Course 3 Chapter 6 Lesson 3

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Preview
Warm Up
California Standards
Lesson Presentation
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Warm Up Estimate each square root to nearest
hundredth. 1. v30 2. v14 3. v55 4. v48
5.48
3.74
7.42
6.93
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(No Transcript)
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Vocabulary
Pythagorean Theorem leg hypotenuse
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The Pythagorean Theorem shows that a special
relationship exists between the sides of a right
triangle.
You can use the theorem to find the length of any
side of a right triangle.
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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
Use the Pythagorean Theorem.
a2 b2 c2
Substitute for a and b.
122 162 c2
Evaluate the powers.
144 256 c2
Add.
400 c2
Take the square root of both sides.
20 c
The length of the hypotenuse is 20 cm.
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Additional Example 1B Calculating the Length of
a Missing Side of a Right Triangle
Use the Pythagorean Theorem to find the missing
measure.
b
5 cm
13 cm
Use the Pythagorean Theorem.
a2 b2 c2
Substitute for a and c.
52 b2 132
25 b2 169
Evaluate the powers.
25
25
Subtract 25 from each side.
b2 144
Take the square root of both sides.
b 12
The length of the missing leg is 12 cm.
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Check It Out! Example 1A
Use the Pythagorean Theorem to find the missing
measure.
c
11 cm
15 cm
a2 b2 c2
Use the Pythagorean Theorem.
Substitute for a and b.
112 152 c2
Evaluate the powers.
121 225 c2
Add.
346 c2
Take the square root of both sides.
18.6 ? c
The length of the hypotenuse is about 18.6 cm.
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Check It Out! Example 1B
Use the Pythagorean Theorem to find the missing
measure.
b
3 cm
5 cm
Use the Pythagorean Theorem.
a2 b2 c2
Substitute for a and c.
32 b2 52
9 b2 25
Evaluate the powers.
9
9
Subtract 9 from each side.
b2 16
Take the square root of both sides.
b 4
The length of the missing leg is 4 cm.
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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.
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Additional Example 2 Continued
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.
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Additional Example 2 Continued
You can use the Pythagorean Theorem to write an
equation.
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Additional Example 2 Continued
a2 b2 c2
Use the Pythagorean Theorem.
Substitute for the known variables.
752 752 c2
5,625 5,625 c2
Evaluate the powers.
11,250 c2
Add.
Take the square roots of both sides.
106.066 ? c
Round.
106.1 ? c
The distance from one corner of the field to the
opposite corner is about 106.1 feet.
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Additional Example 2 Continued
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.
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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.
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Check It Out! Example 2 Continued
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 fields are legs, and they are
33 yards long and 100 yards long.
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Check It Out! Example 2 Continued
You can use the Pythagorean Theorem to write an
equation.
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Check It Out! Example 2 Continued
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.
105.304 ? 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.
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Check It Out! Example 2 Continued
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.
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Additional Example 3A Identifying a Right
Triangle
Tell whether the given side lengths form a right
triangle.
A. 12, 35, 37
?
a2 b2 c2
Compare a2 to b2 to c2.
?
122 352 372
Substitute the longest side length for c.
?
Simplify the powers.
144 1225 1369
Add.
1369 1369
?
The side lengths form a right triangle.
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Additional Example 3B Identifying a Right
Triangle
Tell whether the given side lengths form a right
triangle.
B. 8, 12, 16
?
a2 b2 c2
Compare a2 to b2 to c2.
?
82 122 162
Substitute the longest side length for c.
?
64 144 256
Simplify the powers.
Add.
208 ? 256
?
The side lengths do not form a right triangle.
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Check It Out! Example 3A
Tell whether the given side lengths form a right
triangle.
A. 10, 15, 20
?
a2 b2 c2
Compare a2 to b2 to c2.
?
102 152 202
Substitute the longest side length for c.
?
100 225 400
Simplify the powers.
Add.
325 ? 400
?
The side lengths do not form a right triangle.
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Check It Out! Example 3B
Tell whether the given side lengths form a right
triangle.
B. 8, 15, 17
?
a2 b2 c2
Compare a2 to b2 to c2.
?
82 152 172
Substitute the longest side length for c.
?
Simplify the powers.
64 225 289
Add.
289 289
?
The side lengths form a right triangle.
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Lesson Quiz
Use the Pythagorean Theorem to find each missing
measure.
2.
1.
21 in.
40 m
3. 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
Tell whether the given side lengths form a right
triangle.
yes
no
5. 33, 56, 65
4. 2.5, 3, 4.5