Exploring Square Roots and Irrational Numbers

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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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Exploring Square Roots and Irrational Numbers
LESSON 3-1
Check Skills Youll 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. 2 3. 2 4. 6 5. 10
Check Skills Youll Need
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Exploring Square Roots and Irrational Numbers
LESSON 3-1
Check Skills Youll Need
Solutions 1. exponent 2.
22 2 2 4 3. (2)2 (2) (2)
4 4. (6)2 (6) (6) 36
5. 102 10 10 100
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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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Exploring Square Roots and Irrational Numbers
LESSON 3-1
Additional Examples
Estimate the value of 70 to the nearest
integer.
Quick Check
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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.
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Exploring Square Roots and Irrational Numbers
LESSON 3-1
Additional Examples
(continued)
It takes about 1.7 seconds for the ball to fall
48 ft.
Quick Check
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Exploring Square Roots and Irrational Numbers
LESSON 3-1
Additional Examples
Identify each number as rational or irrational.
Explain.
Rational the decimal repeats.
Irrational 90 is not a perfect square.
Irrational the decimal does not terminate or
repeat.
d. 6.36366366636666. . .
Quick Check
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Exploring Square Roots and Irrational Numbers
LESSON 3-1
Lesson Quiz
1. Find the two square roots of 400. 2. Estimate
34 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
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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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The Pythagorean Theorem
LESSON 3-2
Check Skills Youll 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. 2. 3.
4. 5.
Check Skills Youll Need
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The Pythagorean Theorem
LESSON 3-2
Check Skills Youll Need
Solutions 1. a number that when multiplied
by itself is equal to the given number 2. 7 .
7 49 and 8 . 8 64 ? 8 3. 10
. 10 100 and 11 . 11 121 ? 11
4. 8 . 8 64 and 9 . 9 81 ? 9
5. 4 . 4 16 and 5 . 5 25 ? 5
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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.
The length of the hypotenuse is 10 ft.
Quick Check
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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.
Quick Check
The length of the ramp is about 17 feet.
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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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Using the Pythagorean Theorem
LESSON 3-3
Problem of the Day
Find a number that is halfway between and
.
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Using the Pythagorean Theorem
LESSON 3-3
Check Skills Youll 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 Youll Need
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Using the Pythagorean Theorem
LESSON 3-3
Check Skills Youll 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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Using the Pythagorean Theorem
LESSON 3-3
Additional Examples
Find the missing leg length of the triangle.
The length of the other leg is 5 cm.
Quick Check
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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?
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Using the Pythagorean Theorem
LESSON 3-3
Additional Examples
Quick Check
(continued)
The distance from the top of the ladder to the
ground is about 9.7 ft.
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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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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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Graphing in the Coordinate Plane
LESSON 3-4
Check Skills Youll 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, 3
3. 9, 2, 4, 6 4. 8, 6, 0,
10 5. 2, 7, 5,
4









Check Skills Youll Need
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Graphing in the Coordinate Plane
LESSON 3-4
Check Skills Youll 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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Graphing in the Coordinate Plane
LESSON 3-4
Additional Examples
Graph point P (3, 2 ) on a coordinate plane.
Quick Check
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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?
The distance from the mall to the roller skating
rink is 13 miles.
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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 same coordinate plane. 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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Equations, Tables, and Graphs
LESSON 3-5
Problem of the Day
Write in scientific notation a. 3,700 b. 9,700,0
00 c. 257,000
3.7 ? 103
9.7 ? 106
2.57 ? 105
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Equations, Tables, and Graphs
LESSON 3-5
Check Skills Youll 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 21
3. 13 2a 4. 5a
8

Check Skills Youll Need
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Equations, Tables, and Graphs
LESSON 3-5
Check Skills Youll Need
Solutions 1. variable 2. 6(4) 21
24 21 3 3. 13 2(4) 13 8 21
4. 5(4) 8 20 8 28


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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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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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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 5
1.50x, 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.
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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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Translations
LESSON 3-6
Check Skills Youll 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 Youll Need
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Translations
LESSON 3-6
Check Skills Youll Need
Solutions 1. Quadrant II 2. (4,
2) 3. (2, 1) 4. (5, 2) 5. (1, 1)
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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?
Quick Check
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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.
Quick Check
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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)
P '(3, 5), Q'(2, 3), R'(6, 2)
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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.
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Reflections and Symmetry
LESSON 3-7
Check Skills Youll 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 3. up 4
units 4. down 1 unit, left 4 units
5. up 2 units, right 3 units
Check Skills Youll Need
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Reflections and Symmetry
LESSON 3-7
Check Skills Youll Need
Solutions 2. 3. 4. 5.
1. distance
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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 .
Quick Check
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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 .
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Reflections and Symmetry
LESSON 3-7
Additional Examples
Draw the lines of symmetry in the figure below.
Quick Check
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Reflections and Symmetry
LESSON 3-7
Lesson Quiz
1. 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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Rotations
LESSON 3-8
Problem of the Day
yes yes no
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Rotations
LESSON 3-8
Check Skills Youll 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 3. 150 4. 95
5.
20 6. 35
7. 90
Check Skills Youll Need
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Rotations
LESSON 3-8
Check Skills Youll Need
Solutions
1. matches 2. straight 3. obtuse 4.
obtuse 5. acute 6. acute 7. right
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Rotations
LESSON 3-8
Additional Examples
Find the angle of rotation for the figure below.
The angle of rotation is 45.
Quick Check
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Rotations
LESSON 3-8
Additional Examples
Draw the image of rectangle ABCD after a
rotation of 90 about the origin.
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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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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
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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
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