1 Lesson 3.3.1 The Pythagorean Theorem. 2 Lesson 3.3.1 The Pythagorean Theorem California Standard: Measurement and Geometry 3.3 Know and understand the

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Lesson 3.3.1Lesson 3.3.1

The Pythagorean TheoremThe Pythagorean Theorem

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Lesson

3.3.1The Pythagorean TheoremThe Pythagorean Theorem

California Standard:Measurement and Geometry 3.3Know and understand the Pythagorean theorem and its converse and use it to find the length of the missing side of a right triangle and the lengths of other line segments and, in some situations, empirically verify the Pythagorean theorem by direct measurement.

What it means for you:You’ll learn about an equation that you can use to find a missing side length of a right triangle.

Key words:• Pythagorean theorem• right triangle• hypotenuse• legs• right angle

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The Pythagorean TheoremThe Pythagorean TheoremLesson

3.3.1

You will have come across right triangles before — they’re just triangles that have one corner that’s a 90° angle.

Well, there’s a special formula that links the side lengths of a right triangle — it comes from the Pythagorean theorem.

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The Pythagorean Theorem is About Right Triangles

Lesson

3.3.1

A right triangle is any triangle that has a 90° angle (or right angle) as one of it corners.

The hypotenuse is the longest side of the triangle. It’s the side directly opposite the right angle. Right

angle

The other two sides of the triangle are called the legs.

c b

a

You need to know the names of the

parts of a right triangle:

90°

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3.3.1

In diagrams of right triangles, the hypotenuse is usually labeled as c, and the two legs as a and b. It doesn’t matter which leg you label a, and which you label b.

Leg = b

Leg = a

Hypotenuse = c

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Guided Practice

Solution follows…

Lesson

3.3.1

Complete the missing labels on the diagram.

1. Hypotenuse

2.

3. 90°

Legs

Right angle

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Guided Practice

Solution follows…

Lesson

3.3.1

In Exercises 4–7 say which side of the right triangle is the hypotenuse.

4. 5.

6. 7.

I II

III

I II

III

I II

III

I II

III

I II

II III

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The Theorem Links Side Lengths of Right Triangles

Lesson

3.3.1

Pythagoras was a Greek mathematician who lived around 500 B.C. A famous theorem about right triangles is named after him. It’s called the Pythagorean theorem:

For any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the two legs.

This all sounds very complicated, but it’s not so bad once you know what it actually means.

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Look again at the right triangle. Now add three squares whose side lengths are the same as the side lengths of the triangle:

Lesson

3.3.1

b

a

c

Area = c2

Area = b2

Area = a2

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3.3.1

What the Pythagorean theorem is saying is that the area of the red square is the same as the area of the blue square plus the area of the green square.

c2

b2

a2+=

b

a

c

Area = c2

Area = b2

Area = a2

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3.3.1

So this is what the Pythagorean theorem looks like written algebraically:

For any right triangle:c2 = a2 + b2

It means that if you know the lengths of two sides of a right triangle, you can always find the length of the other side using the equation.

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You Can Check the Theorem Using a Right Triangle

Lesson

3.3.1

You can check for yourself that the theorem works by measuring the side lengths of right triangles, and putting the values into the equation.

Don’t forget — the Pythagorean theorem only works on right triangles. It won’t work on any other type.

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Example 1

Solution follows…

Lesson

3.3.1

Use the right triangle shown to verify the Pythagorean theorem.

3 units

4 units5 units

Solution

a = 3 units, b = 4 units, c = 5 units

Substitute the values

Simplify the equation

25 = 25

52 = 32 + 42

25 = 9 + 16

c2 = a2 + b2 Write out the formula

Read the side lengths from the diagram

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Guided Practice

Solution follows…

Lesson

3.3.1

Use the right triangles in Exercises 8–9 to verify the Pythagorean theorem.

8. 9.

9 units

12 units15 units

5 units

12 units

13 units

152 = 122 + 92

225 = 144 + 81 225 = 225

132 = 122 + 52

169 = 144 + 25 169 = 169

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Guided Practice

Solution follows…

Lesson

3.3.1

Use the right triangles in Exercises 10–11 to verify the Pythagorean theorem.

10. 11.

4.1 units

4 units

2.4 units

2.5 units0.7 units

0.9 units

4.12 = 42 + 0.92

16.81 = 16 + 0.81 16.81 = 16.81

2.52 = 2.42 + 0.72

6.25 = 5.76 + 0.49 6.25 = 6.25

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Independent Practice

Solution follows…

Lesson

3.3.1

In Exercises 1–3 say whether the triangle is a right triangle or not.

1. 2. 3.

Yes

No

Yes

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Solution follows…

Lesson

3.3.1

In Exercises 4–6 say which side of the right triangle is the hypotenuse.

4. 5.

6.

III

II

II II

III

I III

II

II I

II

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Solution follows…

Lesson

3.3.1

Use the triangles in Exercises 7–8 to verify the Pythagorean theorem.

7. 8.

6 units

17 units10 units

8 units

15 units

8 units

102 = 82 + 62

100 = 64 + 36 100 = 100

172 = 152 + 82

289 = 225 + 64 289 = 289

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Solution follows…

Lesson

3.3.1

Use the triangles in Exercises 9–10 to verify the Pythagorean theorem.

9. 10.

1 unit1.25 units

0.75 units

1.252 = 12 + 0.752

1.5625 = 1 + 0.5625 1.5625 = 1.5625

22 = 1.62 + 1.22

4 = 2.56 + 1.44 4 = 4

1.2 units

1.6 units

2 units

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Solution follows…

Lesson

3.3.1

11. Victor used the triangle shown on the right to try to verify the Pythagorean theorem. Explain why his work is wrong.

Victor’s work: 92 = 122 + 152

81 = 144 + 225 81 = 369

Victor has not correctly identified the hypotenuse. The theorem says c2 = a2 + b2 where c is the longest side, or hypotenuse. The first line of his work should be 152 = 92 + 122.

9 units

12 units15 units

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Round UpRound Up

Lesson

3.3.1

The Pythagorean theorem describes the relationship between the lengths of the hypotenuse and the legs of a right triangle.

It means that when you know the lengths of two of the sides of a right triangle, you can always find the length of the third side.

You’ll get a lot of practice at using the Pythagorean theorem in the next few Lessons.

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1 Lesson 3.3.1 The Pythagorean Theorem. 2 Lesson 3.3.1 The Pythagorean Theorem California Standard: Measurement and Geometry 3.3 Know and understand the