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CHAPTER

3 Pythagorean Relationship

GET READY 104

Math Link 106

3.1 Warm Up 107

3.1 Squares and Square Roots 108

3.2 Warm Up 117

3.2 Exploring the Pythagorean Relationship 118

3.3 Warm Up 125

3.3 Estimating Square Roots 126

3.4 Warm Up 132

3.4 Using the Pythagorean Relationship 133

3.5 Warm Up 139

3.5 Applying the Pythagorean Relationship 140

Chapter Review 147

Practice Test 151

Wrap It Up! 154

Key Word Builder 155

Math Games 156

Challenge in Real Life 157

Answers 158

104 MHR ● Chapter 3: Pythagorean Relationship

Name: _____________________________________________________ Date: ______________

The factors are the same.

What 2 numbers can you multiply to

get 60?

Factors

factor ● any of the numbers that form a product when multiplied together ● example: 1 × 10 = 10 2 × 5 = 10 factors factors prime factor ● a factor that is a prime number ● example: 2 × 5 = 10 prime factors

A factor tree shows the prime factors for a composite number. Here are 2 possible factor trees for the number 48.

× ×

×

48

6 8

32 × 42

× ×2 3 × 22×2

× ×

× ×

48

4 12

22

2 2 × 22

×

×

43

3

×

Write 48 as a product of its prime factors: 48 = 2 × 2 × 2 × 2 × 3

1. Use a factor tree to find the prime factors.

a) 24

4 ×

b) 60

2 ×

Perimeter and Area

perimeter area ● the distance around a shape ● the space a 2-D shape covers

P = 2l + 2w A = length × width P = (2 × 4) + (2 × 3) A = 4 × 3 P = 8 + 6 A = 12 P = 14 The area is 12 cm2. The perimeter is 14 cm.

4 cm

3 cm

A prime number has only 2 factors,

1 and itself.

Get Ready ● MHR 105

Name: _____________________________________________________ Date: ______________

5 + 7 = 22x – 7 = – 7

5 = 15x

2.

11 m

3 m

a) Find the perimeter. b) Find the area. ← Formula →

← Substitute →

← Solve → Solving Equations

equation ● 2 expressions joined by an equal sign ● examples: 2x + 1 = 5 and 6w = 24 variable ● a letter in an equation that holds the place of a number

To solve an equation: ● get the variable by itself on one side of the equal sign ● use the reverse order of operations to undo the operations (– and +, then × and ÷)

5x + 7 = 22

5x + 7 – 7 = 22 – 7 Subtract 7 from both sides to undo adding 7.

5 15 = 5 5x Divide both sides by 5 to undo multiplying by 5.

x = 3 3. Solve for x.

a) 3x = 18

3 18 = x

x =

b) 4x + 1 = 13 4x + 1 − = 13 − 4x =

4 12 = x

x =


Name: _____________________________________________________ Date: ______________

Game Design Playing Leader is a peg board game that uses squares. Players move 14 pegs 1 hole at a time along the lines. Player 1 has 13 black pegs. They can move left, right, or down. Player 2 has the grey leader peg. It can move left, right, up, or down. Start with all of the pegs at the top of the board. Player 1 uses the black pegs to try to surround the leader peg so that it cannot move.

• Playing Leader board • coloured counters

Player 2 tries to capture the black pegs by jumping the leader peg over it to an empty space. Remove captured pegs from the board. a) Start at a hole at the top of the board. Move down 1 space at a time. What is the greatest number

of moves you can make with 1 peg in a vertical (↕) straight line?

b) The horizontal (↔) or vertical (↕) distance between 2 peg holes is 5 cm.

● Use a coloured pencil to outline all of the 5 cm × 5 cm squares. How many are there? ● Use a different coloured pencil to outline all of the 10 cm × 10 cm squares. How many are there? ● How many 5 cm and 10 cm squares are there?

+ =

c) Find the area of the game board.

● What is the area of a 5 cm × 5 cm square? × = ● How many 5 cm × 5 cm squares are on the board? ● What is the total area of the board? × =

leader peg

The squares cannot cross each other.

3.1 Warm Up ● MHR 107

Name: _____________________________________________________ Date: ______________

Use a multiplication table.

3.1 Warm Up 1. List the factors of each number.

a) 8 1 × = 8 2 × = 8 Factors of 8:

b) 16 1 × = 2 × = × = Factors of 16:

c) 20

Factors of 20:

d) 32

Factors of 32:

2. Calculate.

a) 52

= 5 × 5 =

b) 22

= × =

c) 82 d) 122

3. Solve using mental math.

a) 25 = 5 ×

b) 4 = 2 ×

c) 64 = × 8

d) 100 = × 10


Name: _____________________________________________________ Date: ______________

2 pairs of 3

3.1 Squares and Square Roots

Working Example 1: Identify Perfect Squares

a) What is the prime factorization of 24 and 81?

prime factorization ● a number written as the product of its prime factors ● example: 2 × 2 × 3 = 12

Solution

Use a factor tree.

24

6

81

9

× 2 ×

×

× 3 ××

×

×

The prime factorization of 24 is × × 2 × .

The prime factorization of 81 is × × 3 × .

b) Is 24 or 81 a perfect square?

perfect square ● a number that is the square of a whole number ● examples: 12 = 1 × 1 = 1 22 = 2 × 2 = 4

32 = 3 × 3 = 9 42 = 4 × 4 = 16 So, 1, 4, 9, 16, … are perfect squares.

Solution

To be a perfect square, there must be pairs of each prime factor in the factor tree. 81 = 3 × 3 × 3 × 3 There are four factors of . So, 81 is a perfect square. 24 = × × × There are three factors of 2, and factor of 3. 24 is not a perfect square, because the factors 2 and 3 each appear an number

of times. (even or odd)

4

4

3.1 Squares and Square Roots ● MHR 109

Name: _____________________________________________________ Date: ______________

c) Draw the square and label the side length for 81. Solution

To find the side length of a perfect square, look at the factor tree. The product of the prime factors is 81 = 3 × 3 × 3 × 3 . 81 = 9 × 9 So, the square will be units on each side.

Draw a factor tree for each number.

a) 45 b) 100

Which number is not a perfect square? Explain how you know. _______________________________________________________ __________________________________________________________________________

9

9


Name: _____________________________________________________ Date: ______________

Working Example 2: Determine the Square of a Number

Find the area of a square picture with a side length of 13 cm. Solution A = s2 A = 2

A = 13 × A = The area is cm2.

● read 132 as thirteen squared ● 132 = 13 × 13

Find the area of a square with a side length of 16 mm.

Draw a diagram: Use the formula: Formula → A = Substitute → A = Solve → A = × A =

Sentence: ___________________________________________________________________

13 cm

13 cm

area of a square = side length × side length A = s × s A = s2


Name: _____________________________________________________ Date: ______________

What number multiplied by itself equals 144?

Working Example 3: Determine the Square Root of a Perfect Square

The square case for a computer game has an area of 144 cm2. What is the side length of the case? Solution

Method 1: Use Inspection A = s × s 144 = 12 × The square root of 144 is , or 144 = . The side length is cm.

square root ● a number that when multiplied by itself equals a given value ● 6 is the square root of 36 because 6 × 6 = 36 ● the symbol for square root is

● read the 6 as the square root of six Method 2: Use Guess and Check 10 × = 100 Too low 13 × = 169 Too high 12 × = 144 Correct! = 144 The side length is cm.

Method 3: Use Prime Factorization

144

2 72

2 8 9

2 2 4 3 3

2 2 2 2 3 3

×

× ×

×

× × × × ×

× × ×

The prime factorization of 144 is 2 × 2 × × × × . Rearrange the prime factors into 2 equal groups: 144 = 2 × 2 × 3 × 2 × 2 × 3 144 = ×

144 = The side length is cm.

144 cm2


Name: _____________________________________________________ Date: ______________

Find the length of the side of a square that has an area of 196 cm2. a) Use Guess and Check.

Side Length (Guess) Check Too High or Too Low

10 cm 10 × 10 = 100 Too low

11 cm 11 × 11 =

Sentence: _____________________________________________________________

b) Use prime factorization to find the side length of the square.

Sentence: _____________________________________________________________

196

2 98×


Name: _____________________________________________________ Date: ______________

1. a) What does it mean to square a number?

__________________________________________________________________________

b) Explain how to square the number 7. __________________________________________________________________________

2. How would you use a factor tree to find the square root of 225?

_____________________________________________________________________________ _____________________________________________________________________________

3. A shape has an area of 4 square centimetres (cm2).

a) Draw the factor tree for 4. b) Is 4 a perfect square? Circle YES or NO.

Why or why not?

_________________________________ _________________________________

c) Draw a square with an area of 4 cm2 and label its side length.


Name: _____________________________________________________ Date: ______________

A = s2

4. A shape has an area of 36 square metres (m2).

a) Draw the factor tree for 36. b) Is 36 a perfect square? Circle YES or NO. Why or why not? _________________________________

_________________________________

c) Draw a square with an area of 36 m2

and label its side length.

5. What is the square of each number?

a) 3 32 = × =

b) 11

6. Find the area of a square with each side length.

a) 10 units b) 16 units

← Formula →

← Substitute →

← Solve → 7. Find the square root.

a) 81 b) 64

8. What is the side length of each square?

a) 49 mm2

b)

100 cm2

Find 49.


Name: _____________________________________________________ Date: ______________

9. A fridge magnet has an area of 54 mm2.

Is 54 a perfect square? Use prime factorization to find the answer.

Sentence: ________________________________ ________________________________________

10. A square floor mat for gymnastics has a side length of 14 m.

What is the area of the floor mat in square metres?

Formula →

Substitute →

Solve →

Sentence: ____________________________________________________________________ 11. Adam has instructions for building a shed.

One page of the instructions is not very clear.

a) What is the area of the rectangle?

Area = ×

= ×

=

b) What is the side length of the square?

Area of square = Area of square = s2

= s2

2= s

= s

Area of rectangle = area of square. Use your answer from part a).

9 m

area of rectangle = area of square

4 m

54

2 × 27

2 ××

2 × ××


Name: _____________________________________________________ Date: ______________

Chess is played on a square board with 32 white squares and 32 dark squares. a) The total number of squares on a chessboard is + = . b) You decide to make your own chessboard. You use a piece of wood that is 32 cm × 40 cm. Each square on the chessboard must be a perfect square. Each square must be larger than 8 cm2. Complete the table to find all of the possible dimensions of the chessboard. Round your answers to the nearest hundredth (2 decimal places).

An example is done for you.

Dimensions of Square

Board (cm)

Area of

Board (cm2)

Number of

Squares

Area of Each Square (cm2)

Perfect Square?

32 × 32 32 × 32 = 1024 64 1024 ÷ 64 = 16 16 4=

Yes

31 × 31 64

30 × 30 64

29 × 29 64

28 × 28 64

27 × 27 64

c) List all of the possible dimensions for the chessboard.

____________________________________________________________________________

32 cm

40 cm


Name: _____________________________________________________ Date: ______________

3.2 Warm Up 1. a) Find the squares of 1 to 4. 12 = 22 = 32 = 42 =

b) What are the first 4 perfect squares?

2. Use a factor tree to find the prime factorizations of 48 and 64.

Circle the number if it is a perfect square.

48

6 × 8

×× ××

4× ××

64

8 ×

3. What is the area of a square picture with a side length of 13 cm?

Formula →

Substitute →

Solve → 4. Find the value of each square.

a) 72 = b) 102 = 5. Find the square root.

a) 36 = b) 121 =

5 × 5 = 25, so 25 is a perfect square.

A = s2


Name: _____________________________________________________ Date: ______________

Which 2 areas add together to equal the other area?

hypotenuse

3.2 Exploring the Pythagorean Relationship

Working Example 1: Describe Relationships in Right Triangles

right triangle • a triangle with a right angle (90°) • the right angle is marked with a small square • the 2 shorter sides are called the legs • the longest side is called the hypotenuse

a) What is the area of each square?

r = 5 cmp =3 cm

q = 4 cm

Solution

p = 3 cm q = 4 cm r = 5 cm A = s2

A = s2 A = s2

A = 32 A = A = A = × A = × A = × A = cm2 A = cm2 A = cm2

b) Which side is the hypotenuse of the triangle?

Solution

Side is the hypotenuse. c) Write an addition statement to show how the areas are related.

Solution

p = 9 q = 16 r = 25=+

+ = 25

hypotenuse

leg

leg

3.2 Exploring the Pythagorean Relationship ● MHR 119

Name: _____________________________________________________ Date: ______________

d) Describe the relationship between the side lengths of the triangle. Use words and symbols.

Solution

Words: The sum of the areas of the squares attached to legs p

and equals the area of the square attached to hypotenuse r. Symbols: p2 + 2 = r2

Pythagorean relationship • the relationship between the lengths of the sides of a right triangle • a2 + b2 = c2, where c is the hypotenuse

The sides of a right triangle are 9 cm, 12 cm, and 15 cm.

a) Sketch the triangle. Draw a square on each side of the triangle.

b) What is the area of each square?

p = 9 A = s2

= 92

= × = cm2

q = 12 r = 15

c) Write an addition statement using the areas of the 3 squares.

+

=

+ =

r = 5 cmp =3 cm

q = 4 cm

a2

c2b2


Name: _____________________________________________________ Date: ______________

Working Example 2: Identify a Right Triangle

A triangle has side lengths of 5 cm, 7 cm, and 9 cm.

a) What are the areas of the 3 squares that can be drawn on the sides of the triangle?

Solution

A = 5 × 5 A = 7 × 7 A = × A = cm2 A = cm2 A = cm2

b) Is the triangle a right triangle? Explain.

Solution

Step 1: Add the areas of the 2 smaller squares.

+A = 25 A = 49

25 + 49 =

Step 2: Does the sum of the areas of the smaller squares equal the area of the large square?

74 cm2 ≠ 81 cm2

The triangle a right triangle. (is or is not)

● ≠ means is not equal to

A triangle has side lengths of 12 cm, 16 cm, and 20 cm.

a) What are the areas of the 3 squares that can be drawn on the sides of the triangle?

A = 12 × 12 A = × A = × A = cm2 A = cm2 A = cm2

b) Is the triangle a right triangle? Explain.

Area of 2 smaller squares = +

=

_________________________________________________________________________

_________________________________________________________________________


Name: _____________________________________________________ Date: ______________

1. Describe in words the relationship among the areas of the 3 squares.

_________________________________________________

_________________________________________________

_________________________________________________

_________________________________________________

2. For this triangle, Kendra wrote the Pythagorean relationship as r2 = p2 + q2. Is she correct? Circle YES or NO. Give 1 reason for your answer.

___________________________________________________________________________

___________________________________________________________________________

3. What are the areas of the 3 squares?

e = f = g = A = × A = × A = × A = mm2 A = A =

225 cm2

64 cm2

289 cm2

p

q

r

e = 30 mmg = 50 mm

f = 40 mm


Name: _____________________________________________________ Date: ______________

4. A right triangle has side lengths of 9 mm, 12 mm, and 15 mm.

a) Draw a square on each side of the triangle.

b) What is the area of each square?

Square 1 Square 2 Square 3

c) Is this a right triangle? Show your work.

Area of 2 small squares = area of large square + = The triangle a right triangle. (is or is not)

5. a) Write an addition statement using the areas of the 3 squares.

Area of 2 small squares = area of large square + = b) What is the side length of each square?

Square 1 Square 2 Square 3 A = 25 cm2 A = A = = 25 = = = cm = =

c) Use symbols to describe the relationship between the side lengths of each square.

Use = or ≠.

144 cm2

169 cm2

25 cm2


Name: _____________________________________________________ Date: ______________

Use the Pythagorean relationship.

6. Use the Pythagorean relationship to find the unknown area of each square.

a)

20 cm2

cm2

32 cm2

b)

100 mm2

576 mm2

mm2

Area of 2 small squares = + = Area of 2 squares = area of large square The area of the large square is .

7. Construction workers are digging a hole for a swimming pool. They want to check that the angle they have dug is 90°. The diagonal of the rectangle is also the hypotenuse of a right angle triangle. The diagonal measures 9.5 m. Is the angle 90°? Explain. Sentence: ___________________________________________________________________

?

6 m

8 m

diagonal


Name: _____________________________________________________ Date: ______________

Which pieces complete this Pythagorean puzzle? The missing pieces are 3 squares and 1 right triangle.

50 cm2

25 cm24 cm

5 cm41 cm2

62 cm2

5 cm

6 cm5 cm

5 cm4 cm

3 cm

16 cm2

B

DA

C

6 cm

6 cm

#5

#2

#3 #7

#6#9

#10

#8

#4

#1

36 cm2

a) Fill in the table to help you find the missing pieces of the puzzle. Use each piece only once. Triangle 2 is shown in the first row as an example.

Triangle

Length of Triangle Side (cm)

Area of Square (cm2)

Square With This

Area

Length of Second

Triangle Side (cm)

Area of Square on

Second Side (cm2)

Square With This Area

Sum of the Areas

of the Squares

(cm2)

Square With This Sum

5 cm

6 cm

#2

5 5 × 5 = 25

25 cm2#8

6 6 × 6 = 36

6 cm

6 cm

#7

25 + 36 = 61

none

5 cm5 cm #4

4 cm5 cm

#5

4 cm

3 cm#9

b) Circle the triangle that fits with 3 different square pieces to complete the puzzle. Triangle 4 Triangle 5 Triangle 9 c) Which 3 square pieces fit together with this triangle? Square Square Square


Name: _____________________________________________________ Date: ______________

Find the square root of each area.

3.3 Warm Up 1. a) List the areas of the 3 squares, from smallest to largest.

, , b) Write an addition statement with the areas of the 3 squares. c) Calculate the side length of each of the squares. The side lengths of the squares are , , and .

2. Is the triangle a right triangle? Show your work. This triangle a right triangle. (is or is not) 3. Round to the nearest tenth.

a) 2.34 ≈ b) 10.56 ≈ c) 5.98 ≈ d) 30.01 ≈

4. Label 1.3, 2.5, and 3.8 on the number line.

0 1 2 3 4

64 cm2

36 cm2

100 cm2

56.25cm2

20.25cm2 30.25

cm2

?


Name: _____________________________________________________ Date: ______________

3.3 Estimating Square Roots

Working Example 1: Estimate the Square Root of a Number

Felicity wants to know if a wading pool will fit in her yard. She estimates the length of the sides to see if it will fit. The pool is square and has an area of 7 m2.

a) What is a reasonable estimate for the side length of the pool? Use perfect squares to estimate. Round your answer to 1 decimal place.

Solution

The side length of the pool is 7 . What number do you square to get 7? 22 = 4, 32 = The perfect squares closest to 7 are 4 and 9:

The square root of 7 is closer to the square root of 9.

9 = , so 7 will be a little less than 3. A reasonable estimate for 7 is 2.7 m.

b) Use a calculator to find the side length of the pool. Round your answer to 1 decimal place. Solution C 7 √⎯ = Round your answer to 1 decimal place. 2.645751311 m ≈ m Is this answer close to the estimate? Circle YES or NO.

Use perfect squares to estimate the square root. Round your answer to 1 decimal place. Check your answer with a calculator.

a) 18 b) 35 Perfect squares on either side of 18:

and

The closer square root is . A reasonable estimate is . Check with a calculator:

2 2.5 3

√4 √7 √9

3.3 Estimating Square Roots ● MHR 127

Name: _____________________________________________________ Date: ______________

Working Example 2: Identify a Number With a Square Root Between 2 Numbers

a) Name a whole number that has a square root between 6 and 7.

Solution

Step 1: Find the square of 6: 62 = . Find the square of 7: 72 =

Step 2: Plot these numbers on a number line:

Step 3: Estimate a number between 36 and 49 . 40 is between 36 and 49 . The value is between 6 and 7.

Step 4: Check: C 40 √⎯ =

6.32455532 is between 6 and 7, so it is a possible answer.

b) How many whole numbers have a square root between 6 and 7? Solution List the whole numbers larger than 36 and smaller than 49: 37, , ,

, , , , , , , , 48

There are whole numbers that have square roots between 6 and 7.

a) Find a whole number with a square root between 8 and 9. Step 1: 82 = 92 =

Step 2: Plot these numbers on a number line. √ √

Step 3: Estimate a number between the 2 square roots: Step 4: Check: b) How many whole numbers have a square root between 8 and 9? _______________________________________________________________________

62 = 36 72 = 49

6 7

√36 √ ? √49


Name: _____________________________________________________ Date: ______________

1. Juan is explaining how to estimate 28 to 1 decimal place without using a calculator. Finish Juan’s explanation.

Step 1: Find the perfect squares on either side of 28. They are 25 and .

Step 2: The perfect square that is closer to 28 is .

Step 3: _____________________________________________________________________

2. Explain how to find a whole number that has a square root between 3 and 4.

Step 1: Find the squares of 3 and 4. 32 = , 42 =

Step 2: Draw a number line.

Step 3:

3. Estimate the square root of each number.

Round your answer to 1 decimal place. Check your answer with a calculator.

a) 32 Perfect squares on either side of 32: 52 = _________ 62 = _________ 32 is between ______ and ______.

The closer square root is .

An estimate is .

Check with a calculator:

b) 55 Perfect squares on either side of 55:

72 = _________ 2

= _________

55 is between ______ and ______.

The closer square root is .

An estimate is .

Check with a calculator:

3.3 Estimating Square Roots ● MHR 129

Name: _____________________________________________________ Date: ______________

4. Estimate each value. Round your answer to 1 decimal place.

Check your answer with a calculator.

a) 14 Perfect squares on either side of 14: , The closer square root is . An estimate is . Check with a calculator:

b) 86

5. What are all the whole numbers with a square root between 2 and 3?

6. Kai uses 1 whole can of paint on a square wall. One can covers 27 m2. Estimate the side length of the wall. Round your answer to 1 decimal place.

m

m

Sentence: ___________________________________________________________________

22 = __________ 32 = __________


Name: _____________________________________________________ Date: ______________

7. The square has an area of 20 cm2.

a) Estimate the side length to 1 decimal place. Perfect squares on either side of 20:

2

24 ______________ ______________= =

20 is between and .

The closer square root is . An estimate is . Check with a calculator: b) Use a ruler to measure the side of the square.

Measure to the nearest tenth of a centimetre (1 decimal place). The side of the square is cm.

8. Write the numbers in order from least to greatest: 7, 46 , 5.8, 27 , 6.3 46 = 27 = ____________________________________________________________________________ 9. Alex is thinking of a number.

a) What numbers have a square root between 2 and 3? _________________________________________ _________________________________________ b) What number is Alex thinking about?

Give 1 reason for your answer. _________________________________________________________________________

5.0 5.5 7.06.56.0

Find the whole numbers between 22 and 32.

20 cm2

3.3 Math Link ● MHR 131

Name: _____________________________________________________ Date: ______________

You have created a peg board game called Mind Buster. The area of the square game board is 134 cm2. You go to the store to get a box to store the game in. You find 5 boxes.

a) Estimate the side length of the square game board with an area of 134 cm2. Round your answer to 1 decimal place. b) Check your answer with a calculator. c) Which boxes cannot store a game board with the dimensions you found in part b)? _______________________________________________________________________ d) Which box or boxes can store the game board? Give 1 reason for your answer. _______________________________________________________________________ _______________________________________________________________________ e) What is the smallest box that will hold the game board? f) What are the lengths of the sides of this box? g) Which box would you choose? Box . Why? _______________________________________________________________________ _______________________________________________________________________

11.9 cm

11.9 cm

Box E

11.3 cm

11.3 cm

Box A Box B 11.3 c m

11.9 cm

11.7 cm

11.4 cm

Box C Box D 11.7 cm

11.6 cm


Name: _____________________________________________________ Date: ______________

3.4 Warm Up 1. Estimate 28 to 1 decimal place.

Perfect squares on either side of 28:

2

25 ______________ ______________= =

28 is between and . The closer square root is . An estimate is . 2. Use a calculator to check your answer to #1. 3. List the numbers with a square root between 10 and 11. 102 = 112 = The numbers are 101, , 120. 4. Use prime factorization to calculate 196 .

196 4 × 49

196 is . 5. Find the square root.

a) 36 = b) 16 = c) 100 = d) 64 = e) 49 = f) 81 =

3.4 Using the Pythagorean Relationship ● MHR 133

Name: _____________________________________________________ Date: ______________

3.4 Using the Pythagorean Relationship Working Example 1: Determine the Length of the Hypotenuse of a Right Triangle

Find the length of hypotenuse c. Round your answer to the nearest tenth of a metre (1 decimal place). Solution Use the Pythagorean relationship, c2 = a2 + b2. a = b = c2 = a2 + b2 = 72 + 102

= +

=

c =

c ≈ The length of the hypotenuse is about m.

Find the length of the hypotenuse for the right triangle. Round your answer to the nearest centimetre. f = g = h2 = f 2 + g2

= 2

+ 2

= +

=

h =

h ≈

The length of the hypotenuse is about cm.

The length of the hypotenuse is c. The lengths of the legs are a and b.

f = 6 cm

g = 10 cm

h

a = 7 m

b = 10 m

c


Name: _____________________________________________________ Date: ______________

Working Example 2: Determine the Length of a Leg of a Right Triangle

What is the length of leg e?

Solution Use the Pythagorean relationship. The letter for the length of the hypotenuse is .

The letters for the lengths of the legs are and . So, f 2 = d2 + e2 Substitute d, e, and f into the equation. 412 = 92 + e2

= + e2 1681 – 81 = 81 + e2 – 81 Subtract 81 from both sides.

= e2

= e

= e The length of the leg is mm.

Find the length of leg s. The letter for the length of the hypotenuse is .

The letters for the lengths of the legs are and . t2 = r2 + s2

2

= 2

+ s2

= + s2

– = + s2 –

= s2

= s

= s

The length of the leg is cm.

d = 9 mmf = 41 mm

e

t = 52 cmr = 20 cm

s


Name: _____________________________________________________ Date: ______________

1. Jack wants to find the length of the missing side of a triangle. He decides to draw it and measure it as shown. Do you agree with his method? Circle YES or NO.

Give 1 reason for your answer. ___________________________________________

2. Kira calculates the missing side of the right triangle.

w = 5 cmx = 13 cm

y

a) Write the Pythagorean relationship.

2 2

2w= +

b) Is Kira’s calculation correct? Circle YES or NO.

Give 1 reason for your answer.

________________________________

________________________________

3. Find the length of each hypotenuse.

a) a = 12 cm

b = 16 cm

c

b) p = 16 m

q = 30 m

r

c2 = a2 + b2

c2 = 2

+ 2

c2 = +

c =

c = cm

r2 = +

y2 = 52 + 132

y2 = 25 + 169 y2 = 194 y ≈ 13.9

The length of side y is approximately 13.9 cm.


Name: _____________________________________________________ Date: ______________

4. a) What is the area of each square attached to the legs of the right triangle? b) What is the area of the square attached to the hypotenuse? c) What is the length of the hypotenuse?

5. Find the missing length of the leg of the right triangle.

c2 = a2 + b2

2

= 2

+ b2

= + b2

− = + b2 −

= b2

= b

= b The length of the leg is cm.

8 cm

6 cm

a = 7 cmc = 25 cm

b


Name: _____________________________________________________ Date: ______________

What side of the triangle does the

diagonal represent?

The diameter is the distance across the circle

through the centre.

6. Find the missing length of the leg for each triangle. Round your answer to the nearest tenth of a millimetre (1 decimal place).

a) i = 9 mm

g = 5 mm

h

b) q = 11 mm

r = 15 mmp

7. Tina wants to build a path across the diagonal of her yard. How long will the path be? Round your answer to the nearest tenth of a metre (1 decimal place). Formula → Substitute → Solve → ___________________________________________________________________________ 8. The hypotenuse of the triangle cuts the circle in half. What is the length of the diameter of this circle? Round your answer to the nearest tenth of a centimetre (1 decimal place). ___________________________________________________________________________

b = 12 m

a = 6 mdiagonal

5 cm

7 cm

cm


Name: _____________________________________________________ Date: ______________

The diagonal is the hypotenuse of the right triangle.

a) Follow the steps to find the distance between A and B. Look at 1 small square.

5 cm

5 cm

Calculate the length of the diagonal to the nearest tenth.

c2 = 2

+ 2

c2 = + c2 =

c =

c =

Use a red pencil to trace a direct line from A to B. How many small diagonals are between A and B?

Find the length of AB. Length of 1 diagonal × number of diagonals

between A and B

= ×

=

b) Follow the steps to find the shortest distance between C and D.

Use a green pencil to trace the shortest distance between C and D.

You must follow the lines on the game board. How many small diagonals are on the green line from C to D?

Find the length of the diagonals from C to D. Length of 1 diagonal × number of diagonals on green line = × =

How many legs of triangles are on the green line from C to D? Find the length of the legs from C to D.

Length of legs × number of legs on green line = × =

Find the total length of the green line.

________________________________________________________________________

5 cm

A

C

D

B

5 cm


Name: _____________________________________________________ Date: ______________

3.5 Warm Up 1. What is the length of the hypotenuse in the triangle? Round your answer to the nearest tenth of a centimetre

(1 decimal place).

Formula → Substitute → Solve →

2. What is the length of the leg in the triangle? Round your answer to the nearest tenth of a metre

(1 decimal place). Formula → Substitute → Solve →

3. A square garden has an area of 36 m2.

a) What is the length of each side?

b) How long is a path running diagonally across the garden? Round your answer to the nearest hundredth of a metre (2 decimal places).

4. Calculate.

a) 92 = b) 122 =

b = 6 cm

a = 3 cm?

a = 8 mc = 16 m

?

diagonal


Name: _____________________________________________________ Date: ______________

Difference means subtract.

3.5 Applying the Pythagorean Relationship

Working Example 1: Determining Distances With Right Triangles

a) Anthony and Shalima are canoeing on a lake between 2 boat ramps. How far is it by canoe between the boat ramps?

Solution

The 2 roads leading from the boat ramps make the legs of a triangle.

The distance by canoe is the hypotenuse, c. Use the Pythagorean relationship.

c2 = a2 + b2

= 15002 + 8002

= + =

c =

c = Anthony and Shalima canoe m. b) Samantha and Nicole walk on the roads from ramp A to ramp B. How much farther do Samantha and Nicole walk than Anthony and Shalima canoe? Solution

Find the total distance by road between the boat ramps. 1500 + 800 = The total distance by road is m. Find the difference between the distance walked and the distance canoed. – 1700 = Samantha and Nicole walked m more than Anthony and Shalima canoed.

boat ramp A

1500 m

800 m

boat ramp B

hypotenuse C

3.5 Applying the Pythagorean Relationship ● MHR 141

Name: _____________________________________________________ Date: ______________

A ship travels west for 10 km. Then, it turns and travels north. If the ship is 25 km from its starting point, how far north did it travel? Round your answer to the nearest tenth of a kilometre (1 decimal place). a) c2 = a2 + b2 = + b2

= + b2

– = – + b2

= b2

= b

= b Sentence: _________________________________________________________________

b) The ship did not take the shortest route.

How much farther did the ship travel than the shortest route? _______________________________________________________________________

starting point

10 km

25 km


Name: _____________________________________________________ Date: ______________

Verify means check.

Working Example 2: Verify a Right Angle Triangle

Danielle is trying to hang a corner shelf. She thinks the 2 walls do not meet at a right angle because the shelf does not fit. She measures 30 cm from the corner along each wall. Then, she measures the hypotenuse to be 41 cm. Do the walls meet at a right angle? Solution

First, draw a diagram.

41 cm30 cm

30 cm?

Use the Pythagorean relationship to find out if the triangle is a right triangle. To be a right triangle, the sum of the areas of the 2 small squares must equal the area of the large square. a2 + b2 = c2

Left Side: a2 + b2

= 302 + ____________

= 900 + ____________

= ____________

Right Side: c2 = 412

= ____________

1800 cm2 ≠ 1681 cm2 The triangle is not a right triangle, so the walls do not meet at a right angle.


Name: _____________________________________________________ Date: ______________

A construction company is digging a rectangular hole with a width of 17 m and a length of 20 m. A worker measures the diagonal length to be 26.25 m. a) Label the rectangle with the dimensions. b) Is the corner a right angle? Explain your answer. a2 + b2 = c2

Left Side: a2 + b2

= 2

+ 2

= + =

Right Side:

c2 = 2

c =

The left side is to the right side. (equal or not equal) The corner a right angle. (is or is not)

1. Explain how you can use the Pythagorean relationship to calculate distance. Use an example from real life.

_____________________________________________________________________________

2. Ilana used the following method to find out if the diagram shows a right triangle.

Left Side: Right Side: The large square is 61 cm. 11 + 60 = 71 The two smaller squares are 71 cm. 61 cm ≠ 71 cm The triangle is not a right triangle.

Is Ilana’s method correct? Circle YES or NO. Explain how you know.

_____________________________________________________________________________

11 cm

60 cm

61 cm

a =

b =

c =


Name: _____________________________________________________ Date: ______________

3. Walter and Maria live beside a rectangular field.

a) Maria walked around 2 sides of the field. How far did Maria walk? + = Sentence: __________________________________________________________________ b) Walter walked across the field in a

diagonal line. How far did Walter walk? Round your answer to the nearest metre.

Formula → c2 = a2 + b2 Substitute → Solve →

c) Who walked farther? By how much?

4. A wire is attached to the top of a pole and to the ground.

Find the height of the pole. Round your answer to the nearest tenth of a metre (1 decimal place). c2 = a2 + b2

= + b2

= + b2

– = – + b2

= b2

= b

= b Sentence: ____________________________________________________________________

120 m

300 m

2 m

10 m

wire


Name: _____________________________________________________ Date: ______________

Use the Pythagorean relationship.

5. You are checking the design plans for a baseball diamond. Is the triangle a right triangle? Explain. _____________________________________________________________________________

6. What is the height, h, of the wheelchair ramp?

Round your answer to the nearest tenth of a centimetre (1 decimal place).

Sentence: _______________________________________________________________ 7. The size of a computer monitor is based on the

length of the diagonal of the screen.

a) How long does the ad say the diagonal is?

b) Shahriar thinks that the diagonal is not as large as the ad says. Is he correct? Calculate the length of the diagonal to find out. Sentence: __________________________________________________________________

79 cm

80 cmh

27 m

27 m

37.1 m ?

42-cm monitoron sale!

GREAT DEAL

30 cm

25 cm


Name: _____________________________________________________ Date: ______________

The diagram shows the plans for a board game. The board is made up of 1 square and 4 identical right triangles.

If the square has an area of 225 cm2, what is the perimeter (distance around the outside) of the game board? Round your answer to the nearest tenth (1 decimal place).

Find the side length of the square. A = s2

= s2

= s

cm = s

Use the Pythagorean relationship to find the length of the hypotenuse of each triangle.

The side length of the square is equal to the leg of the right triangle.

So, c2 = s2 + s2

c2 = 2

+ 2

c2 = +

c2 =

c =

c ≈

Around the perimeter of the game board above: • label all of the leg lengths with s • label the hypotenuse lengths with c

Add the lengths together to find the perimeter of the game board.

What is the length of each leg (s)?

How many s’s are in the perimeter?

What is the length of each hypotenuse (c)?

How many c’s are in the perimeter? Total perimeter = sum of all sides

=

=

Chapter Review ● MHR 147

Name: _____________________________________________________ Date: ______________

3 Chapter Review Key Words For #1 to #5, fill in the blanks. Use the word list.

hypotenuse perfect square prime factorization Pythagorean relationship square root 1. The of 36 is 6. 2. The number 25 is a because it is the

product of the same two factors, 5 × 5 = 25. 3. In a right triangle, the longest side is called the . 4. The sides of a right triangle are a, b, and c. The longest side is c. The equation c2 = a2 + b2 is known as the . 5. The of 18 is 2 × 3 × 3. 3.1 Squares and Square Roots, pages 108–116 6. Find the square of each number.

a) 62 b) 112

7. Find each square root.

a) 100 b) 144 8. Lisa needs at least 17 m2 of fabric to make curtains. Is this square piece of fabric large enough? Show how you can prove your answer.

4 m


Name: _____________________________________________________ Date: ______________

3.2 Exploring the Pythagorean Relationship, pages 118–124 9. A triangle has squares on each of its sides.

a) What is the length of each of the 3 sides of the triangle?

b) How could you show if this triangle is a right triangle? __________________________________________________________________________

10. A triangle has side lengths x = 9 cm, y = 12 cm, and z = 15 cm. Is it a right triangle? x = y = z =

x2 = 2

y2 = 2

z2 = 2

= = =

The sum of the area of the 2 small squares = + = Does this sum equal the area of the large square? Circle YES or NO. It a right triangle. (is or is not) 3.3 Estimating Square Roots, pages 126–131 11. What is an estimate for 10 ? Round your answer to 1 decimal place.

Perfect squares on either side of 10:

32 = 2

=

10 is between and

The closer square root is . An estimate is .

16 cm2

16 cm2

36 cm2

Chapter Review ● MHR 149

Name: _____________________________________________________ Date: ______________

The perimeter is the sum of all the sides.

A = s2

25 = s2

25 = s

12. Cliffmount School is creating square invitations for its 50th anniversary party.

There are 3 possible designs.

a) Estimate a whole number area for

the middle invitation: b) What is the side length of the smallest invitation? c) What is the side length of the largest invitation? d) Estimate the side length of the middle invitation.

3.4 Using the Pythagorean Relationship, pages 133–138 13. Round each answer to the nearest tenth of a

centimetre where appropriate. a) What is the length of the hypotenuse in ∆ABC? AB = units BC = units AC2 = AB2 + BC2

AC2 = 2

+ 2

b) What is the perimeter of ∆ABC AC2 = +

AC2 =

AC =

AC =

36 cm2 25 cm2

Cliffmount School

You are invited to our 50thanniversarycelebration!

Clif

fmou

nt S

choo

l


CliffmountSchool


cm2

0

A B

C

x

y

−2

2 4 6

2

4

6


Name: _____________________________________________________ Date: ______________

14. Find the missing side length of each triangle.

a)

b = 5 m

a = 12 m c

b) t = 9 cm

w = 15 cm

v

c2 = a2 + b2 ← Formula → ← Substitute → ← Solve →

3.5 Applying the Pythagorean Relationship, pages 140–146 15. A 4-m ladder is being used in Romeo and Juliet. The bottom of the ladder will be placed 1 m from the base of Juliet’s house.

a) How far up the wall will the ladder reach? Show your work.

Sentence: __________________________________________________________________ b) The height from the base of the building to Juliet’s window is 3.9 m. Will the ladder reach the window?

__________________________________________________________________________

ac =

b =

Practice Test ● MHR 151

Name: _____________________________________________________ Date: ______________

3 Practice Test

For #1 to #5, circle the best answer. 1. Which number is a perfect square?

A 10 B 20 C 50 D 100

2. What is the side length of the square?

A 6 mm B 9 mm C 12 mm D 18 mm

3. A square has a side length of 7 cm.

What is the area of the square?

A 14 cm2 B 21 cm2 C 28 cm2 D 49 cm2

4. A right triangle has squares on each of its sides.

What is the area of the black square?

A 4 m2 B 14 m2

C 16 m2 D 28 m2

5. The value of 51 is closest to which whole number?

A 7 B 8 C 49

D 51

Complete the statement. 6. For a right triangle with sides a, b, and c, the Pythagorean relationship is c2 = a2 + b2. The variable that represents the length of the hypotenuse is .

81 mm2

22 m2

6 m2


Name: _____________________________________________________ Date: ______________

The sum of the areas of the 2 smaller squares = the area

of the large square

Short Answer 7. The length of the rectangular pool at Wild Water World measures 15 m and a diagonal

measures 17 m. What is the width of the pool? 8. a) Name a whole number that has a square root between 7 and 8.

b) List all of the whole numbers that have a square root between 7 and 8.

72 = 82 = List the numbers between 72 and 82.

__________________________________________________________________________

9. A triangle has sides that are 6 mm and 8 mm, and a hypotenuse that is 10 mm.

a) Label the diagram with the dimensions.

? b) Use the Pythagorean relationship to determine whether this is a right triangle.

Show your work.

Sentence: __________________________________________________________________

15 m

17 m

Practice Test ● MHR 153

Name: _____________________________________________________ Date: ______________

10. Josie and Han are skating on a rectangular skating rink.

a) Josie skated diagonally across the rink. How far did she skate? b) Han skated along the 2 sides of the rink to the opposite corner. How far did he skate? c = , a = ?, b = c2 = a2 + b2

Distance Han skated = + = Sentence: __________________________________________________________________ c) Who skated farther? Circle JOSIE or HAN. By how much? Sentence: __________________________________________________________________

25 m

20 m


Name: _____________________________________________________ Date: ______________

Create a game of your own. These game boards use squares and right triangles in their designs. 1. Draw a game board on a separate sheet of paper. Use squares and right triangles. 2. Your game must include 2 of the following: • calculating the square of a number • calculating the square root of a perfect square • using the Pythagorean relationship to find if a triangle

is a right triangle • finding the missing side length of a right triangle 3. Write the rules for your game and how to play it.

_____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________ _____________________________________________________________________________

Key Word Builder ● MHR 155

Name: _____________________________________________________ Date: ______________

Fill in the blanks with key words from Chapter 3. Then, write them in the crossword puzzle below. Across 2. The number 16 is a because 4 × 4 = 16. 4. One way of showing the is p2 = q2 + r2. Down 1. The number 4 is the of 16. 2. This example shows : 28 = 2 × 2 × 7 3. Side p is known as the of this right triangle. 5. Sides q and r are known as of this right triangle.

4

3

1 2

5

rp

q


Name: _____________________________________________________ Date: ______________

If you roll a 5 and then a 2, your number is 52.

×

××

52

2 26

2 2 13

2 + 2 + 13 = 17

Prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, etc.

If you roll a 41, your score is 0 because 41 is a prime number.

You roll a 1 and a 6. 16 is a perfect square. The prime factorization of 16 is 2 × 2 × 2 × 2.

The sum is 2 + 2 + 2 + 2 = 8. Your score is 10 bonus points + 8 points = 18 points.

Math Games

It’s Prime Time Play It’s Prime Time with a partner.

● 2 dice per pair of student ● It’s Prime Time BLM

Rules: ● Each player rolls 1 die. The person with the highest number

plays first. If there is a tie, roll again. ● For each turn, roll 1 die twice. The result of the first roll gives the first digit of a 2-digit number. The result of the second roll gives the second digit of the number. ● Find the prime factorization of the 2-digit number.

● Find the sum of the factors in the prime factorization.

● Score 0 points for the turn if the 2-digit number is a prime number.

● Score 10 bonus points if the 2-digit number you roll is a perfect square.

● The first player to reach 100 points wins.

Challenge in Real Life ● MHR 157

Name: _____________________________________________________ Date: ______________

A truss is the framework of a roof.

Challenge in Real Life

Framing a Roof Carpenters need to use mental math skills. They study the blueprint of a house plan to help them build walls, stairs, and roofs. You be the carpenter! Calculate the lengths of the supports used to build a roof truss. The truss is made up of 7 pieces of wood. All 7 pieces work together to create triangles that support each other. 1. Using the Pythagorean relationship, calculate the length of c. c2 = a2 + b2

c2 = 2

+ 2

2. What is the total length of piece a?

c2 = a2 + b2

2

= a2 + 2

= a2 +

– = a2 + – = a2

= a

= a

3. What is the total length of f, the bottom piece of the truss?

2 × a = f 2 × =

a

c

f

300 cm500 cm150 cm

200 cm

300 cm

a

500 cm


Answers Get Ready, pages 104–106

1. a) 2 × 2 × 2 × 3 b) 2 × 2 × 3 × 5

2. a) 28 m b) 33 m2 3. a) 6 b) 3 Math Link

a) 6 b) 20; 5; 25 c) A = 500 cm2 3.1 Warm Up, page 107

1. a) 1, 2, 4, 8 b) 1, 2, 4, 8, 16 c) 1, 2, 4, 5, 10, 20 d) 1, 2, 4, 8, 16, 32 2. a) 25 b) 4 c) 64 d) 144 3. a) 5 b) 2 c) 8 d) 10 3.1 Squares and Square Roots, pages 108–116

Working Example 1: Show You Know

a) 45

95

335

×

× ×

b) 100

502

2 × 2 × 25

2 × 2 × 5 × 5

45. The prime factors do not form pairs.


256 mm2 Working Example 3: Show You Know

a) 14 × 14 = 196 b) Prime factorization is 2 × 2 × 7 × 7. Side length is 14 cm. Communicate the Ideas

1. Answers may vary. Examples: a) To square a number, multiply the number by itself. b) Multiply 7 by itself. 7 × 7 = 49

2. Answers may vary. Example: Arrange the prime factors into two pairs of 3 × 5. Since 3 × 5 = 15, the square root of 225 is 15.

Practice

3. a) b) YES. The 2 prime factors can be paired. c) 4. a) 36

6 6

2 × 3 × 2 × 3

×

b) YES. There are pairs of prime factors (2 and 3).c)

5. a) 9 b) 121 6. a) 100 units2 b) 256 units2 7. a) 9 b) 8 8. a) 7 mm b) 10 cm Apply

9. No, 54 is not a perfect square. 10. 196 m2 11. a) 36 m2 b) 6 m

Math Link

a) 64 b)

Dimensions of Square

Board (cm)

Area of Board (cm2)

Number of Squares

Area of Each Square (cm2)

Perfect Square?

32 × 32 1024 64 16 Yes 31 × 31 961 64 15.02 No 30 × 30 900 64 14.06 No 29 × 29 841 64 13.14 No 28 × 28 784 64 12.25 No 27 × 27 729 64 11.39 No 26 × 26 676 64 10.56 No 25 × 25 625 64 9.77 No 24 × 24 576 64 9 Yes 23 × 23 529 64 8.27 No

c) 32 × 32, 24 × 24 3.2 Warm Up, page 117

1. a) 1, 4, 9, 16 b) 1, 4, 9, 16 2.

3. 169 cm2

4. a) 49 b) 100 5. a) 6 b) 11

3.2 Exploring the Pythagorean Relationship, pages 118–124


a)

r = 15 cmp = 9 cm

q = 12 cm

b) p = 81 cm2, q = 144 cm2, r = 225 cm2 c) 81 + 144 = 225


a) 144 cm2, 256 cm2, and 400 cm2 b) Yes. The sum of the areas of the 2 smaller squares equals the area of the large square.

Communicate the Ideas

1. The areas of the 2 smaller squares equal the area of the largest square. 2. NO. Kendra has mixed up the variables. It should be p2 = r2 + q2. Practise

3. 900 mm2, 1600 mm2, 2500 mm2 4. a)

15 mm

12 mm

9 mm

b) 81 mm2, 144 mm2, and 225 mm2

c) The triangle is a right triangle.

5. a) 25 + 144 = 169 b) 5 cm, 12 cm, 13 cm c) 52 + 122 = 132

24

64

322 2

×

× ××

48

6 8

22 2 × 23

2 3 4 2

×

×

× ×

×

60

302

2 152

2 2 3 5

×

×

× × ×

×

4

2 × 2 2 cm

2 cm

64

8

4 2

×

×

8

4 2× ×

2 22 ×× × ×22 2×

6 cm

6 cm

54

2 ×

2 ×

27

3 3

3 9

2 × ×

×

×3

Answers ● MHR 159

Apply

6. a) 52 cm2 b) 676 mm2 7. The angle is not a right angle because the sum of the 2 smaller squares

does not equal 90.25 m2. Math Link a) #4: 5, 25, #8, 5, 25, #8, 50, #1 #5: 4, 25, #10, 5, 25, #8, 41, #3 #9: 4, 16, none, 4, 16, #10, 25, #8 b) Triangle 5 c) Squares 10, 8, and 3 3.3 Warm Up, page 125

1. a) 36 cm2, 64 cm2, 100 cm2 b) 36 + 64 = 100 c) 6, 8, 10 2. This triangle is not a right triangle. 3. a) 2.3 b) 10.6 c) 6.0 d) 30.0 4. 3.3 Estimating Square Roots, pages 126–131


a) Estimate: 4.1; Calculate: 4.2 b) Estimate: 5.9; Calculate: 5.9 Working Example 2: Show You Know

a) Answers will vary. Example: 73 b) 17 Communicate the Ideas

1. Step 1: 36; Step 2: 25; Step 3: Find a reasonable estimate to 1 decimal place. The number should be greater than 25. A reasonable estimate is 5.2.

2. Step 1: 9, 16; Step 3: Find a reasonable estimate. The number should be a whole number between 9 and 16.

Practise

3. Estimates may vary. a) Estimate: 5.8; Calculate: 5.7 b) Estimate: 7.6; Calculate: 7.4

4. Estimates may vary. a) Estimate: 3.7; Calculate: 3.7 b) Estimate: 9.2; Calculate: 9.3

5. 5, 6, 7, 8 Apply

6. 5.1 m 7. a) 4.5 cm b) 4.5 cm

8. 27, 5.8, 6.3, 46, 7

9. a) 5, 6, 7, 8 b) Alex is thinking about 6 because it is also divisible by 3. Math Link

a) Estimates will vary. Example: 11.5 b) 11.6 c) A, B, C d) D, E. The sides are equal to or greater than the length of the sides of the game board. e) D f) 11.7, 11.6 g) Answers may vary. Example: E. It is larger and the board would be easier to fit.

3.4 Warm Up, page 132

1. 5.2 2. 5.3 3. 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116,

117, 118, 119 4. 196

494

7722

×

× ××

14

5. a) 6 b) 4 c) 10 d) 8 e) 7 f) 9 3.4 Using the Pythagorean Relationship, pages 133–138


11.7 cm Working Example 2: Show You Know

48 cm

Communicate the Ideas

1. Answers may vary. Example: It would be difficult to measure. 2. a) x2 = w2 + y2 b) NO. She used the wrong value for the hypotenuse. Practise

3. a) 20 cm b) 34 m 4. a) 36 cm2, 64 cm2 b) 100 cm2 c) 10 cm 5. 24 cm 6. a) 7.5 mm b) 10.2 mm Apply

7. 13.4 m 8. 8.6 cm Math Link

a) 28.4 cm b) 24.2 cm 3.5 Warm Up, page 139

1. 6.7 cm 2. 13.9 m 3. a) 6 m b) 8.5 m 4. Estimates will vary. Example: 7.1 5. a) 81 b) 144 3.5 Applying the Pythagorean Relationship, pages 140–146


a) 22.9 km b) 7.9 km Working Example 2: Show You Know

a) a = 17 m, b = 20 m, c = 26.25 m b) The left side is equal to the right side. The corner is a right angle. Communicate the Ideas

1. Answers will vary. Example: You can find the length of the hypotenuse to see how far a shortcut between 2 points is.

2. NO. She must find the areas of the squares to use the Pythagorean relationship, not the lengths of the sides of the squares.

Practise

3. a) 420 m b) 323 m c) Maria walked 97 m farther. 4. 9.8 m 5. No. The hypotenuse should be 38.2 m. Apply

6. 12.6 cm 7. a) 42 cm b) Shahriar is correct, because the diagonal is 39.1 cm. Math Link

144.8 cm Chapter Review, pages 147–150

1. square root 2. perfect square 3. hypotenuse 4. Pythagorean relationship 5. prime factorization 6. a) 36 b) 121 7. a) 10 b) 12 8. No. It is only 16 m2. 9. a) 4 cm, 4 cm, 6 cm b) Add the areas of the 2 smallest squares to see if

the sum equals the area of the largest square. If it does not, the angle is not a right angle. This is not a right triangle because 162 + 162 ≠ 362.

10. It is a right triangle. 11. Estimates may vary. Example: 3.2 12. a) Estimates may vary. Example: 30 cm2 b) 5 cm c) The side length of

the largest invitation is 6 cm. d) Estimates may vary. Example: 5.5 cm 13. a) 5.4 cm b) 12.4 cm 14. a) 13 m b) 12 cm 15. a) 3.9 m b) Yes.

1 2 3 4

1.3 2.5 3.8


Practice Test, pages 151–153

1. D 2. B 3. D 4. C 5. A 6. c 7. 8 m 8. a) Answers will vary. Example: 50 b) 50, 51, 52, 53, 54, 55, 56, 57, 58,

59, 60, 61, 62, 63 9. a)

8 mm

6 mm 10 mm

b) Yes, this is a right triangle.

10. a) 25 m b) 35 m c) HAN. 10 m Wrap It Up!, page 154

Answers will vary. Key Word Builder, page 155

Across 2. perfect square 4. Pythagorean relationship Down 1. square root 2. prime factorization 3. hypotenuse 5. legs Challenge in Real Life, page 157

1. c = 250 cm 2. a = 400 cm 3. f = 800 cm

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