1. Exploring Square Roots and Irrational Numbers
LESSON 3-1
Problem of the Day
Convert 9 ft 8 in. to inches. Name the operations you use and give the answer.
multiplication and addition; 116 in.
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2. Exploring Square Roots and Irrational Numbers
LESSON 3-1
Check Skills You’ll Need
(For help, go to Lesson 2-7.)
1. Vocabulary Review In a power, the ? tells how many
times a base is used as a factor.
Evaluate the expression x2 for each value of x.
–2 4. –6
5. 10
Check Skills You’ll Need
3-1

3. Exploring Square Roots and Irrational Numbers
LESSON 3-1
Check Skills You’ll Need
Solutions
1. exponent = 2 • 2 = 4
3. (–2)2 = (–2) • (–2) = (–6)2 = (–6) • (–6) = 36
= 10 • 10 = 100
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4. Exploring Square Roots and Irrational Numbers
LESSON 3-1
Additional Examples
Find the two square roots of 81.
9 • 9 = 81
–9 • (–9) = 81
The two square roots of 81 are 9 and –9.
Quick Check
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5. Exploring Square Roots and Irrational Numbers
LESSON 3-1
Additional Examples
Estimate the value of – to the nearest integer.
Since 70 is closer to 64 than it is to 81, – –8.
Quick Check
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6. Exploring Square Roots and Irrational Numbers
LESSON 3-1
Additional Examples
The math class drops a small ball from the top of a stairwell. They measure the distance to the basement as 48 feet. Use the formula d = 16t2 to find how long it takes the ball to fall.
48 = 16t2 Substitute 48 for d.
d = 16t2 Use the formula.
= t2 Divide each side by 16. 48 16
3 = t2 Simplify.
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7. Exploring Square Roots and Irrational Numbers
LESSON 3-1
Additional Examples
(continued)
3 = t Find the positive square root.
Use a calculator. 3
Round to the nearest tenth. t
It takes about 1.7 seconds for the ball to fall 48 ft.
Quick Check
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8. Exploring Square Roots and Irrational Numbers
LESSON 3-1
Additional Examples
Identify each number as rational or irrational. Explain.
a. –9.3333
Rational; the decimal repeats.
b. 4 7 9
Rational; the number can be written as the ratio . 43 9 c
Irrational; 90 is not a perfect square.
Irrational; the decimal does not terminate or repeat. d
Quick Check
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9. Exploring Square Roots and Irrational Numbers
LESSON 3-1
Lesson Quiz
1. Find the two square roots of 400.
2. Estimate to the nearest integer.
3. Using d = 16t 2, find how long it takes a skydiver to fall 676 ft from
an airplane.
4. Is rational or irrational? Explain.
20 and –20 6
6.5 s 64 5
Rational; it can be written as . 8 5
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10. The Pythagorean Theorem
LESSON 3-2
Problem of the Day
Describe the pattern in this sequence of numbers and find the next two numbers. 5, 8, 4, 9, 3, . . .
description: +3, –4, +5, –6; next two numbers: 10, 2
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11. The Pythagorean Theorem
LESSON 3-2
Check Skills You’ll Need
(For help, go to Lesson 3-1.)
1. Vocabulary Review What is the square root of
a number?
Estimate the value of each expression to the nearest integer. 60
111 80 22
Check Skills You’ll Need
3-2

12. The Pythagorean Theorem
LESSON 3-2
Check Skills You’ll Need
Solutions
1. a number that when multiplied by itself is equal to the given number
= 49 and = 64;  8
= 100 and = 121;  11
= 64 and = 81;  9
= 16 and = 25;  5 60
111 80 22
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13. The Pythagorean Theorem
LESSON 3-2
Additional Examples
Find the length of the hypotenuse of a right triangle whose legs are 6 ft and 8 ft.
a2 + b2 = c2 Use the Pythagorean Theorem.
= c2 Substitute a = 6, b = 8.
= c2 Simplify.
100 = c2 Add.
100 = c2 Find the positive square root of each side.
10 = c Simplify.
The length of the hypotenuse is 10 ft.
Quick Check
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14. The Pythagorean Theorem
LESSON 3-2
Additional Examples
A wheelchair ramp that leads into an apartment building doorway is 5 feet above the ground. The horizontal distance from the entrance to the end of the ramp is 16 feet. What is the length in feet of the ramp? Round to the nearest foot.
a2 + b2 = c2 Use the Pythagorean Theorem.
= c2 Substitute 5 for a and 16 for b.
= c2 Simplify.
281 = c2 Add.
281 = c2 Find the positive square root of each side.
= c Simplify.
17 feet c Round to the nearest foot.
Quick Check
The length of the ramp is about 17 feet.
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15. The Pythagorean Theorem
LESSON 3-2
Lesson Quiz
1. Find the hypotenuse of a right triangle with legs of 9 in.
Round to the nearest inch.
2. A right triangle has legs of 5 cm and 18 cm. What is
the length of its hypotenuse to the nearest centimeter?
3. A staircase is 20 ft high. The horizontal distance from one
end of the staircase to the other end is 24 ft. What is the
distance from the top of the staircase to the bottom of the
staircase? Round to the nearest foot.
4. A book is leaning with one end at the top edge of a
bookend.The bookend is 6 in. high. The distance along
the shelf from the edge of the book to the bottom of the
bookend is 4 in. How long is the book? Round to the
nearest inch.
13 in.
19 cm
31 ft
7 in.
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16. Using the Pythagorean Theorem
LESSON 3-3
Problem of the Day 7 10 11 14
Find a number that is halfway between and or
104
140 26 35
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17. Using the Pythagorean Theorem
LESSON 3-3
Check Skills You’ll Need
(For help, go to Lesson 3-2.)
1. Vocabulary Review State the Pythagorean Theorem.
Find the length of the hypotenuse given the lengths of the
two legs, a and b. Round to the nearest tenth.
2. a =3, b= 4
3. a =7, b= 5
Check Skills You’ll Need
3-3

18. Using the Pythagorean Theorem
LESSON 3-3
Check Skills You’ll Need
Solutions
1. The Pythagorean Theorem states that in any right triangle, the sum
of the squares of the lengths of the legs (a and b) is equal to the
square of the length of the hypotenuse. a2 + b2 = c2
2. 5
3. 8.6
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19. Using the Pythagorean Theorem
LESSON 3-3
Additional Examples
Find the missing leg length of the triangle.
a2 + b2 = c2 Use the Pythagorean Theorem.
a = 132 Substitute 12 for b and 13 for c.
a = 169 Simplify.
a2 = 25 Subtract.
a2 = 25 Find the positive square root of each side.
a = 5 Simplify.
The length of the other leg is 5 cm.
Quick Check
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20. Using the Pythagorean Theorem
LESSON 3-3
Additional Examples
The bottom of a 10-foot ladder is 2.5 ft from the side of a wall. How far, to the nearest tenth, is the top of the ladder from the ground?
The diagram shows a right triangle with hypotenuse 10 ft and leg 2.5 ft. The distance from the top of the ladder to the ground is a.
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21. Using the Pythagorean Theorem
LESSON 3-3
Additional Examples
(continued)
Quick Check
a2 + b2 = c2
Use the Pythagorean Theorem.
a2 + (2.5)2 = 102
Substitute b = 2.5 and c = 10.
a = 100
Multiply.
a2 = 93.75
Subtract 6.25 from each side.
Use a calculator.
a =
Find the positive square root.
a Round to the nearest tenth.
The distance from the top of the ladder to the ground is about 9.7 ft.
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22. Using the Pythagorean Theorem
LESSON 3-3
Lesson Quiz
1. A triangle has a hypotenuse of 17 in. and one of its legs is
8 in. What is the length of the other leg?
2. The bottom of a 12-ft ladder is 4 ft from the side of a
house. Find the height of the top of the ladder above the
ground to the nearest tenth.
3. An artist is measuring a rectangular canvas. Its length is
30 in. The distance from one corner of the canvas to the
other (along the diagonal) is 34 in. What is its width?
4. The legs of a right triangle have the same length. Its
hypotenuse is 30 ft. How long is each leg? If necessary,
round to the nearest foot.
15 in.
11.3 ft
16 in.
21 ft
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23. Graphing in the Coordinate Plane
LESSON 3-4
Problem of the Day
If October 4th falls on a Saturday, on what day of the week will
November 4th fall? December 4th?
Tuesday; Thursday
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24. Graphing in the Coordinate Plane
LESSON 3-4
Check Skills You’ll Need
(For help, go to Lesson 1-2.)
1. Vocabulary Review How can you tell whether two
numbers on a number line are opposites?
Order the integers in each set from least to greatest.
2. 3, 5, 1, , 2, 4, 6
4. 8, 6, 0, , 7, 5, 4 – – – – – – – – –
Check Skills You’ll Need
3-4

25. Graphing in the Coordinate Plane
LESSON 3-4
Check Skills You’ll Need
Solutions
1. They are the same distance from zero on a number line but on opposite sides of zero.
2. –5, –3, 1, 3
3. –6, –4, 2, 9
4. –10, –8, 0, 6
5. –5, –2, 4, 7
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26. Graphing in the Coordinate Plane
LESSON 3-4
Additional Examples
Graph point P (–3, 2 ) on a coordinate plane. 1 2
Quick Check
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27. Graphing in the Coordinate Plane
LESSON 3-4
Additional Examples
Quick Check
The mall is 5 miles north of the library. The roller
skating rink is 12 miles east of the library. To the nearest
mile, how far is the mall from the roller skating rink?
a2 + b2 = c2 Use the Pythagorean Theorem.
= c2 Substitute.
= c2 Simplify.
169 = c2 Add.
169 = 25 Find the positive square root of each side.
13 = c Simplify.
The distance from the mall to the roller skating rink is 13 miles.
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28. Graphing in the Coordinate Plane
LESSON 3-4
Lesson Quiz
1. Graph the points A(2, 2), B( 3, 1), and C(2, 1.5) on the
same coordinate plane.
2. Graph the points F(2, 1), G(6.5, 1), and H(6.5, 5) on the
3. Find the length of the hypotenuse of ∆FGH.
4. On a soccer field, one goalpost is 25 yards west of a
second goalpost. The gymnasium is 20 yards north of
the second goalpost. How far is the gymnasium from
the first goalpost? – – –
7.5 units
32 yards
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29. Equations, Tables, and Graphs
LESSON 3-5
Problem of the Day
Write in scientific notation:
a. 3,700
b. 9,700,000
c. 257,000
3.7  103
9.7  106
2.57  105
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30. Equations, Tables, and Graphs
LESSON 3-5
Check Skills You’ll Need
(For help, go to Lesson 1-1.)
1. Vocabulary Review What do you call a symbol that
stands for one or more numbers?
Evaluate for a = 4.
2. 6a a
4. 5a + 8 –
Check Skills You’ll Need
3-5

31. Equations, Tables, and Graphs
LESSON 3-5
Check Skills You’ll Need
Solutions
1. variable
2. 6(4) 21 = = 3
(4) = = 21
4. 5(4) + 8 = = 28 – –
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32. Equations, Tables, and Graphs
LESSON 3-5
Additional Examples
Suppose you buy a bag of food for your pet dog every week. Dog food costs $4 per bag. Make a table and write an equation to represent the total cost of buying dog food for any number of weeks.
The equation c = 4w models the total cost of buying dog food.
Quick Check
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33. Equations, Tables, and Graphs
LESSON 3-5
Additional Examples
Graph the linear equation y = –x + 3, where y
represents the pressure inside a deflating balloon after
x seconds.
Each point (x, y) on the graph represents a solution of the equation. For example, the point (1, 2) means that after 1 second the pressure inside the balloon is 2 units of pressure.
Quick Check
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34. Equations, Tables, and Graphs
LESSON 3-5
Lesson Quiz
1. Suppose you make $8 per hour at an after-school job.
Make a table and write an equation to represent your total
pay after 6 hours of work.
2. Membership at a video store costs $5 per month, plus
$1.50 to rent each movie. Graph the linear equation
y = x, where y represents the total cost in a
month and x represents the number of movies rented
each month.
3. Suppose you rent 6 movies in a month, in the
situation above. Make a table that represents
your total costs.
4. The air pressure in a tire is 32 pounds per square
inch. Every hour, air is leaking out at the rate of
3 pounds per square inch. Write an equation that describes
this situation.
p = 8h
Sample: p = h –
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35. Translations
LESSON 3-6
Problem of the Day
Find how long it will take 4 painters to paint half a duplex if it takes 8 painters 8 days to paint both sides of the duplex.
8 days
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36. 1. Vocabulary Review In what quadrant is ( 3, 5) located? –
Translations
LESSON 3-6
Check Skills You’ll Need
(For help, go to Lesson 3-4.)
1. Vocabulary Review In what quadrant is ( 3, 5)
located? –
Name the coordinates of each point in the graph.
2. A 3. B 4. C
5. D
Check Skills You’ll Need
3-6

37. Solutions 1. Quadrant II 2. (4, 2) 3. (2, 1) 4. (5, –2) 5. (1, –1)
Translations
LESSON 3-6
Check Skills You’ll Need
Solutions
1. Quadrant II
2. (4, 2)
3. (2, 1)
4. (5, –2)
5. (1, –1)
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38. ABC has vertices A (1, –3), B (3, 0), and C (4,–2). Graph
Translations
LESSON 3-6
Additional Examples
ABC has vertices A (1, –3), B (3, 0), and C (4,–2). Graph
ABC and its image after a translation to the left 3 units and up 2 units.
What are the coordinates of its images?
Connect the images of the vertices.
The coordinates of its image are
A'(-2, -1), B'(0, 2), C'(1,0).
Slide each vertex left 3 units and up
2 units.
Quick Check
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39. Write a rule to describe the translation of G (–5, 3) to G (–1, –2).
Translations
LESSON 3-6
Additional Examples
Write a rule to describe the translation of G (–5, 3) to
G (–1, –2).
Point G is moved 4 units to the right and
5 units down.
So, the translation adds 4
to the x-coordinate and subtracts 5 from
the y-coordinate.
The rule is (x, y)     (x + 4, y – 5).
Quick Check
3-6

40. 1. Three points form a triangle: P(0, 4), Q(–1, 2), and
Translations
LESSON 3-6
Lesson Quiz
1. Three points form a triangle: P(0, 4), Q(–1, 2), and
R(3, –3). What are the coordinates of its image after
a translation left 4 units and down 3 units?
2. Write a rule to describe the translation of K(–6, 4) to
K '(1, –8).
3. What are the coordinates of the image of ∆ PQR (from
question 1) after a translation right 3 units and up 1 unit?
4. Write a rule to describe the translation in quiz question 1.
P '(–4, 1), Q '(–5, –1), R' (–1, –6)
(x, y) (x + 7, y – 12)
P '(3, 5), Q'(2, 3), R'(6, – 2)
(x, y) (x – 4, y – 3)
3-6

41. Reflections and Symmetry
LESSON 3-7
Problem of the Day
Describe the pattern. Then give the next three terms: 3, 6, 12, 24, 48, . . .
Starting with 3, each number is two times the one before it: 96, 192, 384.
3-7

42. Reflections and Symmetry
LESSON 3-7
Check Skills You’ll Need
(For help, go to Lesson 3-6.)
1. Vocabulary Review A translation moves each point in
a figure the same ? in the same direction.
Graph the point A(2, 4) and its image after the given translation.
2. left 2 units up 4 units
4. down 1 unit, left 4 units up 2 units, right 3 units
Check Skills You’ll Need
3-7

43. Reflections and Symmetry
LESSON 3-7
Check Skills You’ll Need
Solutions
2. 3.
4. 5.
1. distance
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44. Reflections and Symmetry
LESSON 3-7
Additional Examples
Graph the point H (–4, 5). Then graph its image after it is reflected over the y-axis. Name the coordinates of H .
Since H is 4 units to the left of the y-axis,
The coordinates of H are (4,5).
H is 4 units to the right of the y-axis.
Quick Check
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45. Reflections and Symmetry
LESSON 3-7
Additional Examples
Quick Check
BCD has vertices B (–3, 1), C (–2, 5), and D (–5, 4).
Graph BCD and its image after a reflection over the x-axis.
Name the coordinates of the vertices of B C D .
Reflect the other vertices.
Draw B C D .
Since B is 1 unit above the x-axis,
B is 1 unit below the x-axis.
The coordinates of the vertices are B (–3, –1), C (–2, –5), and D (–5, –4).
3-7

46. Reflections and Symmetry
LESSON 3-7
Additional Examples
Draw the lines of symmetry in the figure below.
There is one line of symmetry.
Quick Check
3-7

47. Reflections and Symmetry
LESSON 3-7
Lesson Quiz
VQM has vertices V(–3, 1), Q(0, 0), and M(4, 4). Name the coordinates of the vertices of V Q M after a reflection
over the x-axis.
2. List all capital letters of the alphabet that have two or more lines of symmetry.
(–3, –1), (0, 0), (4, –4)
H, I, O, X
3. Name the coordinates of point S(–5, 2) after a reflection
about the line that passes through (–1, 4) and (–1, 0).
(3, 2)
4. How many lines of symmetry does a regular
hexagon have? 6
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48. Rotations
LESSON 3-8
Problem of the Day
Use mental math to decide whether or not each of these pairs of fractions is equal: and , and , and 36 42 4 5 6 15 7 16 20 18
yes; yes; no
3-8

49. 1. Vocabulary Review When a figure has reflectional
Rotations
LESSON 3-8
Check Skills You’ll Need
(For help, go to the Skills Handbook page 640.)
1. Vocabulary Review When a figure has reflectional
symmetry, one half ? the other half exactly.
Classify each angle as acute, right, obtuse, or straight.
2. 180° °
4. 95° °
6. 35° °
Check Skills You’ll Need
3-8

50. Solutions 1. matches 2. straight 3. obtuse 4. obtuse 5. acute 6. acute
Rotations
LESSON 3-8
Check Skills You’ll Need
Solutions
1. matches
2. straight
3. obtuse
4. obtuse
5. acute
6. acute
7. right
3-8

51. Find the angle of rotation for the figure below.
Rotations
LESSON 3-8
Additional Examples
Find the angle of rotation for the figure below.
The image matches the original after of a complete rotation.
• 360° = 45°. 1 8
The angle of rotation is 45°.
Quick Check
3-8

52. • Draw rectangle ABCD on a piece of graph paper.
Rotations
LESSON 3-8
Additional Examples
Draw the image of rectangle ABCD after a rotation of 90° about the origin.
Step 1 Draw and trace.
• Draw rectangle ABCD on a piece of graph paper.
Place a piece of tracing paper over your graph.
• Trace the vertices of the rectangle,
the x-axis, and the y-axis.
• Place your pencil at the origin to rotate the paper.
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53. Step 2 Rotate and mark each vertex.
Rotations
LESSON 3-8
Additional Examples
(continued)
Step 2 Rotate and mark each vertex.
• Rotate the tracing paper 90° counterclockwise.
The axes should line up.
• Mark the position of each vertex by pressing
your pencil through the paper.
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54. Step 3 Complete the new figure. • Remove the tracing paper.
Rotations
LESSON 3-8
Additional Examples
(continued)
Step 3 Complete the new figure.
• Remove the tracing paper.
• Draw the rectangle.
• Label the vertices to complete the figure.
Quick Check
3-8

55. 2. Graph (1, 6). Rotate it 90°, 180°, and 270° about the origin and
Rotations
LESSON 3-8
1. A regular hexagon has six equal sides. If the figure has rotational symmetry, find the angle of rotation.
Lesson Quiz
60°
2. Graph (1, 6). Rotate it 90°, 180°, and 270° about the origin and
name the coordinates of each image.
(–6, 1), (–1, –6), (6, –1)
3. The points T(0, 0), U(–3, 0), and V(–3, 5) form a triangle.
Name the coordinates of the image of ∆T 'U 'V ' after a
rotation of 90° about the origin.
4. What is the angle of rotation for a square?
T '(0, 0), U '(0, –3), V '(–5,–3)
90°
3-8