1. 9-8 The Pythagorean Theorem
Warm Up
Problem of the Day
Lesson Presentation
Lesson Quizzes

2. 9-8 The Pythagorean Theorem
Warm Up
Estimate each square root to the nearest
whole number. Use a calculator to check the
reasonableness of your answers.
1. √18 4
2. √26 5
3. √86 9
4. √125 11

3. 9-8 The Pythagorean Theorem
Problem of the Day
A shipping carton measures 12 in. by 15
in. by 16 in. What is the longest rod that
can be shipped in it?
25 in.

4. 9-8 The Pythagorean Theorem
Learn to use the Pythagorean Theorem to
find the length of a side of a right triangle.

5. 9-8 The Pythagorean Theorem
Vocabulary
leg
hypotenuse
Pythagorean Theorem

6. 9-8 The Pythagorean Theorem
In a right triangle, the two
sides that form the right Hypotenuse
angle are called legs. The Leg
side opposite the right angle
Leg
is called the hypotenuse.
One of the first people to recognize the
relationship between the sides of a right triangle
was the Greek mathematician Pythagoras. This
special relationship is called the Pythagorean
Theorem.

7. 9-8 The Pythagorean Theorem
You can use the Pythagorean Theorem to find the
length of any side of a right triangle.

8. 9-8 The Pythagorean Theorem
Additional Example 1A: Calculating the Length of a
Side of a Right Triangle
Use the Pythagorean Theorem to find the
missing measure.
c
12 cm
16 cm
a2 + b2 = c2 Use the Pythagorean Theorem.
122 + 162 = c2 Substitute for a and b.
144 + 256 = c2 Evaluate the powers.
400 = c2 Add.
√400 = √c2 Take the square root of both sides.
20 = c
The length of the hypotenuse is 20 cm.

9. 9-8 The Pythagorean Theorem
Additional Example 1B: Calculating the Length of a
Missing Side of a Right Triangle
Use the Pythagorean Theorem to find the
missing measure.
b
13 cm 5 cm
a2 + b2 = c2 Use the Pythagorean Theorem.
52 + b2 = 132 Substitute for a and c.
25 + b2 = 169 Evaluate the powers.
–25 –25 Subtract 25 from each side.
b2 = 144
√b2 = √144 Take the square root of both sides.
b = 12
The length of the missing leg is 12 cm.

10. 9-8 The Pythagorean Theorem
Check It Out: Example 1A
Use the Pythagorean Theorem to find the
missing measure.
11 cm c
15 cm
a2 + b2 = c2 Use the Pythagorean Theorem.
112 + 152 = c2 Substitute for a and b.
121 + 225 = c2 Evaluate the powers.
346 = c2 Add.
√346 = √c2 Take the square root of both sides.
18.6 ≈ c
The length of the hypotenuse is about 18.6 cm.

11. 9-8 The Pythagorean Theorem
Check It Out: Example 1B
Use the Pythagorean Theorem to find the
missing measure.
b
5 cm 3 cm
a2 + b2 = c2 Use the Pythagorean Theorem.
32 + b2 = 52 Substitute for a and c.
9 + b2 = 25 Evaluate the powers.
–9 –9 Subtract 9 from each side.
b2 = 16
√b2 = √ 16 Take the square root of both sides.
b = 4
The length of the missing leg is 4 cm.

12. 9-8 The Pythagorean Theorem
Additional Example 2: Problem Solving
Application
A square field has sides of 75 feet. About
how far is it from one corner of the field
to the opposite corner of the field?
Round your answer to the nearest tenth.

13. 9-8 The Pythagorean Theorem
Additional Example 2 Continued
1 Understand the Problem
Rewrite the question as a statement.
• Find the distance from one corner of the field
to the opposite corner of the field.
List the important information:
• Drawing a segment from one corner of the
field to the opposite corner of the field divides
the field into two right triangles.
• The segment between the two corners is
the hypotenuse.
• The sides of the field are legs, and they
are each 75 feet long.

14. 9-8 The Pythagorean Theorem
Additional Example 2 Continued
2 Make a Plan
You can use the Pythagorean Theorem to
write an equation.

15. 9-8 The Pythagorean Theorem
Additional Example 2 Continued
3 Solve
a2 + b2 = c2 Use the Pythagorean Theorem.
752 + 752 = c2 Substitute for the known variables.
5,625 + 5,625 = c2 Evaluate the powers.
11,250 = c2 Add.
106.066012 ≈ c Take the square roots of both sides.
106.1 ≈ c Round.
The distance from one corner of the field to the
opposite corner is about 106.1 feet

16. 9-8 The Pythagorean Theorem
Additional Example 2 Continued
4 Look Back
The hypotenuse is the longest side of a right
triangle. Since the distance from one corner of
the field to the opposite corner is greater than
the length of a side of the field, the answer is
reasonable.

17. 9-8 The Pythagorean Theorem
Check It Out: Example 2
A rectangular field has a length of 100
yards and a width of 33 yards. About how
far is it from one corner of the field to the
opposite corner of the field? Round your
answer to the nearest tenth.
1 Understand the Problem
Rewrite the question as a statement.
• Find the distance from one corner of

18. 9-8 The Pythagorean Theorem
Check It Out: Example 2 Continued
List the important information:
• Drawing a segment from one corner of the field
to the opposite corner of the field divides the field
into two right triangles.
• The segment between the two corners is
the hypotenuse.
• The sides of the fields are legs, and they are 33
yards long and 100 yards long.
2 Make a Plan
You can use the Pythagorean Theorem to
write an equation.

19. 9-8 The Pythagorean Theorem
Check It Out: Example 2 Continued
3 Solve
a2 + b2 = c2 Use the Pythagorean Theorem.
332 + 1002 = c2 Substitute for the known variables.
1089 + 10,000 = c2 Evaluate the powers.
11,089 = c2 Add.
105.3043208 ≈ c Take the square roots of both sides.
105.3 ≈ c Round.
The distance from one corner of the field to the
opposite corner is about 105.3 yards.

20. 9-8 The Pythagorean Theorem
Check It Out: Example 2 Continued
4 Look Back
The hypotenuse is the longest side of a right
triangle. Since the distance from one corner
of the field to the opposite corner is greater
than the length of either side of the field, the
answer is reasonable.

21. 9-8 The Pythagorean Theorem
Lesson Quizzes
Standard Lesson Quiz
Lesson Quiz for Student Response Systems

22. 9-8 The Pythagorean Theorem
Lesson Quiz: Part I
Use the Pythagorean Theorem to find each
missing measure.
1. 2.
40 m 21 in.
3. a = , b = 30, c = 34 16
4. a = 20, b = 21, c = 29

23. 9-8 The Pythagorean Theorem
Lesson Quiz: Part II
5. Each rectangular section of a fence is braced
by a board nailed on the diagonal of the
section. The fence is 6 ft tall and the brace is
10 ft long. What is the length of the section?
8 ft

24. 9-8 The Pythagorean Theorem
Lesson Quiz for Student Response Systems
1. Use the Pythagorean Theorem to identify the
missing measure.
A. 50 ft 30 ft
B. 60 ft
40 ft
C. 70 ft
D. 80 ft

25. 9-8 The Pythagorean Theorem
Lesson Quiz for Student Response Systems
2. Use the Pythagorean Theorem to identify the
missing measure.
A. 27 m
B. 45 m
C. 56 m
D. 75 m

26. 9-8 The Pythagorean Theorem
Lesson Quiz for Student Response Systems
3. Use the Pythagorean Theorem to identify the
missing measure. a = 40, b = ___ , c = 58
A. 18
B. 42
C. 70
D. 98

27. 9-8 The Pythagorean Theorem
Lesson Quiz for Student Response Systems
4. A rectangular field measures 11 feet by 60 feet.
What is the length of the irrigation pipe that has to
be placed along the diagonal of the field?
A. 11 ft
B. 12 ft
C. 60 ft
D. 61 ft