
closer to 102
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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?
Is 225 a perfect square?
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
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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
If the area of a square is 16 square units, then the side length of the square is units.
�___
16 �
c) �___
81
If the area of a square is 81 square units, then the side length of the square is units.
�___
81 �
Practising
3. a) Complete the factor rainbow.
441 ÷ 7 � , so 7 � � 441.
The factor partner for 7 is .
441 ÷ 9 � , so 9 � � 441.
The factor partner for 9 is .
441 ÷ 21 � , so � 21 � 441.
The factor partner for 21 is .
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
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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.
• If the sum of the digits is divisible by 3, then 3 is a factor.
• If the sum of the digits is divisible by 9, then 9 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.
�
� �
�
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1.4Student book pages 16–20
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
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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.
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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.
The area of a square with side length 4 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
The area of a square with side length 4 units is square units.
The area of a square with side length 5 units is square units.
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).
�
�__
�__
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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
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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
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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.
?
?
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1.6Student book pages 26–31
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
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Page 18
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.
>
=
<
<
<
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Page 19
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
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Page 20
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
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1.7Student book pages 32–35
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
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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
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Page 23
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
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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
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Page 25
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
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