FeatureLesson Geometry Lesson Main (For help, go to the Skills Handbook, page 753.) 1.2. 3.4. Lesson 8-1 The Pythagorean Theorem and Its Converse Square.

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(For help, go to the Skills Handbook, page 753.)

1. 2.

3. 4.

Lesson 8-1

The Pythagorean Theorem and Its Converse The Pythagorean Theorem and Its Converse

Square the lengths of the sides of each triangle. What do you notice?

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Check Skills You’ll Need

8-1

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1. 32 = (3)(3) = 9; 42 = (4)(4) = 16; 52 = (5)(5) = 25; 9 + 16 = 25

2. 52 = (5)(5) = 25; 122 = (12)(12) = 144; 132 = (13)(13) = 169; 25 + 144 = 169

3. 62 = (6)(6) = 36; 82 = (8)(8) = 64; 102 = (10)(10) = 100; 36 + 64 = 100

4. 42 = (4)(4) = 16; (4 2)2 = (4 2)(4 2) = 16 (2)(2) = 16(2) = 32; 16 + 16 = 32.

Solutions

Lesson 8-1

The Pythagorean Theorem and Its Converse The Pythagorean Theorem and Its Converse

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8-1

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Lesson 8-1

The Pythagorean Theorem and Its Converse The Pythagorean Theorem and Its Converse

Notes

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2 2 2c a b

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Lesson 8-1

The Pythagorean Theorem and Its Converse The Pythagorean Theorem and Its Converse

Notes

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A Pythagorean triple is a set of nonzero whole numbers a, b, and c that satisfy the equation a2 + b2 = c2.

Some common Pythagorean triples are:3, 4, 5 5, 12, 13 8, 15, 17 7, 24, 25

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Lesson 8-1

The Pythagorean Theorem and Its Converse The Pythagorean Theorem and Its Converse

Notes

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Lesson 8-1

The Pythagorean Theorem and Its Converse The Pythagorean Theorem and Its Converse

Notes

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Lesson 8-1

The Pythagorean Theorem and Its Converse The Pythagorean Theorem and Its Converse

Notes

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obtuse. is implies 222 bac

acute. is implies 222 bac

right. is implies 222 bac

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A right triangle has legs of length 16 and 30. Find

the length of the hypotenuse. Do the lengths of the sides

form a Pythagorean triple?

a2 + b2 = c2 Use the Pythagorean Theorem.

162 + 302 = c2 Substitute 16 for a and 30 for b.

256 + 900 = c2 Simplify.

1156 = c2

34 = c Take the square root.

The length of the hypotenuse is 34.

The lengths of the sides, 16, 30, and 34, form a Pythagorean triple because they are whole numbers that satisfy a2 + b2 = c2. Notice that each length is twice the common Pythagorean triple of 8, 15, and 17.

Lesson 8-1

The Pythagorean Theorem and Its Converse The Pythagorean Theorem and Its Converse

Quick Check

Additional Examples

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Pythagorean Triples

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a2 + b2 = c2 Use the Pythagorean Theorem.

x2 + 102 = 122 Substitute x for a, 10 for b, and 12 for c.

x2 + 100 = 144 Simplify.

x2 = 44 Subtract 100 from each side.

x = 4(11) Take the square root of each side.

x = 2 11 Simplify.

Find the value of x. Leave your answer in simplest radical

form.

Lesson 8-1

The Pythagorean Theorem and Its Converse The Pythagorean Theorem and Its Converse

Quick Check

Additional Examples

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Using Simplest Radical Form

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c = 16,200 Take the square root.

c 127.27922 Use a calculator.

a2 + b2 = c2 Use the Pythagorean Theorem.

902 + 902 = c2 Substitute 90 for a and for b.

8,100 + 8,100 = c2 Simplify.

16,200 = c2

The distance to home plate from second base is about 127 ft.

Use the information to draw a baseball diamond.

A baseball diamond is a square with 90-ft sides. Home plate

and second base are at opposite vertices of the square. About how far

is home plate from second base?

Lesson 8-1

The Pythagorean Theorem and Its Converse The Pythagorean Theorem and Its Converse

Quick Check

Additional Examples

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Real-World Connection

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Is this triangle a right triangle?

Lesson 8-1

The Pythagorean Theorem and Its Converse The Pythagorean Theorem and Its Converse

52 ≠ 49

Because a2 + b2 ≠ c2, the triangle is not a right triangle.

Quick Check

a2 + b2 c2

42 + 62 72 Substitute 4 for a, 6 for b, and 7 for c.

16 + 36 49 Simplify.

Additional Examples

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Is It a Right Triangle?

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The numbers represent the lengths of the sides of atriangle. Classify each triangle as acute, obtuse, or right.

Lesson 8-1

The Pythagorean Theorem and Its Converse The Pythagorean Theorem and Its Converse

a.15, 20, 25

c2 a2 + b2 Compare c2 with a2 + b2.

625 225 + 400 Simplify.

625 = 625

Because c2 = a2 + b2, the triangle is a right triangle.

252 152 + 202 Substitute the greatest length for c.

Additional Examples

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Classifying Triangles as Obtuse, Acute, or Right

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(Continued)

Lesson 8-1

The Pythagorean Theorem and Its Converse The Pythagorean Theorem and Its Converse

b. 10, 15, 20

400 325

Because c2 a2 + b2, the triangle is obtuse.

Quick Check

20

2 102 + 152 Substitute the greatest length for c.

400 100 + 225 Simplify.

c 2 a2 + b

2 Compare c 2 with a2 + b 2.

Additional Examples

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2 33

1. Find the value of x.

2. Find the value of x. Leave your answer in simplest radical form.

3. The town of Elena is 24 mi north and 8 mi west of Holberg. A train runs on a straight track between the two towns. How many miles does it cover? Round your answer to the nearest whole number.

4. The lengths of the sides of a triangle are 5 cm, 8 cm, and 10 cm. Is it acute, right, or obtuse?

15

25 mi

obtuse

Lesson 8-1

The Pythagorean Theorem and Its Converse The Pythagorean Theorem and Its Converse

Lesson Quiz

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