Check out the

Dynamic Classroom.
Learning Target
Understand the Pythagorean
Laurie’s Notes BigIdeasMath.com

COMMON
STATE STANDARDS
Theorem. CORE 8.EE.A.2. 8.G.B.6,
B.G.B.7, 8.G.B.8
Success Criteria
● Explain the Pythagorean Preparing to Teach
Theorem. ● MP2 Reason Abstractly and Quantitatively: In the exploration, students will
● Use the Pythagorean analyze the relationship between the two models they create. This represents
Theorem to find unknown an “informal proof” of the Pythagorean Theorem.
side lengths of triangles.
● Use the Pythagorean Motivate
Theorem to find distances ● Share information about Pythagoras, who was born in Greece around 570 B.C.
between points in a ● He is known as the Father of Numbers.
coordinate plane. ● He traveled extensively in Egypt, learning math, astronomy, and music.

● Pythagoras urged the citizens of Cretona to follow his religious, political,

and philosophical goals.

● His followers were known as Pythagoreans. They observed a rule of silence

Warm Up called echemythia. One had to remain silent for five years before he could
contribute to the group. Breaking this silence was punishable by death!
Cumulative, vocabulary, and
prerequisite skills practice
opportunities are available in Exploration 1
the Resources by Chapter or at
● Suggestions: Use 1-centimeter grid paper for ease of manipulating the cut
BigIdeasMath.com.
pieces. Grid paper is available online at BigIdeasMath.com. Suggest that
students draw the original triangle in the upper left of the grid paper, and then
make a working copy of the triangle towards the middle of the paper. This
gives enough room for the squares to be drawn on each side of the triangle.
ELL Support ● Vertices of the triangle need to be on lattice points. You do not want every
student in the room to use the same triangle. Suggest other lengths for the
Students may know the word theory
shorter sides (3 and 4, 3 and 6, 2 and 4, 2 and 3, and so on).
from science. Have them share
● Model: Drawing the square on the longest side of the triangle is the
what they understand about a
challenging step. Model one technique for accomplishing the task using a
theory. If necessary, explain that a
right triangle with shorter side lengths of 2 units and 5 units.
theory is an idea that tries to explain
● Notice that the longest side has a slope of “right 5 units, up 2 units.”
events after having carefully
● Place your pencil on the upper right endpoint and rotate the paper
viewed them. It comes from a Latin
90°clockwise. Move your pencil right 5 units and up 2 units. Mark a point.
and Greek root meaning “view.”
● Repeat rotating and moving “right 5 units, up 2 units” until you get back to
Theorem comes from the same
the longest side of the triangle.
root, and literally means “that which
● Use a straightedge to connect the four points (two that you marked and
is viewed.” In science and math, a
two on the endpoints of the longest side) to form the square.
theorem is a rule, or theory, that can
● Before students cut, check that they have 3 squares of the correct size.
be proved.
● Give students time to discover how to
arrange the figures on the grid paper.
a2
Here is one example. You may want to
show students one arrangement and c2
b2
Exploration 1 then have them find the other.
● Big Idea: The two large squares have
a. See Additional Answers. equal area. Referring to areas, if
b. Answer should include, but is c2 + (4 triangles) = a2 + b2 + (4 triangles), then c2 = a2 + b2 by subtracting
not limited to: The two values the 4 triangles from each side of the equation.
should be close to each other. ● The work in this exploration constitutes an “informal proof” of the
Pythagorean Theorem. There are many proofs of this theorem, and this
version is generally understood by middle school students.
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 9.2 The Pythagorean Theorem
Learning Target: Understand the Pythagorean Theorem.
Success Criteria: • I can explain the Pythagorean Theorem.
• I can use the Pythagorean Theorem to find unknown side lengths of triangles.
• I can use the Pythagorean Theorem to find distances between points in a coordinate plane.

Pythagoras was a Greek mathematician and philosopher

who proved one of the most famous rules in mathematics.
In mathematics, a rule is called a theorem. So, the rule
that Pythagoras proved is called the Pythagorean Theorem.

Pythagoras
(c. 570–c. 490 B.C.)

EXPLORATION 1 Discovering the Pythagorean Theorem

Work with a partner.
• On grid paper, draw a right triangle with one horizontal side and
one vertical side.
• Label the lengths of the two shorter sides a and b. Label the length
of the longest side c.
• Draw three squares that each share a side with your triangle. Label
the areas of the squares a 2, b 2, and c 2.
• Cut out each square. Then make eight copies of the right
triangle and cut them out.

c2

c a a2

b2

Math Practice
Construct
Arguments a. Arrange the figures to show how a 2 and b 2 relate to c 2. Use an equation
Is the relationship to represent this relationship.
among a2, b2, and
c2 true for all right b. Estimate the side length c of your triangle. Then use the relationship in
triangles? Explain. part (a) to find c. Compare the values.

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 9.2 Lesson

Key Vocabulary
theorem, p. 381
Key Ideas
legs, p. 382
Sides of a Right Triangle
hypotenuse, p. 382
Pythagorean The sides of a right triangle have special names.
Theorem, p. 382

hypotenuse, c The hypotenuse

leg, a is the side opposite
the right angle.
The legs are the
two sides that form
leg, b
the right angle.

The Pythagorean Theorem

ngle,
In a right tria Words In any right triangle, the sum of the squares of the
e
the legs are th lengths of the legs is equal to the square of the length
s a n d the
shorter side ays of the hypotenuse.
a lw
hypotenuse is e.
d
the longest si Algebra a2 + b2 = c2

EXAMPLE 1 Finding the Length of a Hypotenuse

Find the length of the hypotenuse of the triangle.
5m
a2 + b2 = c2 Write the Pythagorean Theorem.
52 + 122 = c 2 Substitute 5 for a and 12 for b.

c
25 + 144 = c 2 Evaluate the powers.
12 m
169 = c 2 Add.
13 = c Take the positive square root of each side.

The length of the hypotenuse is 13 meters.

Try It Find the length of the hypotenuse of the triangle.

1. 2. 3 in.
10
c
8 ft
2
in. c
5
15 ft

382 Chapter 9 Real Numbers and the Pythagorean Theorem Multi-Language Glossary at BigIdeasMath.com

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 Laurie’s Notes
Scaffolding Instruction Scaffold instruction to support all
● Students have investigated a visual “proof” of the Pythagorean Theorem. students in their learning. Learning is
They will now use the Pythagorean Theorem to find missing side lengths of individualized and you may want to
right triangles. group students differently as they move
● Emerging: Students may confuse the legs and the hypotenuse of a right in and out of these levels with each skill
triangle. They will need guided instruction for the Key Ideas and examples. and concept. Student self-assessment
● Proficient: Students understand the difference between the legs and the and feedback help guide your
hypotenuse of a right triangle. They can use substitution to find missing instructional decisions about how and
side lengths. These students should proceed to Examples 3 and 4 before when to layer support for all students to
completing the Self-Assessment exercises. become proficient learners.

Key Ideas
● Draw a right triangle and label the legs and the hypotenuse. The hypotenuse
is always opposite the right angle and is the longest side of a right triangle.
● Try not to have all right triangles in the same orientation.
● Write the Pythagorean Theorem.
● Common Error: Students often forget that the Pythagorean Theorem is a
relationship that is only true for right triangles.

EXAMPLE 1 Extra Example 1

Find the length of the hypotenuse of the
● Draw and label the triangle. Review the symbol used to show that an angle is triangle.
a right angle.
“What information is known for this triangle?” The legs are 5 meters and
12 meters.
● Substitute and solve as shown. Explain that you disregard the negative c
3 in.
square root because length is always positive.
● Note that the values for a and b could be interchanged.

Try It 4 in.
● Give time for students to work through the problems. Knowing perfect squares
5 in.
is helpful.
● MP2 Reason Abstractly and Quantitatively: In Exercise 2, if students
recognize that the decimal equivalents of the given fractions are 0.3 and 0.4,
Try It
finding the hypotenuse may be quick for them. 1. 17 ft
1
2. — in.
2

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 Laurie’s Notes
Extra Example 2 EXAMPLE 2
Find the missing length of the triangle.
“What information is known for this triangle?” One leg is 2.1 centimeters and
25 ft the hypotenuse is 2.9 centimeters.
7 ft ● Substitute and solve as shown.
b
● Common Error: Students need to be careful with decimal multiplication. It is
very common for students to multiply the decimal by 2 instead of multiplying
24 ft the decimal by itself.
● FYI: The triangle is similar to a 20-21-29 right triangle.
Try It
3. 30 yd
Try It
● Think-Pair-Share: Students should read each exercise independently and
4. 4 m then work in pairs to complete the exercises. Then have each pair compare
their answers with another pair and discuss any discrepancies.

Extra Example 3 EXAMPLE 3

Find the slant height of the ● Students need to analyze the diagram before solving. It may look complicated,
square pyramid. but this problem is very similar to Example 1, in which students found the
0.9 m
hypotenuse of a right triangle.
“What information is known?” The legs of a right triangle are 6.4 inches
and 4.8 inches.
“How do you find the slant height?” Substitute the values for a and b into the
Pythagorean Theorem and solve for c.
1.2 m ● Remind students that the values for a and b could be interchanged.
1.5 m
Try It
Try It ●

●
Remind students to analyze the diagram before solving.
Neighbor Check: Have students work independently and then have their
5. 15 yd neighbors check their work. Have students discuss any discrepancies.
6. 11 ft
ELL Support
After demonstrating Examples 2 and 3, have students work in pairs to discuss
and complete Try It Exercises 3–6. Have one student ask another, “What values
do you substitute for a, b, and/or c? What is the equation? What is the length
of the unknown side?” Have students alternate the roles of asking and
answering questions.
Beginner: Write the steps used to solve the problem.
Intermediate: Use phrases or simple sentences to answer and explain.
Advanced: Use detailed sentences to answer and explain.

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 EXAMPLE 2 Finding the Length of a Leg
Find the missing length of the triangle.

a2 + b2 = c2 Write the Pythagorean Theorem.

a 2.1 cm a 2 + 2.12 = 2.9 2 Substitute 2.1 for b and 2.9 for c.

a 2 + 4.41 = 8.41 Evaluate the powers.
a2 = 4 Subtract 4.41 from each side.
2.9 cm
a =2 Take the positive square root of each side.

The missing length is 2 centimeters.

Try It Find the missing length of the triangle.

3. 4.
9.6 m
a
34 yd
16 yd
10.4 m

You can use right triangles and the Pythagorean Theorem to find lengths
of three-dimensional figures.

EXAMPLE 3 Finding a Length of a Three-Dimensional Figure

Find the slant height of the square pyramid.
6.4 in.
a2 + b2 = c2 Write the Pythagorean Theorem.
6.42 + 4.82 = c 2 Substitute 6.4 for a and 4.8 for b.
4.8 in. 40.96 + 23.04 = c 2 Evaluate the powers.
64 = c 2 Add.
8 =c Take the positive square root of each side.

The slant height is 8 inches.

Try It Find x.
5. 6. 8.8 ft
x
x
9 yd
6.6 ft
12 yd

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 You can use right triangles and the Pythagorean Theorem to find distances
between points in a coordinate plane.

EXAMPLE 4 Finding a Distance in a Coordinate Plane

(−2, 5)
5
y
Find the distance between (−2, 5) and (4, −3).
4
Plot the points in a coordinate plane. Then draw a right triangle with a
2 hypotenuse that represents the distance between the points.
8 units 1
Use the Pythagorean Theorem to find the length of the hypotenuse.
−4 −3 −1 1 3 4 5 6 x

−2 a2 + b2 = c2 Write the Pythagorean Theorem.

6 units (4, −3) 82 + 62 = c2 Substitute 8 for a and 6 for b.

−4
−5 64 + 36 = c 2 Evaluate the powers.
100 = c 2 Add.
10 = c Take the positive square root of each side.

The distance between (22, 5) and (4, 23) is 10 units.

Try It Find the distance between the points.

7. (3, 6) and (7, 9) 8. (23, 24) and (2, 8)

Self-Assessment for Concepts & Skills

Solve each exercise. Then rate your understanding of the success criteria
in your journal.

FINDING A MISSING LENGTH Find x.

9. 10. 1.6 cm

4 ft x
2.4 ft

x 1.2 cm

11. FINDING A DISTANCE Find the distance between (25, 2) and (7, 27).

12. DIFFERENT WORDS, SAME QUESTION Which is different?

Find “both” answers.
a c
Which side is the hypotenuse? Which side is the longest?

Which side is a leg? Which side is opposite the right angle?

b

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 Laurie’s Notes
EXAMPLE 4 Extra Example 4
Find the distance between (−3, −5) and
● Have students plot the points in a coordinate plane and then draw a right (2, 7). 13 units
triangle with a hypotenuse that represents the distance between the points.
“Is there enough information to use the Pythagorean Theorem? Explain.”
Yes, the lengths of both legs of the right triangle can be found and then used
to find the length of the hypotenuse (the distance between the points).
“How can you find the length of each leg?” Students will likely count the
grid squares.
“Can you count the grid squares for the hypotenuse? Explain.” No, the
hypotenuse is neither vertical nor horizontal, so the increments do not
represent 1 unit.
● Work through the remainder of the solution.
● Note that the values for a and b could be interchanged.
● Have students check the value of c in the Pythagorean Theorem.
● Remind students that in the context of distance, they should only consider

●
positive values.
FYI: This example previews the distance formula, which students will
Try It
study in high school. Deriving the distance formula is not required in this 7. 5 units
course; however, in Practice Exercise 36, students will unknowingly derive 8. 13 units
the formula.

Try It
● Neighbor Check: Have students work independently and then have their ELL Support
neighbors check their work. Have students discuss any discrepancies.
Allow students to work in pairs for
the Self-Assessment for Concepts
Formative Assessment Tip & Skills exercises. Have each
pair display their answers for
Fact-First Questioning
Exercises 9–11 on a whiteboard
This is a higher-order questioning technique that goes beyond asking straight
for your review. Have two pairs
recall questions. Instead, this strategy allows you to assess students’ growing
form a group to discuss and come
understanding of a concept or skill.
to an agreement on their answers
First, state a fact. Then ask students why, or how, or to explain. Student for Exercise 12. Then have each
thinking is activated and you gain insight into the depth of students’ conceptual group present their explanation to
understanding. Example: Make the statement, “If you are given two side lengths of the class.
a right triangle, you can find the third side length.” Then ask, “Why is this true?”

Self-Assessment for Concepts & Skills Self-Assessment

for Concepts & Skills
● Students should work independently and then discuss their answers
with a partner. 9. 3.2 ft
Fact-First Questioning: Make the statement, “The Pythagorean Theorem 10. 2 cm
can be used to find the distances between points in a coordinate plane.”
Then ask, “Why this is true?” 11. 15 units
12. Which side is a leg?; a or b ; c
The Success Criteria Self-Assessment chart can be found in the
Student Journal or online at BigIdeasMath.com.

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 Laurie’s Notes
Extra Example 5
EXAMPLE 5
You and your cousin are planning to go
to an amusement park. You live 36 miles ● Ask a student to read the problem.
south of the amusement park and “Given the compass directions stated, what is a reasonable way to represent
15 miles west of your cousin. How far this information?” in a coordinate plane
away from the amusement park does ● MP4 Model with Mathematics: Explain that east is the positive x-direction
your cousin live? 39 mi and north is the positive y-direction. Draw the situation in a coordinate plane.
“Is there enough information to use the Pythagorean Theorem? Explain.”
Yes, the lengths of both legs of the right triangle can be found and then used
to find the length of the hypotenuse (the distance between you and the other
team’s base).

Self-Assessment Self-Assessment for Problem Solving

for Problem Solving
● The goal for all students is to feel comfortable with the problem-solving
13. 8.4 mi2; 2.42 + x2 = 7.42, plan. It is important for students to problem-solve in class, where they
so x = 7 mi. Then, may receive support from you and their peers.
1 ● Think-Pair-Share: Students should read each exercise independently
A = — (2.4)(7) = 8.4 mi2.
2 and then work in pairs to complete the exercises. Then have each pair
14. 9 p.m.; 225 miles is compare their answers with another pair and discuss any discrepancies.
equivalent to 15 units on the The Success Criteria Self-Assessment chart can be found in the
coordinate plane. The cargo Student Journal or online at BigIdeasMath.com.
ship is 9 units north of the
shipyard, so it will be
15 units away when it is
12 units east of the shipyard Closure
because 92 + 122 = 152. It is ● Exit Ticket: You draw a right triangle with shorter side lengths of 4 and 6 on
traveling 1 unit per hour, so grid paper. What is the area of the square drawn on the longest side of the
it will take 12 hours to travel
triangle? 52 square units
12 units. 12 hours after
9 a.m. is 9 p.m.

Learning Target
Understand the Pythagorean
Theorem.

Success Criteria
● Explain the Pythagorean
Theorem.
● Use the Pythagorean
Theorem to find unknown
side lengths of triangles.
● Use the Pythagorean
Theorem to find distances
between points in a
coordinate plane.

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 EXAMPLE 5 Modeling Real Life
You play capture the flag. You are 50 yards north and 20 yards east of
your team’s base. The other team’s base is 80 yards north and 60 yards
east of your base. How far are you from the other team’s base?

N Step 1: Draw the situation in a coordinate plane. Let the origin represent
90
Other Base (60, 80) your team’s base. From the descriptions, you are at (20, 50) and the
80
70
other team’s base is at (60, 80).
30 yd
60 Step 2: Draw a right triangle with a hypotenuse that represents the distance
You
50
(20, 50) 40 yd between you and the other team’s base. The lengths of the legs are
40
30
30 yards and 40 yards.
20 Step 3: Use the Pythagorean Theorem to find the length of the hypotenuse.
10
Your Base
a2 + b2 = c2 Write the Pythagorean Theorem.
W 10 20 30 40 50 60 70 80 E
S 302 + 402 = c 2 Substitute 30 for a and 40 for b.
900 + 1600 = c 2 Evaluate the powers.
2500 = c 2 Add.
50 = c Take the positive square root of each side.

So, you are 50 yards from the other team’s base.

Self-Assessment for Problem Solving

Solve each exercise. Then rate your understanding of the success criteria
in your journal.

13. A zookeeper knows that an escaped red panda is hiding somewhere in

the triangular region shown. What is the area (in square miles) that the
zookeeper needs to search? Explain.
x

2.4 mi
7.4 mi

14. DIG DEEPER Objects detected by radar are plotted in a coordinate

plane where each unit represents 15 miles. The point (0, 0) represents
the location of a shipyard. A cargo ship is traveling at a constant speed
and in a constant direction parallel to the coastline. At 9 A.m., the
radar shows the cargo ship at (0, 9). At 10 a.m., the radar shows the
cargo ship at (1, 9). At what time will the cargo ship be 225 miles from
the shipyard? Explain.

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 9.2 Practice Go to BigIdeasMath.com to get
HELP with solving the exercises.

Review & Refresh

Solve the equation.
—

√
1000
1. 7z = 252
2
2. 0.75q = 1082
3. — = n − 54
2
10

4. What is the solution of the system of linear equations y = 4x + 1 and

2x + y = 13?

A. (1, 5) B. (5, 3) C. (2, 9) D. (9, 2)

Concepts, Skills, & Problem Solving

USING GRID PAPER Find c. (See Exploration 1, p. 381.)
5. 6.

c
c 5
6

12
8

FINDING A MISSING LENGTH Find the missing length of the triangle.

7. 20 km 8. 9. 5.6 in.
c
7.2 ft
a 10.6 in.
21 km c
9.6 ft

10. 9 mm
11. 12. a
1.1 yd
b 26 cm
10 cm
6.1 yd

15 mm b

13. YOU BE THE TEACHER Your friend

finds the missing length of the triangle. Is
25 ft a2 + b2 = c2
your friend correct? Explain your reasoning. 7 ft
7 2 + 252 = c2
674 = c2
—
√ 674 = c

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 Check out the Dynamic
Assessment System.
Assignment Guide BigIdeasMath.com
and Concept Check
Scaffold assignments to support all students in their learning progression.
Review & Refresh
The suggested assignments are a starting point. Continue to assign additional 1. z = −6 and z = 6
exercises and revisit with spaced practice to move every student toward
2. q = −12 and q = 12
proficiency.
3. n = −8 and n = 8
Level Assignment 1 Assignment 2 4. C

Emerging
1, 4, 5, 7, 9, 11, 13, 14, 15,
18, 19
8, 10, 12, 16, 17, 21, 23, 24,
26, 28, 36 Concepts, Skills,
2, 4, 6, 8, 10, 12, 13, 15, 16, 17, 25, 26, 27, 28, 30, 31,
& Problem Solving
Proficient
18, 20, 22, 24 32, 34, 35, 36 5. 10
3, 4, 6, 8, 10, 12, 13, 15, 16, 17, 25, 26, 29, 30, 31, 32, 6. 13
Advanced
19, 22, 24 33, 34, 35, 36 7. 29 km
● Assignment 1 is for use after students complete the Self-Assessment for 8. 12 ft
Concepts & Skills.
9. 9 in.
● Assignment 2 is for use after students complete the Self-Assessment for
Problem Solving. 10. 12 mm
● The red exercises can be used as a concept check. 11. 24 cm
12. 6 yd
Review & Refresh Prior Skills
Exercises 1–3 Solving Equations Using Square Roots 13. no; The length of the
Exercise 4 Choosing a Solution Method for a System of Linear Equations hypotenuse should be
substituted for c, not b.

Common Errors
● Exercises 7–12 Students may substitute the given lengths in the wrong part
of the formula. For example, in Exercise 7, they may write 202 + c2 = 212
instead of 202 + 212 = c2. Remind students that the side opposite the right
angle is the hypotenuse c.
● Exercises 7–12 Students may multiply each side length by two instead of
squaring the side length. Remind them of the definition of exponents.

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 Concepts, Skills, Common Errors
& Problem Solving ● Exercises 24 and 25 Students may think that there is not enough information
to find the value of x. Tell them that is possible to find x; however, they may
14. 26 ft have to make an extra calculation before writing an equation for x.
15. 50 in.
16. 1.2 m
17. 6 cm
18. 15 units
19. 5 units
20. 41 units
21. 29 units
22. 37 units
23. 30.5 units
24. 16 cm
25. 37 mm
26. 57 ft
27. yes; The distance from the
player's mouth to the referee's
ear is 25 feet.

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 FINDING LENGTHS OF THREE-DIMENSIONAL FIGURES Find x.
14. 15. x
24 ft

x 14 in.

48 in.

10 ft

16. 1.5 m 17.

x

x 6.5 cm

0.9 m

2.5 cm

FINDING DISTANCES IN THE COORDINATE PLANE Find the distance between the points.
18. (0, 0), (9, 12) 19. (1, 2), (−3, 5)

20. (—18, 9), (22, 0) 21. (−7, −2), (13, −23)

22. (15, −17), (−20, −5) 23. (−13, −3.5), (17, 2)

FINDING A MISSING LENGTH Find x.

24. 25. 5 mm
20 cm x
13 mm

x 35 mm
12 cm

26. MODELING REAL LIFE The figure shows the

x
location of a golf ball after a tee shot. How many feet Hole
H o
from the hole is the ball?

27. MODELING REAL LIFE A tennis player asks the

referee a question. The sound of the player’s voice travels
30 feet. Can the referee hear the question? Explain. 180 yd

24 ft

12 ft Hole 13
5 ft Par 3
Tee 181 yards

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 28. PROBLEM SOLVING You are cutting a rectangular piece of fabric
in half along a diagonal. The fabric measures 28 inches wide and
1
1— yards long. What is the length (in inches) of the diagonal?
4
A
29. PROJECT Measure the length, width, and height of a
rectangular room. Use the Pythagorean Theorem to height
find the distance from B to C and the distance from C
A to B. width
length B
30. STRUCTURE The legs of a right triangle have
lengths of 28 meters and 21 meters. The hypotenuse
has a length of 5x meters. What is the value of x ?

31. PRECISION You and a friend stand back-to-back. You run 20 feet
forward, then 15 feet to your right. At the same time, your friend runs 16 feet
forward, then 12 feet to her right. She stops and hits you with a snowball.
a. Draw the situation in a coordinate plane.
b. How far does your friend throw the snowball?

y 32. MODELING REAL LIFE The coordinate plane

30
shows dig sites for archaeological research. Each unit
25
Site A Site B on the grid represents 1 square foot. What is the
20
15 distance from Site A to Site C?
10
5 33. PRECISION A box has a length of 30 inches, a
Site C
5 10 15 20 25 30 x
width of 40 inches, and a height of 120 inches. Can
a cylindrical rod with a length of 342.9 centimeters
fit in the box? Explain your reasoning.

34. MODELING REAL LIFE A green roof is like a

traditional roof but covered with plants. Plants 8 ft
used for a green roof cost $0.75 per square foot. The
roof at the right is 40 feet long. How much does it cost 15 ft
to cover both sides of the roof? Justify your answer. 15 ft

y 35. CRITICAL THINKING A triangle has coordinates

(x2, y2) A(2, 1), B(2, 4), and C(5, 1). Write an expression for
—. Use a calculator to find the length
the length of BC
d — to the nearest hundredth.
of BC
(x1, y1)
36. DIG DEEPER Write an equation for the distance d
x between the points (x1, y1) and (x2, y2). Explain how
you found the equation.

388 Chapter 9 Real Numbers and the Pythagorean Theorem

2022_g8_se_09.indb 388 2/9/21 4:06 PM

 For Your Information Concepts, Skills,
● Exercise 31 There is more than one correct drawing for this exercise. & Problem Solving
Encourage students to start at the origin and move along an axis to begin.
28. 53 in.
Mini-Assessment 29. Check students' work.
Find the missing length of the triangle. 30. 7
1.
c 31. a. Sample answer:
14 ft y 15 You
20

48 ft 20
50 ft
−20 −10 10 20 x
2.
−10 16

a Friend 12
51 mm

b. 45 ft
45 mm
32. 25 ft
24 mm
33. no; The rod is 135 inches long,
3. and the diagonal from a top
27 in. corner to the opposite bottom
b
corner is 130 inches long.
34. $1020; 152 + 82 = c 2, so c = 17 ft.
45 in. Then, the area of the entire roof
36 in. ⋅
is 2(40 17) = 1360 ft2, and

Find the distance between the points. —

⋅
1360 $0.75 = $1020.
35. √ 18 ; 4.24 units
4. (0, 4), (3, 0) 5 units ——
5. (−3, −1), (5, 5) 10 units 36. d = √ (x2 − x1)2 + (y2 − y1)2 ;
Draw a right triangle with d as
the hypotenuse. The length of the
horizontal leg is equal to x2 − x1
and the length of the vertical
leg is equal to y2 − y1. Using the
Pythagorean Theorem,
(x2 − x1)2 + (y2 − y1)2 = d 2,
——
so d = √ (x2 − x1)2 + (y2 − y1)2 .

Section Resources
Surface Level Deep Level
Resources by Chapter Resources by Chapter
• Extra Practice • Enrichment and Extension
• Reteach Graphic Organizers
• Puzzle Time Dynamic Assessment System
Student Journal • Section Practice
• Self-Assessment
• Practice
Differentiating the Lesson
Tutorial Videos
Skills Review Handbook
Skills Trainer

T-388

2022_g8_cc_te_09.indb 388 4/27/21 4:13 PM