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M3: Chapter 9 Notes
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Academic
Chapter 9 Notes
Real Numbers &
Right Triangles
Name____________________Pd.____
M3: Chapter 9 Notes
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Vocabulary Words
Sections 9.1 – 9.4
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Square root
Perfect square
Radical expression
Hypotenuse
Legs
Pythagorean Theorem
Irrational number
Real number
M3: Chapter 9 Notes
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Section 9.1: Square Roots
Learning Goal: We will find and approximate square roots of numbers.
Vocabulary:
 Square root – the square root of a number n is a number m such
2
that m  n . The radical sign,
, represents a nonnegative
square root.
Example 1: Finding a Square Root
a. A square courtyard has an
area of 400 square yards.
Find the length of one side
of the courtyard.
b. The base of the Eiffel Tower
is a square with an area of
15,625 square feet. What is
the length of a side of the
base?
ON YOUR OWN:
 Perfect square – a number that is the square of an integer
Example 2: Approximating a Square Root
a. Approximate 67 to the
b. Approximate 51 to the
nearest integer.
nearest integer.
M3: Chapter 9 Notes
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ON YOUR OWN:
Example 3: Using a Calculator
a. Use a calculator to
approximate 632 . Round to
the nearest tenth.
b. Use a calculator to
approximate 515 . Round to
the nearest tenth.
 Radical expression – an expression that involves a radical sign,
Example 4: Evaluating a Radical Expression
a. Evaluate g (h  2) when
b. Evaluate 2 a  b 2 when
a  11 and b  5 .
g  5 and h  3 .
ON YOUR OWN:
M3: Chapter 9 Notes
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Example 5: Solving an Equation Using Square Roots
A construction worker building a skyscraper accidentally drops a bolt
from a height of 500 feet. Use the equation d  16t 2 to determine the
time t in seconds that it takes a dropped object to fall a distance of d
feet. How long does the free fall part of the ride take?
M3: Chapter 9 Notes
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Section 9.3: The Pythagorean Theorem
Learning Goal: We will use the Pythagorean Theorem to solve problems.
Vocabulary:
 Hypotenuse – the side of a right triangle that is opposite the
right angle
 Leg – the two sides of a right triangle that form the right angle
Example 1: Finding the Length of a Hypotenuse
Kyle is building a triangular model of a mountain for the scenery for a
play. Find the length of the base of the model, to the nearest foot.
Example 2: Finding the Length of a Leg
Find the unknown length b.
M3: Chapter 9 Notes
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ON YOUR OWN:
Find the unknown length. Round your answer to the nearest tenth if
necessary.
Converse of the Pythagorean Theorem:
Example 3: Identifying Right Triangles
Determine whether the triangle with the given side lengths is a right
triangle.
a. 2, 4, 5
b. 9, 40, 41
c. 3, 5, 7
d. 15, 8, 17
M3: Chapter 9 Notes
EXTRA PRACTICE:
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M3: Chapter 9 Notes
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Section 9.4: Real Numbers
Learning Goal: We will compare an order real numbers.
Vocabulary:
 Irrational number – a number that cannot be written as a quotient
of two numbers. (a decimal that neither terminates nor repeats)
 Real number – the set of all rational numbers and irrational
numbers
Example 1: Classifying Real Numbers
Number
Decimal Form
Decimal Type
5
a.
8
5
b.
6
c. 19
ON YOUR OWN:
Type
M3: Chapter 9 Notes
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Example 2: Comparing Real Numbers
11
Copy and complete 5 _____ using <, >, or =.
4
ON YOUR OWN:
Example 3: Ordering Real Numbers
Use a number line to order the numbers  3 ,
to greatest.
ON YOUR OWN:
7,
21
9
,  from least
8
5
M3: Chapter 9 Notes
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Example 4: Using Irrational Numbers
Starting at their campsite, Alaina and Bob walk two separate paths
that are each 500 feet long. Alaina walks 400 feet east and then 100
feet south. Bob walks 200 feet south and then 300 feet east. Who is
farther from the campsite and by how many feet?