Pre-Algebra Interactive Chalkboard
Copyright © by The McGraw-Hill Companies, Inc.
Send all inquiries to:
GLENCOE DIVISION
Glencoe/McGraw-Hill
8787 Orion Place
Columbus, Ohio 43240
Click the mouse button or press the
Space Bar to display the answers.
Lesson 9-1 Squares and Square Roots
Lesson 9-2 The Real Number System
Lesson 9-3 Angles
Lesson 9-4 Triangles
Lesson 9-5 The Pythagorean Theorem
Lesson 9-6 The Distance and Midpoint Formulas
Lesson 9-7 Similar Triangles and Indirect Measurement
Lesson 9-8 Sine, Cosine, and Tangent Ratios
Example 1 Find Square Roots
Example 2 Calculate Square Roots
Example 3 Estimate Square Roots
Example 4 Use Square Roots to Solve a Problem
Find
.
indicates the positive square root of 64.
Since
Answer: 8
Find
.
indicates the negative square root of 121.
Since
Answer: –11
Find
.
indicates both square roots of 4.
Since
Answer: 2 and –2
Find each square root.
a.
Answer: 5
b.
Answer: –12
c.
Answer: 4 and –4
Use a calculator to find
nearest tenth.
2nd
[
] 23
ENTER
to the
4.79583152
Use a calculator.
Round to the
nearest tenth.
Check
Since
, the
answer is reasonable.
Answer: 4.8
Use a calculator to find
nearest tenth.
2nd
[
] 46
ENTER
to the
6.78232998
Use a calculator.
Round to the
nearest tenth.
Answer: –6.8
Check
Since
, the
answer is reasonable.
Use a calculator to find each square root to the
nearest tenth.
a.
Answer: 8.4
b.
Answer: –6.2
Estimate
to the nearest whole number.
Find the two perfect squares closest to 22. To do this, list
some perfect squares.
1, 4, 9, 16, 25, …
16 and 25 are closest to 22.
16 <
22 <
<
<
<
<
4
25
22 is between 16 and 25.
is between
5
and
and
.
Since 22 is closer to 25 than 16, the best whole number
estimate for
is 5.
Answer: 5
.
Estimate
to the nearest whole number.
Find the two perfect squares closest to 319. To do this,
list some perfect squares.
..., 225, 256, 289, 324, …
289 and 324 are closest to 319.
–324 < –319
<
< –289
–319 is between –324
and –289.
<
and
–18 <
< –17
is between
.
and
.
Since –319 is closer to –324 than –289, the best whole
number estimate for
is –18.
Check
Answer: –18
Estimate each square root to the nearest
whole number.
a.
Answer: 7
b.
Answer: –12
Landmarks The observation deck at the Seattle
Space Needle is 520 feet above the ground. On a
clear day, about how far could a tourist on the deck
see? Round to the nearest tenth.
Use the formula
where D is the distance
in miles and A is the altitude, or height, in feet.
Write the formula.
Replace A with 520.
Evaluate the square
root first.
Multiply.
Answer: On a clear day, a tourist could see about
27.8 miles.
Skyscraper A skyscraper stands 378 feet high. On a
clear day, about how far could an individual standing
on the roof of the skyscraper see? Round to the
nearest tenth.
Answer: On a clear day, an individual could see about
23.7 miles.
Click the mouse button or press the
Space Bar to display the answers.
Example 1 Classify Real Numbers
Example 2 Compare Real Numbers on a Number Line
Example 3 Solve Equations
Name all of the sets of numbers to which the real
number 17 belongs.
Answer: This number is a natural number, a whole
number, an integer, and a rational number.
Name all of the sets of numbers to which the real
number
belongs.
Answer: Since
, this number is an integer
and a rational number.
Name all of the sets of numbers to which the real
number
belongs.
Answer: Since
, this number is a natural
number, a whole number, an integer, and a
rational number.
Name all of the sets of numbers to which the real
number
belongs.
Answer: This repeating decimal is a rational number
because it is equivalent to
.
Name all of the sets of numbers to which the real
number
Answer:
belongs.
It is not the square root
of a perfect square so it is irrational.
Name all of the sets of numbers to which each real
number belongs.
a. 31
Answer: natural number, whole number,
integer, rational number
b.
Answer: integer, rational number
c. 4.375
Answer: rational number
d.
Answer: natural number, whole number,
integer, rational number
e.
Answer: irrational number
Replace  with <, >, or = to make
true statement.
Express each number as a decimal. Then graph
the numbers.
a
Answer: Since
is to the left of
Order
from least to greatest.
Express each number as a decimal. Then compare
the decimals.
Answer: From least to greatest, the order
is
a. Replace  with <, >, or = to make
a true statement.
Answer: >
b. Order
to greatest.
Answer:
from least
Solve
if necessary.
. Round to the nearest tenth,
Write the equation.
Take the square root
of each side.
Find the positive and
negative square root.
Answer: The solutions are 13 and –13.
Solve
. Round to the nearest tenth,
if necessary.
Write the equation.
Take the square root
of each side.
Find the positive and
negative square root.
Use a calculator.
Answer: The solutions are 7.1 and –7.1.
Solve each equation. Round to the nearest tenth,
if necessary.
a.
Answer: 9 and –9
b.
Answer: 4.9 and –4.9
Click the mouse button or press the
Space Bar to display the answers.
Example 1 Measure Angles
Example 2 Draw Angles
Example 3 Classify Angles
Example 4 Use Angles to Solve a Problem
Use a protractor to measure RSW.
Step 1 Place the center point of the protractor’s base
on vertex S. Align the straight edge with side
so that the marker for 0° is on the ray.
Use a protractor to measure RSW.
42°
Step 2 Use the scale that begins with 0° at
. Read
where the other side of the angle,
, crosses
this scale.
Use a protractor to measure RSW.
42°
Answer: The measure of angle RSW is 42°.
Using symbols,
Find the measurements of GUM, SUM,
and BUG.
120°
Answer:
is at 0° on the right.
Find the measurements of GUM, SUM,
and BUG.
32°
Answer:
is at 0° on the right.
Find the measurements of GUM, SUM,
and BUG.
60°
Answer:
is at 0° on the left.
a. Use a protractor to measure ABC.
Answer: 75°
b. Find the measures of FDE, GDE, and HDG.
Answer: FDE = 37°, GDE = 118°, HDG = 62°
Draw R having a measurement of 145°.
R
Step 1
Draw a ray with endpoint R.
Draw R having a measurement of 145°.
R
Step 2
Place the center point of the protractor on R.
Align the mark labeled 0 with the ray.
Draw R having a measurement of 145°.
145°
R
Step 3
Use the scale that begins with 0. Locate
the mark labeled 145. Then draw the
other side of the angle.
Answer:
145°
R
Draw M having a measurement of 47°.
Answer:
Classify the angle as acute, obtuse, right,
or straight.
mKLM < 90.
Answer: KLM is acute.
Classify the angle as acute, obtuse, right,
or straight.
mNPQ = 180.
Answer: NPQ is straight.
Classify the angle as acute, obtuse, right,
or straight.
mRST > 90.
Answer: RST is obtuse.
Classify each angle as acute, obtuse, right,
or straight.
a.
Answer: right
b.
Answer: obtuse
Classify each angle as acute, obtuse, right,
or straight.
c.
Answer: straight
The diagram shows the angle between the back of a
chair and the seat of the chair. Classify this angle.
Answer: Since 95° is greater than
90°, the angle is obtuse.
The diagram shows the angle between the bed of the
truck and the frame of the truck. Classify this angle.
Answer: The angle is acute.
Click the mouse button or press the
Space Bar to display the answers.
Example 1 Find Angle Measures
Example 2 Use Ratios to Find Angle Measures
Example 3 Classify Triangles
Find the value of x in DEF.
The sum of the
measures is 180.
Replace mD with
100 and mE with 33.
Simplify.
Subtract 133 from
each side.
Answer: The measure of F is 47°.
Find the value of x in MNO.
Answer: The measure of N is 57°.
Algebra The measures of the angles of a certain
triangle are in the ratio 2:3:5. What are the measures
of the angles?
Words
The measures of the angles are in the
ratio 2:3:5.
Variables Let 2x represent the measure of one angle,
3x the measure of a second angle, and 5x
the measure of the third angle.
Equation
The sum of the
measures is 180.
Combine like terms.
Divide each side
by 10.
Simplify.
Since
Answer: The measures of the angles are 36°, 54°,
and 90°.
Check
So, the answer is
correct.
Algebra The measures of the angles of a certain
triangle are in the ratio 3:5:7. What are the measures
of the angles?
Answer: The measures of the angles are 36°, 60°,
and 84°.
Classify the triangle by its angles and by its sides.
Angles All angles are acute.
Sides
All sides are congruent.
Answer: The triangle is an acute
equilateral triangle.
Classify the triangle by its angles and by its sides.
Angles The triangle has a right angle.
Sides
The triangle has no
congruent sides.
Answer: The triangle is a
right scalene triangle.
Classify each triangle by its angles and by its sides.
a.
Answer: obtuse scalene
b.
Answer: acute equilateral
Click the mouse button or press the
Space Bar to display the answers.
Example 1 Find the Length of the Hypotenuse
Example 2 Solve a Right Triangle
Example 3 Use the Pythagorean Theorem
Example 4 Identify a Right Triangle
Find the length of the hypotenuse of the
right triangle.
Pythagorean
Theorem
Replace a with 21
and b with 20.
Evaluate 212 and 202.
Add 441 and 400.
Take the square root of each side.
Answer: The length of the hypotenuse is 29 feet.
Find the length of the hypotenuse of the
right triangle.
Answer: The length of the
hypotenuse is 5 meters.
Find the length of the leg of the right triangle.
Pythagorean
Theorem
Replace c with
11 and a with 8.
Evaluate 112 and 82.
Subtract 64 from each side.
Simplify.
Take the square root of
each side.
2nd
[
] 57
ENTER
7.549834435
Answer: The length of the leg is about 7.5 meters.
Find the length of the leg of the right triangle.
Answer: The length of the leg
is about 12.7 inches.
Multiple-Choice Test Item
A building is 10 feet tall. A ladder is positioned
against the building so that the base of the ladder
is 3 feet from the building. How long is the ladder?
A 12.4 feet
C 10.0 feet
B 10.4 feet
D 14.9 feet
Read the Test Item
Make a drawing to illustrate the
problem. The ladder, ground,
and side of the house form a
right triangle.
Solve the Test Item
Use the Pythagorean Theorem to find the length of
the ladder.
Pythagorean Theorem
Replace a with 3 and b with 10.
Evaluate 32 and 102.
Simplify.
Take the square root of
each side.
Round to the nearest tenth.
The ladder is about 10.4 feet tall.
Answer: The answer is B.
Multiple-Choice Test Item
An 18-foot ladder is placed against a building which
is 14 feet tall. About how far is the base of the ladder
from the building?
A 11.6 feet
C 11.3 feet
B 10.9 feet
D 11.1 feet
Answer: The answer is C.
The measures of three sides of a triangle are given.
Determine whether the triangle is a right triangle.
48 ft, 60 ft, 78 ft
Pythagorean Theorem
Replace a with 48, b with 60,
and c with 78.
Evaluate.
Simplify.
The triangle is not a right triangle.
Answer: no
The measures of three sides of a triangle are given.
Determine whether the triangle is a right triangle.
24 cm, 70 cm, 74 cm
Pythagorean Theorem
Replace a with 24, b with 70,
and c with 74.
Evaluate.
Simplify.
The triangle is a right triangle.
Answer: yes
The measures of three sides of a triangle are given.
Determine whether each triangle is a right triangle.
a. 42 in., 61 in., 84 in.
Answer: no
b. 16 m, 30 m, 34 m
Answer: yes
Click the mouse button or press the
Space Bar to display the answers.
Example 1 Use the Distance Formula
Example 2 Use the Distance Formula to Solve a Problem
Example 3 Use the Midpoint Formula
Find the distance between M(8, 4) and N(–6, –2).
Round to the nearest tenth, if necessary.
Use the Distance Formula.
Distance Formula
Simplify.
Evaluate (–14)2
and (–6)2.
Add 196 and 36.
Take the square root.
Answer: The distance between points M and N is
about 15.2 units.
Find the distance between A(–4, 5) and B(3, –9).
Round to the nearest tenth, if necessary.
Answer: The distance between points A and B is
about 15.7 units.
Geometry Find the perimeter of XYZ to the
nearest tenth.
First, use the Distance
Formula to find the length
of each side of the triangle.
Distance Formula
Simplify.
Evaluate powers.
Simplify.
Distance Formula
Simplify.
Evaluate powers.
Simplify.
Distance Formula
Simplify.
Evaluate powers.
Simplify.
Then add the lengths of the sides to find the perimeter.
Answer: The perimeter is about 15.8 units.
Geometry Find the perimeter of ABC to the
nearest tenth.
Answer: The perimeter is
about 21.3 units.
Find the coordinates of the midpoint of
Midpoint Formula
Substitution
Simplify.
Answer: The coordinates of the midpoint of
are (3, 3).
Find the coordinates of the midpoint of
Answer: The coordinates
of the midpoint
of
(1, –1).
are
Click the mouse button or press the
Space Bar to display the answers.
Example 1 Find Measures of Similar Triangles
Example 2 Use Indirect Measurement
Example 3 Use Shadow Reckoning
If RUN ~ CAB, what is the value of x?
The corresponding sides are proportional.
Write a proportion.
Replace UR with 4, AC
with 8, UN with 10, and
AB with x.
Find the cross products.
Simplify.
Mentally divide each side by 4.
Answer: The value of x is 20.
If ABC ~ DEF, what is the value of x?
Answer: The value of
x is 3.
Maps A surveyor wants to find the distance RS
across the lake. He constructs PQT similar to PRS
and measures the distances as shown. What is the
distance across the lake?
Write a proportion.
Substitution
Find the cross products.
Simplify.
Divide each side by 25.
Answer: The distance across the lake is
28.8 meters.
Maps In the figure, MNO is similar to OPQ. Find
the distance across the park.
Answer: The distance across the park is
4.8 miles.
Landmarks Suppose the John Hancock Center in
Chicago, Illinois, casts a 257.5-foot shadow at the
same time a nearby tourist casts a 1.5-foot shadow.
If the tourist is 6 feet tall, how tall is the John
Hancock Center?
Explore You know the lengths of the shadows and the
height of the tourist. You need to find the height
of the John Hancock Center.
Plan
Write and solve a proportion.
Solve
tourist’s shadow
building’s shadow
tourist’s height
building’s height
Find the cross products.
Multiply.
Divide each side by 1.5.
Answer: The height of the John Hancock Center
is 1030 feet.
Building A man standing near a building casts a
2.5-foot shadow at the same time the building casts
a 200-foot shadow. If the man is 6 feet tall, how tall
is the building?
Answer: The height of the building is 480 feet.
Click the mouse button or press the
Space Bar to display the answers.
Example 1 Find Trigonometric Ratios
Example 2 Use a Calculator to Find Trigonometric Ratios
Example 3 Use Trigonometric Ratios
Example 4 Use Trigonometric Ratios to Solve a Problem
Find sin A, cos A, and tan A.
Answer:
Find sin A, cos A, and tan A.
Answer:
Find sin A, cos A, and tan A.
Answer:
Find sin B, cos B, and tan B.
Answer: sin B = 0.8; cos B = 0.6; tan B = 1.3333
Find the value of sin 19° to the nearest
ten thousandth.
SIN
19
ENTER
0.3255681545
Answer: sin 19° is about 0.3256.
Find the value of cos 51° to the nearest
ten thousandth.
COS
51
ENTER
0.629320391
Answer: cos 51° is about 0.6293.
Find the value of tan 24° to the nearest
ten thousandth.
TAN
24
ENTER
0.4452286853
Answer: tan 24° is about 0.4452.
Find each value to the nearest ten thousandth.
a. sin 63°
Answer: 0.8910
b. cos 14°
Answer: 0.9703
c. tan 41°
Answer: 0.8693
Find the missing measure. Round to the
nearest tenth.
The measures of an acute
angle and the side adjacent
to it are known. You need
to find the measure of the
hypotenuse. Use the
cosine ratio.
Write the
cosine ratio.
Substitution
Multiply each side by x.
Simplify.
Divide each side by cos 71°.
12 
COS
71
ENTER
36.85864184
Simplify.
Answer: The measure of the hypotenuse is
about 36.9 units.
Find the missing measure.
Round to the nearest tenth.
Answer: The measure of the missing side is
about 21.4 units.
Architecture A tourist visiting the Petronas Towers in
Kuala Lumpur, Malaysia, stands 261 feet away from
their base. She looks at the top at an angle of 80° with
the ground. How tall are the Towers?
Use the tangent ratio.
Write the
tangent ratio.
Substitution
Multiply each side by 261.
261
X
TAN
80
ENTER
1480.204555
Simplify.
Answer: The height of the Towers is
about 1480.2 feet.
Architecture Jenna stands 142 feet away from the
base of a building. She looks at the top at an angle of
62° with the ground. How tall is the building?
Answer: The building is about 267.1 feet tall.
Explore online information about the
information introduced in this chapter.
Click on the Connect button to launch your browser
and go to the Pre-Algebra Web site. At this site, you
will find extra examples for each lesson in the
Student Edition of your textbook. When you finish
exploring, exit the browser program to return to this
presentation. If you experience difficulty connecting
to the Web site, manually launch your Web browser
and go to www.pre-alg.com/extra_examples.