Download Square Roots

9.1
Square Roots
Goal: Find and approximate square roots of numbers.
Vocabulary
Square
root:
Perfect
square:
Radical
expression:
Example 1
Finding a Square Root
Playground A community is building
a playground on a square plot of land
with an area of 625 square yards.
What is the length of each side of the
plot of land?
2
A 625 yd
Solution
Because length
cannot be negative, it
doesn't make sense
to find the negative
square root.
The plot of land is a square with an area of 625 square
yards, so the length of each side of the plot of land is the
.
625
because
625.
Answer: The length of each side of the plot of land is
.
Checkpoint Find the square roots of the number.
1. 9
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2. 49
3. 169
4. 196
Chapter 9 • Pre-Algebra Notetaking Guide
199
9.1
Square Roots
Goal: Find and approximate square roots of numbers.
Vocabulary
Square A square root of a number n is a number m such that
m2 n.
root:
Perfect A perfect square is a number that is the square of an
square: integer.
A radical expression is an expression that involves
Radical
expression: a radical sign.
Example 1
Finding a Square Root
Playground A community is building
a playground on a square plot of land
with an area of 625 square yards.
What is the length of each side of the
plot of land?
2
A 625 yd
Solution
Because length
cannot be negative, it
doesn't make sense
to find the negative
square root.
The plot of land is a square with an area of 625 square
yards, so the length of each side of the plot of land is the
positive square root of 625 .
625
25 because 252 625.
Answer: The length of each side of the plot of land is 25 yards .
Checkpoint Find the square roots of the number.
1. 9
3, 3
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2. 49
7, 7
3. 169
13, 13
4. 196
14, 14
Chapter 9 • Pre-Algebra Notetaking Guide
199
Example 2
Approximating a Square Root
Approximate 28
to the nearest integer.
The perfect square closest to, but less than, 28 is
square closest to, but greater than, 28 is
and
. So, 28 is between
. This statement can be expressed by the compound
inequality
< 28 <
.
< 28 <
Identify perfect squares closest to 28.
<
< 28
Take positive square root of each number.
<
< 28
Evaluate square root of each perfect square.
Answer: Because 28 is closer to
to
. The perfect
than to
, 28
is closer
. So, to the nearest integer, 28
≈
than to
.
Checkpoint Approximate the square root to the nearest integer.
5. 46
6. 125
Example 3
7. 68.9
8. 87.5
Using a Calculator
Use a calculator to approximate 636
. Round to the
nearest tenth.
Keystrokes
2nd
200
[] 636
Chapter 9 • Pre-Algebra Notetaking Guide
Display
)
Answer
Copyright © Holt McDougal. All rights reserved.
Example 2
Approximating a Square Root
Approximate 28
to the nearest integer.
The perfect square closest to, but less than, 28 is 25 . The perfect
square closest to, but greater than, 28 is 36 . So, 28 is between
25 and 36 . This statement can be expressed by the compound
inequality 25 < 28 < 36 .
25 < 28 < 36
Identify perfect squares closest to 28.
25
< 28
< 36
< 6
5 < 28
Take positive square root of each number.
Evaluate square root of each perfect square.
Answer: Because 28 is closer to 25 than to 36 , 28
is closer
to 5 than to 6 . So, to the nearest integer, 28
≈ 5 .
Checkpoint Approximate the square root to the nearest integer.
5. 46
6. 125
11
7
Example 3
7. 68.9
8. 87.5
9
8
Using a Calculator
Use a calculator to approximate 636
. Round to the
nearest tenth.
Keystrokes
2nd
200
[] 636
Chapter 9 • Pre-Algebra Notetaking Guide
)
Display
Answer
25.21904043
25.2
Copyright © Holt McDougal. All rights reserved.
Checkpoint Use a calculator to approximate the square root.
Round to the nearest tenth.
9. 6
10. 104
11. 819
12. 1874
Evaluating a Radical Expression
Example 4
Evaluate 5
a2 b when a 6 and b 27.
a2 b 5
Substitute for a and for b.
Evaluate expression inside
radical symbol.
Evaluate square root.
Multiply.
Checkpoint Evaluate the expression when a 16 and b 9.
13. a
b
Copyright © Holt McDougal. All rights reserved.
14. b2 2a
15. 2ab
Chapter 9 • Pre-Algebra Notetaking Guide
201
Checkpoint Use a calculator to approximate the square root.
Round to the nearest tenth.
9. 6
10. 104
11. 819
10.2
28.6
2.4
Example 4
12. 1874
43.3
Evaluating a Radical Expression
Evaluate 5
a2 b when a 6 and b 27.
a2 b 5
5
62 27
59
Substitute for a and for b.
Evaluate expression inside
radical symbol.
5p3
Evaluate square root.
15
Multiply.
Checkpoint Evaluate the expression when a 16 and b 9.
13. a
b
5
Copyright © Holt McDougal. All rights reserved.
14. b2 2a
7
15. 2ab
24
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201
9.2
Simplifying Square Roots
Goal: Simplify radical expressions.
Vocabulary
Simplest
form:
Product Property of Square Roots
Algebra
ab
a
p b
, where a ≥ 0 and b ≥ 0
Numbers
9
p 7 9
p 7
37
Example 1
150
Factor using greatest perfect
square factor.
Product property of square roots
Simplify.
Example 2
18t 2 202
Simplifying a Radical Expression
Simplifying a Variable Expression
Factor using greatest perfect
square factor.
Product property of square roots
Simplify.
Commutative property
Chapter 9 • Pre-Algebra Notetaking Guide
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9.2
Simplifying Square Roots
Goal: Simplify radical expressions.
Vocabulary
A radical expression is in simplest form when
(1) no factor of the expression under the radical
sign has any perfect square factor other than 1,
(2) there are no fractions under the radical sign,
and (3) there are no radical signs in the
denominator of any fraction.
Simplest
form:
Product Property of Square Roots
Algebra
ab
a
p b
, where a ≥ 0 and b ≥ 0
Numbers
9
p 7 9
p 7
37
Example 1
Simplifying a Radical Expression
150
25
p6
25
p 6
Product property of square roots
56
Simplify.
Example 2
Simplifying a Variable Expression
18t 2 9 p 2 p
t2
202
Factor using greatest perfect
square factor.
Factor using greatest perfect
square factor.
9
p 2
p t2
Product property of square roots
3 p 2
pt
Simplify.
3t 2
Commutative property
Chapter 9 • Pre-Algebra Notetaking Guide
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Checkpoint Simplify the expression.
1. 75
2. 80
3. 24r 2
4. 56m2
Quotient Property of Square Roots
Algebra
a
, where a ≥ 0 and b > 0
ab b
Numbers
11
11
11
4
2
4
Example 3
17
25
Simplifying a Radical Expression
Quotient property of square roots
Simplify.
Checkpoint Simplify the expression.
5.
Copyright © Holt McDougal. All rights reserved.
5
9
6.
21
64
7.
16 z
49
8.
32y2
81
Chapter 9 • Pre-Algebra Notetaking Guide
203
Checkpoint Simplify the expression.
1. 75
2. 80
53
3. 24r 2
4. 56m2
2r 6
2m14
45
Quotient Property of Square Roots
Algebra
a
, where a ≥ 0 and b > 0
ab b
Numbers
11
11
11
4
2
4
Example 3
17
25
Simplifying a Radical Expression
17
25
17
5
Quotient property of square roots
Simplify.
Checkpoint Simplify the expression.
5.
5
9
5
3
Copyright © Holt McDougal. All rights reserved.
6.
21
64
21
8
7.
16 z
49
4z
7
8.
32y2
81
4y 2
9
Chapter 9 • Pre-Algebra Notetaking Guide
203
Example 4
Using Radical Expressions
The expression
to fall h feet.
h
gives the time (in seconds) it takes an object
16
a. Write the expression in simplest form.
b. Find the length of time it takes for an object to fall 120 feet.
Give your answer to the nearest tenth of a second.
Solution
a.
h
16
Quotient property of square roots
Simplify.
Answer: In simplest form,
b. h
≈
h
16
.
Substitute 120 for h.
Approximate using a calculator.
Answer: The length of time it takes for an object to fall
120 feet is about
.
Checkpoint
9. Find the length of time it takes for an object to fall 155 feet.
Give your answer to the nearest tenth of a second.
204
Chapter 9 • Pre-Algebra Notetaking Guide
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Using Radical Expressions
Example 4
The expression
to fall h feet.
h
gives the time (in seconds) it takes an object
16
a. Write the expression in simplest form.
b. Find the length of time it takes for an object to fall 120 feet.
Give your answer to the nearest tenth of a second.
Solution
a.
h
16
h
16
h
4
Quotient property of square roots
Simplify.
Answer: In simplest form,
h
120
b. 4
h
16
h
.
4
Substitute 120 for h.
4
≈ 2.7
Approximate using a calculator.
Answer: The length of time it takes for an object to fall
120 feet is about 2.7 seconds .
Checkpoint
9. Find the length of time it takes for an object to fall 155 feet.
Give your answer to the nearest tenth of a second.
3.1 sec
204
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Focus On
Operations
Use after Lesson 9.2
Performing Operations
on Square Roots
Goal: Perform operations on square roots.
Example 1
Multiplying and Dividing Radical Expressions
a. 12
· 2
12
·2
·
Product property of square roots
Multiply.
Product property of square roots
Simplify.
b. 18
10
The process of
rewriting a
fraction so that
its denominator
is a rational
number is
called
rationalizing the
denominator.
Quotient property of square roots
Simplify.
Quotient property of square roots
Simplify.
· Rationalize denominator.
Multiply and simplify.
Example 2
Adding and Subtracting Radical Expressions
a. 56
32
46
32
) 32
(
32
b. 93
12
93
·
93
93
(
)
Copyright © Holt McDougal. All rights reserved.
·
Commutative
property
Distributive
property
Simplify.
Factor using perfect square
factor.
Product property of
square roots
Simplify.
Distributive property
Simplify.
Chapter 9 • Pre-Algebra Notetaking Guide
205
Focus On
Operations
Use after Lesson 9.2
Performing Operations
on Square Roots
Goal: Perform operations on square roots.
Example 1
Multiplying and Dividing Radical Expressions
a. 12
· 2
12
·2
24
4
· 6
26
18
b. 18
10
10
9
Quotient property of square roots
Simplify.
5
9
5
3
5
5
3 · 5
5
5
3
5
The process of
rewriting a
fraction so that
its denominator
is a rational
number is
called
rationalizing the
denominator.
Example 2
Product property of square roots
Multiply.
Product property of square roots
Simplify.
Quotient property of square roots
Simplify.
Rationalize denominator.
Multiply and simplify.
Adding and Subtracting Radical Expressions
a. 56
32
46
56
46
32
( 5 4 )
6 32
Commutative
property
Distributive
property
6
32
Simplify.
b. 93
12
93
Factor using perfect square
4·3
factor.
93
4
· 3
Product property of
square roots
93
23
Simplify.
( 9 2 )
3
113
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Distributive property
Simplify.
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205
Checkpoint Simplify the expression.
1. 12
· 6
2. 48
21
3. 58
62
4. 83
311
3
Example 3
Using Radical Expressions
gives the radius (in
Circular plate The expression 3.14
centimeters) of a circular plate with area A (in square centimeters).
About how much longer is the radius of a circular plate with an
area of 628 square centimeters than the radius of a circular plate
with an area of 157 square centimeters?
A
1. Find the radius when A 628.
A
3.14
Substitute
for A.
Divide.
· 2
Product property of square roots
Simplify.
2. Find the radius when A 157.
A
3.14
Substitute
for A.
Divide.
· 2
Product property of square roots
Simplify.
3. Find the difference of the lengths.
102
52
(
Answer about
206
Chapter 9 • Pre-Algebra Notetaking Guide
)
≈
Distributive property
· 1.4 ≈
Simplify and
approximate.
centimeters
Copyright © Holt McDougal. All rights reserved.
Checkpoint Simplify the expression.
1. 12
· 6
2. 48
21
62
47
7
3. 58
62
4. 83
311
3
73
3 11
162
Example 3
Using Radical Expressions
gives the radius (in
Circular plate The expression 3.14
centimeters) of a circular plate with area A (in square centimeters).
About how much longer is the radius of a circular plate with an
area of 628 square centimeters than the radius of a circular plate
with an area of 157 square centimeters?
A
1. Find the radius when A 628.
628
A
3.14
3.14
Substitute 628 for A.
200
Divide.
100
· 2
Product property of square roots
102
Simplify.
2. Find the radius when A 157.
A
3.14
3.14
157
Substitute 157 for A.
50
Divide.
· 2
25
Product property of square roots
52
Simplify.
3. Find the difference of the lengths.
102
52
( 10 5 ) 2
52
≈ 5 · 1.4 ≈ 7
Distributive property
Simplify and
approximate.
Answer about 7 centimeters
206
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9.3
The Pythagorean Theorem
Goal: Use the Pythagorean theorem to solve problems.
Vocabulary
Hypotenuse:
Legs:
Pythagorean Theorem
Words For any right triangle, the sum
of the squares of the lengths of the
legs equals the square of the length of
the hypotenuse.
Example 1
2
2
2 ft
24 ft
a2 b2 c2
Pythagorean theorem
c2
Substitute for a and for b.
c2
Evaluate powers and add.
c
Take positive square root of each side.
≈c
Simplify.
2
Answer: The length of the ramp is about
Copyright © Holt McDougal. All rights reserved.
b
Finding the Length of a Hypotenuse
A building’s access ramp has a horizontal
distance of 24 feet and a vertical distance
of 2 feet. Find the length of the ramp to the
nearest tenth of a foot.
2
c
a
legs
Algebra a b c
2
hypotenuse
feet.
Chapter 9 • Pre-Algebra Notetaking Guide
207
9.3
The Pythagorean Theorem
Goal: Use the Pythagorean theorem to solve problems.
Vocabulary
Hypotenuse:
In a right triangle, the hypotenuse is the side
opposite the right angle.
In a right triangle, the legs are the sides that form the
right angle.
Legs:
Pythagorean Theorem
Words For any right triangle, the sum
of the squares of the lengths of the
legs equals the square of the length of
the hypotenuse.
2
2
A building’s access ramp has a horizontal
distance of 24 feet and a vertical distance
of 2 feet. Find the length of the ramp to the
nearest tenth of a foot.
2
b
Finding the Length of a Hypotenuse
Example 1
2
c
a
legs
Algebra a b c
2
hypotenuse
a2 b2 c2
Pythagorean theorem
24
c2
Substitute for a and for b.
580 c2
Evaluate powers and add.
2
580
c
24.1 ≈ c
2 ft
24 ft
Take positive square root of each side.
Simplify.
Answer: The length of the ramp is about 24.1 feet.
Copyright © Holt McDougal. All rights reserved.
Chapter 9 • Pre-Algebra Notetaking Guide
207
Finding the Length of a Leg
Example 2
Find the unknown length a in simplest form.
a2 b2 c2
a2 2
a2 Pythagorean theorem
2
c 15
a
Substitute.
b 10
Evaluate powers.
a2 Subtract
from each side.
a
Take positive square root of each side.
a
Simplify.
Answer: The unknown length a is
units.
Checkpoint Find the unknown length. Write your answer in
simplest form.
1.
2.
a6
c
a
c 13
3.
a 11
c 20
b 12
b
b8
Converse of the Pythagorean Theorem
The Pythagorean theorem can be written in “if-then” form.
Theorem: If a triangle is a right triangle, then a2 b2 c2.
If you reverse the two parts of the statement, the new statement
is called the converse of the Pythagorean theorem.
Although not all
converses of true
statements are true,
the converse of the
Pythagorean theorem
is true.
208
Converse: If a2 b2 c2, then the triangle is a right triangle.
Chapter 9 • Pre-Algebra Notetaking Guide
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Finding the Length of a Leg
Example 2
Find the unknown length a in simplest form.
a2 b2 c2
a2 10
2
15
Pythagorean theorem
2
c 15
a
Substitute.
a2 100 225
b 10
Evaluate powers.
a2 125
Subtract 100 from each side.
a 125
Take positive square root of each side.
a 55
Simplify.
units.
Answer: The unknown length a is 55
Checkpoint Find the unknown length. Write your answer in
simplest form.
1.
2.
c
a6
a
c 13
3.
a 11
c 20
b 12
b
b8
10
5
331
Converse of the Pythagorean Theorem
The Pythagorean theorem can be written in “if-then” form.
Theorem: If a triangle is a right triangle, then a2 b2 c2.
If you reverse the two parts of the statement, the new statement
is called the converse of the Pythagorean theorem.
Although not all
converses of true
statements are true,
the converse of the
Pythagorean theorem
is true.
208
Converse: If a2 b2 c2, then the triangle is a right triangle.
Chapter 9 • Pre-Algebra Notetaking Guide
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Example 3
Identifying Right Triangles
Determine whether the triangle with the given side lengths is a
right triangle.
a. a 8, b 9, c 12
Solution
a.
a2 b2 c2
2 2
b. a 7, b 24, c 25
b.
2
Answer:
a2 b2 c2
2
2 2
Answer:
Checkpoint Determine whether the triangle with the given
side lengths is a right triangle.
4. a 12, b 9, c 15
Copyright © Holt McDougal. All rights reserved.
5. a 10, b 25, c 27
Chapter 9 • Pre-Algebra Notetaking Guide
209
Example 3
Identifying Right Triangles
Determine whether the triangle with the given side lengths is a
right triangle.
a. a 8, b 9, c 12
Solution
a.
a2 b2 c2
8 2 9 2 12
b.
2
64 81 144
145
b. a 7, b 24, c 25
144
Answer:
a2 b2 c2
7 2 24 2 25
2
49 576 625
625
625
Answer:
Not a right triangle
A right triangle
Checkpoint Determine whether the triangle with the given
side lengths is a right triangle.
4. a 12, b 9, c 15
yes
Copyright © Holt McDougal. All rights reserved.
5. a 10, b 25, c 27
no
Chapter 9 • Pre-Algebra Notetaking Guide
209
Focus On
Geometry
Use after Lesson 9.3
Angles of a Triangle
Goal: Determine the sum of the measures of the angles of a
triangle.
Vocabulary
Acute triangle
60°
45°
75°
Right triangle
55°
35°
Obtuse triangle
40°
105°
Equiangular triangle
35°
60°
60°
60°
Sum of the Measures of the Angles of a Triangle
C
The sum of the measures
of the interior angles
of a triangle is 180°.
B
A
maA 1 maB 1 maC 5 180°
Example 1
Finding an Angle Measure in a Triangle
Find the value of x.
X°
32°
54°
54° +
+ x° 5
+x 5
x 5
210
Chapter 9 • Pre-Algebra Notetaking Guide
Sum of angle measures is 180°.
Add.
Subtract
from each side.
Copyright © Holt McDougal. All rights reserved.
Focus On
Geometry
Use after Lesson 9.3
Angles of a Triangle
Goal: Determine the sum of the measures of the angles of a
triangle.
Vocabulary
Acute triangle
60°
3 acute angles
45°
75°
Right triangle
55°
1 right angle
35°
Obtuse triangle
40°
1 obtuse angle
105°
Equiangular triangle
35°
60°
3 congruent angles
60°
60°
Sum of the Measures of the Angles of a Triangle
C
The sum of the measures
of the interior angles
of a triangle is 180°.
B
A
maA 1 maB 1 maC 5 180°
Example 1
Finding an Angle Measure in a Triangle
Find the value of x.
X°
32°
54°
54° + 32° + x° 5 180°
86 + x 5 180
x 5 94
210
Chapter 9 • Pre-Algebra Notetaking Guide
Sum of angle measures is 180°.
Add.
Subtract 86 from each side.
Copyright © Holt McDougal. All rights reserved.
Classifying a Triangle by its Angle Measures
Example 2
Find the values of x, y, and z. Then classify all triangles in the
diagram by their angle measures.
B
64°
34°
A
x°
102° y°
D
z°
C
1. Write and solve an equation to find the value of x.
x° + 34° +
=
Sum of angle measures is 180°.
x+
=
Add.
x=
Subtract
from each side.
2. Write and solve an equation to find the value of y.
y° +
=
The measure of a straight
angle is 180°.
y=
Subtract
from each side.
3. Write and solve an equation to find the value of z.
z° + 78° +
=
Sum of angle measures is 180°.
z+
=
Add.
z=
Subtract
from each side.
4. TABD has
, so it is a(n)
triangle.
TBCD has
, so it is an
triangle.
The measure of aB is 34° +
=
, so TABC has
and is a(n)
triangle.
Checkpoint Complete the following exercise.
1. Find the values of x, y, and z. Then classify all triangles in the
diagram by their angle measures.
B
x° z°
A
Copyright © Holt McDougal. All rights reserved.
26°
120° y°
D
60°
C
Chapter 9 • Pre-Algebra Notetaking Guide
211
Classifying a Triangle by its Angle Measures
Example 2
Find the values of x, y, and z. Then classify all triangles in the
diagram by their angle measures.
B
64°
34°
A
x°
102° y°
D
z°
C
1. Write and solve an equation to find the value of x.
x° + 34° + 102° = 180°
Sum of angle measures is 180°.
Add.
x + 136 = 180
Subtract 136 from each side.
x = 44
2. Write and solve an equation to find the value of y.
y° + 102° = 180°
The measure of a straight
angle is 180°.
y = 78°
Subtract 102 from each side.
3. Write and solve an equation to find the value of z.
z° + 78° + 64° = 180°
Sum of angle measures is 180°.
Add.
z + 142 = 180
Subtract 142 from each side.
z = 38
4. T ABD has one obtuse angle , so it is a(n) obtuse triangle.
T BCD has three acute angles , so it is an acute triangle.
The measure of aB is 34° + 64° = 98° , so T ABC has
one obtuse angle and is a(n) obtuse triangle.
Checkpoint Complete the following exercise.
1. Find the values of x, y, and z. Then classify all triangles in the
diagram by their angle measures.
B
x = 34, y = 60, z = 60;
T ABD is obtuse,
T BCD is equiangular,
T ABC is obtuse
x° z°
A
Copyright © Holt McDougal. All rights reserved.
26°
120° y°
D
60°
C
Chapter 9 • Pre-Algebra Notetaking Guide
211
9.4
Real Numbers
Goal: Compare and order real numbers.
Vocabulary
Irrational
number:
Real
numbers:
Classifying Real Numbers
Example 1
Number Decimal Form
7
10
7
10
b. 2
9
2
9
c. 5
5
a. The square root of
any whole number
that is not a perfect
square is irrational.
Example 2
Decimal Type
Type
Comparing Real Numbers
6
5
Copy and complete 3
__
? using <, >, or .
Solution
6
5
Graph 3
and on a number line.
3
is to the
0
Answer: 3
212
Chapter 9 • Pre-Algebra Notetaking Guide
1
6
of .
5
2
6
5
Copyright © Holt McDougal. All rights reserved.
9.4
Real Numbers
Goal: Compare and order real numbers.
Vocabulary
Irrational An irrational number is a number that cannot be
number: written as a quotient of two integers.
Real numbers consist of all rational and irrational
Real
numbers: numbers.
Example 1
Classifying Real Numbers
Number Decimal Form
Type
7
10
7
0.7
10
Terminating
Rational
b. 2
9
2
0.222. . . 0.2
9
Repeating
Rational
c. 5
5
2.23606797. . .
Nonterminating,
Irrational
a. The square root of
any whole number
that is not a perfect
square is irrational.
Decimal Type
nonrepeating
Example 2
Comparing Real Numbers
6
5
Copy and complete 3
__
? using <, >, or .
Solution
6
5
Graph 3
and on a number line.
6
5
6
3
is to the right of .
3
5
0
1
2
6
5
Answer: 3
> 212
Chapter 9 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Ordering Real Numbers
Example 3
6
5
Use a number line to order the numbers 2 , 2.2, 2, and
from least to greatest.
22
Graph the numbers on a number line and read them from left
to right.
3
2
1
0
1
2
3
Answer: From least to greatest, the numbers are
.
Checkpoint Tell whether the number is rational or irrational.
8
11
1. 2. 7
3. 49
4.
25
Copy and complete the statement using <, >, or .
3
2
5. __
? 3
Copyright © Holt McDougal. All rights reserved.
6. 10
__
? 3.5
Chapter 9 • Pre-Algebra Notetaking Guide
213
Ordering Real Numbers
Example 3
6
5
Use a number line to order the numbers 2 , 2.2, 2, and
from least to greatest.
22
Graph the numbers on a number line and read them from left
to right.
6
2
2 2 2.2
3
2
1
0
1
5
2
2
3
Answer: From least to greatest, the numbers are
5
6
22
, 2.2, , and .
2
2
Checkpoint Tell whether the number is rational or irrational.
8
11
2. 7
1. rational
irrational
3. 49
4.
rational
25
irrational
Copy and complete the statement using <, >, or .
3
2
5. __
? 3
>
Copyright © Holt McDougal. All rights reserved.
6. 10
__
? 3.5
<
Chapter 9 • Pre-Algebra Notetaking Guide
213
Checkpoint Use a number line to order the numbers from
least to greatest.
27
5
7. 5.9, 35
, , 35
9
2
8. 21
, 4.6, , 25
Example 4
Using Irrational Numbers
Speed After an accident, a police officer finds that the length of
a car’s skid marks is 98 feet. The car’s speed s (in miles per hour)
.
and the length l (in feet) of the skid marks are related by s 27l
Find the car’s speed to the nearest tenth of a mile per hour.
Solution
s 27l
27 p
≈
Write formula.
Substitute value.
Multiply.
Approximate using a calculator.
Answer: The car’s speed was about
214
Chapter 9 • Pre-Algebra Notetaking Guide
miles per hour.
Copyright © Holt McDougal. All rights reserved.
Checkpoint Use a number line to order the numbers from
least to greatest.
27
5
7. 5.9, 35
, , 35
27
, 5.9, 35
, 35
5
9
2
8. 21
, 4.6, , 25
9
2
4.6, 21
, , 25
Example 4
Using Irrational Numbers
Speed After an accident, a police officer finds that the length of
a car’s skid marks is 98 feet. The car’s speed s (in miles per hour)
.
and the length l (in feet) of the skid marks are related by s 27l
Find the car’s speed to the nearest tenth of a mile per hour.
Solution
s 27l
27 p 98
2646
≈ 51.4
Write formula.
Substitute value.
Multiply.
Approximate using a calculator.
Answer: The car’s speed was about 51.4 miles per hour.
214
Chapter 9 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
9.5
Distance and Midpoint Formulas
Goal: Use the distance, midpoint, and slope formulas.
The Distance Formula
Words The distance between two
points in a coordinate plane is equal
to the square root of the sum of the
horizontal change squared and the
vertical change squared.
2
x2 x1 )
(y2
y1 )2
Algebra d (
Example 1
y
(x2, y2)
d
(x1, y1)
y2 y1
x2 x1
x
O
Finding the Distance Between Two Points
Find the distance between the points M(2, 3) and N(1, 5).
2
d (
x2 x1 )
(y2
y1 )2
(1 ) (5 )
2
2
( ) Distance formula
2
2
Substitute.
Subtract.
Evaluate powers.
Add.
Answer: The distance between the points is
units.
Checkpoint Find the distance between the points. Write your
answer in simplest form.
1. (2, 5), (3, 7)
Copyright © Holt McDougal. All rights reserved.
2. (2, 2), (0, 4)
Chapter 9 • Pre-Algebra Notetaking Guide
215
9.5
Distance and Midpoint Formulas
Goal: Use the distance, midpoint, and slope formulas.
The Distance Formula
Words The distance between two
points in a coordinate plane is equal
to the square root of the sum of the
horizontal change squared and the
vertical change squared.
2
x2 x1 )
(y2
y1 )2
Algebra d (
Example 1
y
(x2, y2)
d
(x1, y1)
y2 y1
x2 x1
x
O
Finding the Distance Between Two Points
Find the distance between the points M(2, 3) and N(1, 5).
2
d (
x2 x1 )
(y2
y1 )2
Distance formula
(1 2 ) (5 3 )
2
2
( 1 ) 2
1 4
2
2
5
Substitute.
Subtract.
Evaluate powers.
Add.
units.
Answer: The distance between the points is 5
Checkpoint Find the distance between the points. Write your
answer in simplest form.
1. (2, 5), (3, 7)
13
Copyright © Holt McDougal. All rights reserved.
2. (2, 2), (0, 4)
210
Chapter 9 • Pre-Algebra Notetaking Guide
215
The Midpoint Formula
Words The coordinates of the
midpoint of a segment are the
average of the endpoints’
x-coordinates and the average
of the endpoints’ y-coordinates.
y
y1 y2
2
Example 2
M
(x
1
A(x1, y1)
x2 y1 y2
,
2
2
)
y1
x1 x2 y1 y2
, Algebra M 2
2
B(x2, y2)
y2
O
x1 x1 x2 x2
2
x
Finding a Midpoint
Find the midpoint M of the segment with endpoints (2, 5)
and (6, 3).
x x
2
y y
2
1
2
1
2
,
M Midpoint formula
2
2
, Substitute values.
Simplify.
Checkpoint Find the midpoint of the segment with the
given endpoints.
3. (2, 5), (3, 7)
216
Chapter 9 • Pre-Algebra Notetaking Guide
4. (3, 4), (7, 2)
Copyright © Holt McDougal. All rights reserved.
The Midpoint Formula
Words The coordinates of the
midpoint of a segment are the
average of the endpoints’
x-coordinates and the average
of the endpoints’ y-coordinates.
x1 x2 y1 y2
, Algebra M 2
2
y
B(x2, y2)
y2
y1 y2
2
M
(x
1
A(x1, y1)
x2 y1 y2
,
2
2
)
y1
O
x1 x1 x2 x2
2
x
Finding a Midpoint
Example 2
Find the midpoint M of the segment with endpoints (2, 5)
and (6, 3).
x x
2
y y
2
1
2
1
2
,
M Midpoint formula
2 (6)
5 (3)
, 2
2
(2, 1)
Substitute values.
Simplify.
Checkpoint Find the midpoint of the segment with the
given endpoints.
3. (2, 5), (3, 7)
12, 1
216
Chapter 9 • Pre-Algebra Notetaking Guide
4. (3, 4), (7, 2)
(5, 1)
Copyright © Holt McDougal. All rights reserved.
Slope
If points A(x1, y1) and B(x2, y2) do not lie on a vertical line, you can
use coordinate notation to write a formula for the slope of the line
through A and B.
y y
x2 x1
difference of y-coordinates
difference of x-coordinates
2
1
slope Example 3
Finding Slope
Find the slope of the line through (2, 1) and (1, 2).
y y
x2 x1
2
1
slope Slope formula
Substitute values.
Simplify.
Checkpoint Find the slope of the line through the given points.
5. (2, 6), (3, 9)
Copyright © Holt McDougal. All rights reserved.
6. (6, 2), (4, 7)
Chapter 9 • Pre-Algebra Notetaking Guide
217
Slope
If points A(x1, y1) and B(x2, y2) do not lie on a vertical line, you can
use coordinate notation to write a formula for the slope of the line
through A and B.
y y
x2 x1
difference of y-coordinates
difference of x-coordinates
2
1
slope Example 3
Finding Slope
Find the slope of the line through (2, 1) and (1, 2).
y y
x2 x1
2
1
slope Slope formula
2 (1)
1 2
Substitute values.
3
1
3
Simplify.
Checkpoint Find the slope of the line through the given points.
5. (2, 6), (3, 9)
3
Copyright © Holt McDougal. All rights reserved.
6. (6, 2), (4, 7)
9
10
Chapter 9 • Pre-Algebra Notetaking Guide
217
9.6
Special Right Triangles
Goal: Use special right triangles to solve problems.
45-45-90 Triangle
Words In a 45-45-90 triangle,
the length of the hypotenuse is
the product of the length of a leg
.
and 2
45
a 2
a
45
a
Algebra hypotenuse leg p 2
a2
Example 1
Using a 45-45-90 Triangle
A 45-45-90 triangle used
in mechanical drawing has
10-inch legs. Find the length
of the hypotenuse to the nearest
tenth of an inch.
45
10 in.
45
Solution
10 in.
hypotenuse leg p 2
≈
p 2
Rule for 45-45-90 triangle
Substitute.
Use a calculator.
Answer: The length of the triangle’s hypotenuse is about
inches.
Checkpoint
1. Find the unknown length x. Write
your answer in simplest form.
45
6
x
45
6
218
Chapter 9 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
9.6
Special Right Triangles
Goal: Use special right triangles to solve problems.
45-45-90 Triangle
Words In a 45-45-90 triangle,
the length of the hypotenuse is
the product of the length of a leg
.
and 2
45
a 2
a
45
a
Algebra hypotenuse leg p 2
a2
Example 1
Using a 45-45-90 Triangle
A 45-45-90 triangle used
in mechanical drawing has
10-inch legs. Find the length
of the hypotenuse to the nearest
tenth of an inch.
45
10 in.
45
Solution
10 in.
hypotenuse leg p 2
Rule for 45-45-90 triangle
10 p 2
Substitute.
≈ 14.1
Use a calculator.
Answer: The length of the triangle’s hypotenuse is about
14.1 inches.
Checkpoint
1. Find the unknown length x. Write
your answer in simplest form.
45
6
x
45
62
218
Chapter 9 • Pre-Algebra Notetaking Guide
6
Copyright © Holt McDougal. All rights reserved.
30-60-90 Triangle
In a 30-60-90
triangle, the shorter
leg is opposite the
30 angle, and the
longer leg is opposite
the 60 angle.
Words In a 30-60-90 triangle,
the length of the hypotenuse is
twice the length of the shorter leg.
The length of the longer leg is the
product of the length of the shorter
.
leg and 3
30
2a
a 3
60
a
Algebra hypotenuse 2 p shorter leg 2a
longer leg shorter leg p 3
a3
Using a 30-60-90 Triangle
Example 2
Find the length x of the hypotenuse and
the length y of the longer leg of the triangle.
12
x
60
30
y
The triangle is a 30-60-90 triangle.
The length of the shorter leg is
units.
a. hypotenuse 2 p shorter leg
x2p
Answer: The length x of the hypotenuse is
units.
b. longer leg shorter leg p 3
y
3
3
units.
Answer: The length y of the longer leg is
Checkpoint
2. Find the unknown lengths x and y.
Write your answers in simplest form.
15
60
x
30
y
Copyright © Holt McDougal. All rights reserved.
Chapter 9 • Pre-Algebra Notetaking Guide
219
30-60-90 Triangle
In a 30-60-90
triangle, the shorter
leg is opposite the
30 angle, and the
longer leg is opposite
the 60 angle.
Words In a 30-60-90 triangle,
the length of the hypotenuse is
twice the length of the shorter leg.
The length of the longer leg is the
product of the length of the shorter
.
leg and 3
30
2a
a 3
60
a
Algebra hypotenuse 2 p shorter leg 2a
longer leg shorter leg p 3
a3
Using a 30-60-90 Triangle
Example 2
Find the length x of the hypotenuse and
the length y of the longer leg of the triangle.
12
x
60
30
y
The triangle is a 30-60-90 triangle.
The length of the shorter leg is 12 units.
a. hypotenuse 2 p shorter leg
x 2 p 12
24
Answer: The length x of the hypotenuse is 24 units.
b. longer leg shorter leg p 3
y 12 3
Answer: The length y of the longer leg is 12 3
units.
Checkpoint
2. Find the unknown lengths x and y.
Write your answers in simplest form.
15
60
x
30
y
x 30, y 153
Copyright © Holt McDougal. All rights reserved.
Chapter 9 • Pre-Algebra Notetaking Guide
219
Example 3
Using a Special Right Triangle
An escalator going up to the second floor in a
mall is 224 feet long and makes a 30 angle
with the first floor. Find, to the nearest foot,
the lengths of the triangle’s legs.
60
224 ft
x
30
y
Solution
You need to find the length of the shorter leg first.
1. Find the length x of the shorter leg.
hypotenuse 2 p shorter leg
Rule for 30-60-90 triangle
2x
Substitute.
x
Divide each side by
.
2. Find the length y of the longer leg.
longer leg shorter leg p 3
y
3
y≈
Rule for 30-60-90 triangle
Substitute.
Use a calculator.
Answer: The length of the shorter leg is
length of the longer leg is about
220
Chapter 9 • Pre-Algebra Notetaking Guide
feet and the
feet.
Copyright © Holt McDougal. All rights reserved.
Example 3
Using a Special Right Triangle
An escalator going up to the second floor in a
mall is 224 feet long and makes a 30 angle
with the first floor. Find, to the nearest foot,
the lengths of the triangle’s legs.
60
224 ft
x
30
y
Solution
You need to find the length of the shorter leg first.
1. Find the length x of the shorter leg.
hypotenuse 2 p shorter leg
Rule for 30-60-90 triangle
224 2x
Substitute.
112 x
Divide each side by 2 .
2. Find the length y of the longer leg.
longer leg shorter leg p 3
Rule for 30-60-90 triangle
y 112 3
Substitute.
y ≈ 194
Use a calculator.
Answer: The length of the shorter leg is 112 feet and the
length of the longer leg is about 194 feet.
220
Chapter 9 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
9.7
The Tangent Ratio
Goal: Use tangent to find side lengths of right triangles.
Vocabulary
Trigonometric
ratio:
The Tangent Ratio
The tangent of an acute angle of
a right triangle is the ratio of the
length of the side opposite the
angle to the length of the side
adjacent to the angle.
side opposite aA
side adjacent to aA
B
hypotenuse
c
A
a
b
tan A Example 1
side
a opposite
A
C
b
side adjacent to A
Finding a Tangent Ratio
For T PQR, find the tangent of aP.
opposite
tan P adjacent
P
39
R
89
80
Checkpoint
1. For T PQR in Example 1, find the tangent of aQ.
Copyright © Holt McDougal. All rights reserved.
Chapter 9 • Pre-Algebra Notetaking Guide
221
9.7
The Tangent Ratio
Goal: Use tangent to find side lengths of right triangles.
Vocabulary
Trigonometric A trigonometric ratio is a ratio of the lengths of
two sides of a right triangle.
ratio:
The Tangent Ratio
The tangent of an acute angle of
a right triangle is the ratio of the
length of the side opposite the
angle to the length of the side
adjacent to the angle.
side opposite aA
side adjacent to aA
B
hypotenuse
c
A
a
b
tan A Example 1
side
a opposite
A
C
b
side adjacent to A
Finding a Tangent Ratio
For T PQR, find the tangent of aP.
P
39
opposite
80
tan P 3
9
adjacent
R
89
80
Checkpoint
1. For T PQR in Example 1, find the tangent of aQ.
39
80
Copyright © Holt McDougal. All rights reserved.
Chapter 9 • Pre-Algebra Notetaking Guide
221
Using a Calculator
Example 2
a. tan 24
Keystrokes
When using a
calculator to find a
trigonometric ratio,
make sure the
calculator is in
degree mode. Round
the result to four
decimal places if
necessary.
2nd
Display
Answer
Display
Answer
[TRIG]
24
)
b. tan 55
Keystrokes
2nd
[TRIG]
55
)
Checkpoint Approximate the tangent value to four
decimal places.
2. tan 5
3. tan 38
4. tan 72
Using a Tangent Ratio
Example 3
Find the height h (in feet) of
the roof to the nearest foot.
h
Solution
27
30 ft
30 ft
Use the tangent ratio. In the diagram, the length of the leg opposite
the 27 angle is h. The length of the adjacent leg is 30 feet.
opposite
adjacent
tan 27 Definition of tangent ratio
tan 27 Substitute.
≈
Use a calculator to approximate tan 27.
≈h
Multiply each side by
Answer: The height of the roof is about
222
Chapter 9 • Pre-Algebra Notetaking Guide
.
feet.
Copyright © Holt McDougal. All rights reserved.
Using a Calculator
Example 2
a. tan 24
Keystrokes
When using a
calculator to find a
trigonometric ratio,
make sure the
calculator is in
degree mode. Round
the result to four
decimal places if
necessary.
2nd
Display
[TRIG]
24
Answer
0.445228685
0.4452
)
b. tan 55
Keystrokes
2nd
Display
[TRIG]
55
Answer
1.428148007
1.4281
)
Checkpoint Approximate the tangent value to four
decimal places.
2. tan 5
3. tan 38
0.0875
Example 3
4. tan 72
0.7813
3.0777
Using a Tangent Ratio
Find the height h (in feet) of
the roof to the nearest foot.
h
Solution
27
30 ft
30 ft
Use the tangent ratio. In the diagram, the length of the leg opposite
the 27 angle is h. The length of the adjacent leg is 30 feet.
opposite
adjacent
tan 27 Definition of tangent ratio
h
tan 27 30
Substitute.
h
0.5095 ≈ 30
15 ≈ h
Use a calculator to approximate tan 27.
Multiply each side by 30 .
Answer: The height of the roof is about 15 feet.
222
Chapter 9 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
9.8
The Sine and Cosine Ratios
Goal: Use sine and cosine to find triangle side lengths.
The Sine and Cosine Ratios
The sine of an acute angle of a
right triangle is the ratio of the
length of the side opposite the
angle to the length of the
hypotenuse.
side opposite aA
hypotenuse
a
c
sin A B
hypotenuse
c
A
side
a opposite
A
C
b
side adjacent to A
The cosine of an acute angle of a right triangle is the ratio of the
length of the angle’s adjacent side to the length of the hypotenuse.
side adjacent aA
hypotenuse
b
c
cos A Finding Sine and Cosine Ratios
Example 1
For T PQR, find the sine and cosine of aP.
opposite
sin P hypotenuse
P
39
R
89
80
adjacent
hypotenuse
cos P Checkpoint
1. For T PQR in Example 1, find the sine and cosine of aQ.
Copyright © Holt McDougal. All rights reserved.
Chapter 9 • Pre-Algebra Notetaking Guide
223
9.8
The Sine and Cosine Ratios
Goal: Use sine and cosine to find triangle side lengths.
The Sine and Cosine Ratios
The sine of an acute angle of a
right triangle is the ratio of the
length of the side opposite the
angle to the length of the
hypotenuse.
side opposite aA
hypotenuse
a
c
sin A B
hypotenuse
c
A
side
a opposite
A
C
b
side adjacent to A
The cosine of an acute angle of a right triangle is the ratio of the
length of the angle’s adjacent side to the length of the hypotenuse.
side adjacent aA
hypotenuse
b
c
cos A Finding Sine and Cosine Ratios
Example 1
For T PQR, find the sine and cosine of aP.
P
39
opposite
80
sin P 8
9
hypotenuse
R
89
80
adjacent
39
cos P 8
9
hypotenuse
Checkpoint
1. For T PQR in Example 1, find the sine and cosine of aQ.
39
89
80
89
sin Q , cos Q Copyright © Holt McDougal. All rights reserved.
Chapter 9 • Pre-Algebra Notetaking Guide
223
Using a Calculator
Example 2
a. sin 60
Keystrokes
2nd
60
Answer
Display
Answer
[TRIG]
)
Display
b. cos 45
Keystrokes
2nd
[TRIG]
45
)
Checkpoint Approximate the sine or cosine value to four
decimal places.
2. cos 9
3. cos 78
4. sin 13
5. sin 88
Using a Cosine Ratio
Example 3
Find the value of x in the triangle.
In T DEF, DE
&* is adjacent to aD.
Because you know the length of
the hypotenuse, use cos D and
the definition of the cosine ratio
to find the value of x. Round
your answer to the nearest tenth
of a unit.
adjacent
hypotenuse
cos D 224
Chapter 9 • Pre-Algebra Notetaking Guide
F
25
39
E
D
x
Definition of cosine ratio
Substitute.
≈
Use a calculator to approximate cos 39.
≈x
Multiply each side by
.
Copyright © Holt McDougal. All rights reserved.
Using a Calculator
Example 2
a. sin 60
Keystrokes
2nd
60
[TRIG]
)
Display
Answer
0.866025404
0.8660
Display
Answer
0.707106781
0.7071
b. cos 45
Keystrokes
2nd
[TRIG]
45
)
Checkpoint Approximate the sine or cosine value to four
decimal places.
2. cos 9
3. cos 78
4. sin 13
5. sin 88
0.2079
0.2250
0.9994
0.9877
Example 3
Using a Cosine Ratio
Find the value of x in the triangle.
In T DEF, DE
&* is adjacent to aD.
Because you know the length of
the hypotenuse, use cos D and
the definition of the cosine ratio
to find the value of x. Round
your answer to the nearest tenth
of a unit.
adjacent
hypotenuse
cos D x
cos 39 25
x
25
0.7771 ≈ 19.4 ≈ x
224
Chapter 9 • Pre-Algebra Notetaking Guide
F
25
39
E
x
D
Definition of cosine ratio
Substitute.
Use a calculator to approximate cos 39.
Multiply each side by 25 .
Copyright © Holt McDougal. All rights reserved.
Using a Sine Ratio
Example 4
A ski jump is 140 meters long and makes
an angle of 25 with the ground. To the
nearest meter, estimate the height h of
the ski jump.
140 m
h
25
Solution
To estimate the height of the ski jump, find the length of the side
the 25 angle. Because you know the length of the
hypotenuse, use sin 25.
opposite
hypotenuse
sin 25 Definition of sine ratio
sin 25 Substitute.
≈
Use a calculator to approximate sin 25.
≈h
Multiply each side by
Answer: The height of the ski jump is about
Copyright © Holt McDougal. All rights reserved.
.
meters.
Chapter 9 • Pre-Algebra Notetaking Guide
225
Example 4
Using a Sine Ratio
A ski jump is 140 meters long and makes
an angle of 25 with the ground. To the
nearest meter, estimate the height h of
the ski jump.
140 m
h
25
Solution
To estimate the height of the ski jump, find the length of the side
opposite the 25 angle. Because you know the length of the
hypotenuse, use sin 25.
opposite
hypotenuse
sin 25 Definition of sine ratio
h
sin 25 140
Substitute.
h
140
0.4226 ≈ 59 ≈ h
Use a calculator to approximate sin 25.
Multiply each side by 140 .
Answer: The height of the ski jump is about 59 meters.
Copyright © Holt McDougal. All rights reserved.
Chapter 9 • Pre-Algebra Notetaking Guide
225
9
Words to Review
Give an example of the vocabulary word.
226
Square root
Perfect square
Radical
expression
Simplest form of a radical
expression
Hypotenuse
Leg
Pythagorean theorem
Acute triangle
Chapter 9 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
9
Words to Review
Give an example of the vocabulary word.
Perfect square
Square root
A square root of a number
n is the number m such that
m2 n.
Radical
expression
A number that is the square
of an integer. 4 is a perfect
square because 4 22.
Simplest form of a radical
expression
An expression that involves a
radical sign, .
A radical expression is in
simplest form when (1) no
factor of the expression under
the radical sign is a perfect
square other than 1, and (2)
there are no fractions under
the radical sign and no radical
sign in the denominator of
any fraction.
Leg
Hypotenuse
&* is the
In T DEF, DF
hypotenuse.
&* and EF
&* are legs.
In T DEF, DE
F
F
D
E
E
D
Pythagorean theorem
For any right triangle, the
sum of the squares of the
lengths a and b of the legs
equals the square of the
length c of the hypotenuse:
a2 b2 c 2.
Acute triangle
53°
68°
59°
226
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Right triangle
Obtuse triangle
Equiangular triangle
Irrational number
Real number
Distance formula
Midpoint formula
Slope formula
Sine
Cosine
Copyright © Holt McDougal. All rights reserved.
Chapter 9 • Pre-Algebra Notetaking Guide
227
Right triangle
Obtuse triangle
58°
35°
124°
32°
Equiangular triangle
Irrational number
A number that cannot be
written as a quotient of two
integers. The decimal form of
an irrational number neither
terminates nor repeats.
60°
60°
60°
Real number
Distance formula
The set of all rational numbers
and irrational numbers. For
1
example, 3, 0, , π, 5.17,
3
and 44 are all real numbers.
Midpoint formula
x1 x2 y1 y2
M , 2
2
Sine
If points A(x1, y1) and B(x2, y2)
do not lie on a vertical line,
&( is
then the slope of ^AB
y2 y1
.
x2 x1
P
5
3
R
4
opposite
hypotenuse
3
5
sin Q Copyright © Holt McDougal. All rights reserved.
(x2 x1)2 (y2 y1)2.
d Cosine
P
R
The distance d between two
points (x1, y1) and (x2, y2) is
given by
Slope formula
The coordinates of the
midpoint M of a segment
with endpoints A(x1, y1) and
B(x2, y2) are given by
3
21°
5
4
adjacent
hypotenuse
4
5
cos Q Chapter 9 • Pre-Algebra Notetaking Guide
227
Tangent
Review your notes and Chapter 9 by using the Chapter Review on
pages 526–529 of your textbook.
228
Chapter 9 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Tangent
P
3
R
5
4
opposite
adjacent
3
4
tan Q Review your notes and Chapter 9 by using the Chapter Review on
pages 526–529 of your textbook.
228
Chapter 9 • Pre-Algebra Notetaking Guide
Copyright © Holt McDougal. All rights reserved.