4-9 The Pythagorean Theorem

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4-9 The Pythagorean Theorem

Warm UpEstimate each square root to nearest hundredth.

1. √30

2. √14

3. √55

4. √48

5.48

3.74

7.42

6.93

4-9 The Pythagorean Theorem

MG3.3 Know and understand the Pythagorean theorem and its converse and use it 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.

California Standards

4-9 The Pythagorean Theorem

Pythagorean Theorem

leg

hypotenuse

Vocabulary

4-9 The Pythagorean Theorem

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.

4-9 The Pythagorean Theorem

Use the Pythagorean Theorem to find the missing measure.

Additional Example 1A: Calculating the Length of a Side of a Right Triangle

12 cm

16 cm

a2 + b2 = c2

c

122 + 162 = c2 144 + 256 = c2

400 = c2

The length of the hypotenuse is 20 cm.

Use the Pythagorean Theorem. Substitute for a and b.

Evaluate the powers.Add.Take the square root of both sides.

20 = c

√400 = √c2

4-9 The Pythagorean TheoremAdditional Example 1B: Calculating the Length of a

Missing Side of a Right Triangle Use the Pythagorean Theorem to find the missing measure.

5 cm

b

a2 + b2 = c2

13 cm

52 + b2 = 132 25 + b2 = 169

b2 = 144

The length of the missing leg is 12 cm.

Use the Pythagorean Theorem. Substitute for a and c.Evaluate the powers.

Take the square root of both sides.b = 12

–25 –25 Subtract 25 from each side.

√b2 = √144

4-9 The Pythagorean Theorem

Use the Pythagorean Theorem to find the missing measure.

11 cm

15 cm

a2 + b2 = c2

c

112 + 152 = c2 121 + 225 = c2

346 = c2

The length of the hypotenuse is about 18.6 cm.

Use the Pythagorean Theorem. Substitute for a and b.

Evaluate the powers.Add.Take the square root of both sides.

18.6 c

Check It Out! Example 1A

√346 = √c2

4-9 The Pythagorean Theorem

Use the Pythagorean Theorem to find the missing measure.

3 cm

b

a2 + b2 = c2

5 cm

32 + b2 = 52 9 + b2 = 25

b2 = 16

The length of the missing leg is 4 cm.

Use the Pythagorean Theorem. Substitute for a and c.Evaluate the powers.

Take the square root of both sides.b = 4

–9 –9 Subtract 9 from each side.

Check It Out! Example 1B

√b2 = √ 16

4-9 The Pythagorean TheoremAdditional 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.

4-9 The Pythagorean TheoremAdditional Example 2 Continued

• The segment between the two corners is the hypotenuse.

• The sides of the field are legs, and they are each 75 feet long.

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.

11 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.

4-9 The Pythagorean TheoremAdditional Example 2 Continued

22 Make a PlanYou can use the Pythagorean Theorem towrite an equation.

4-9 The Pythagorean TheoremAdditional Example 2 Continued

Solve33

a2 + b2 = c2

752 + 752 = c2

5,625 + 5,625 = c2

11,250 = c2

106.066 c

The distance from one corner of the field to the opposite corner is about 106.1 feet.

Use the Pythagorean Theorem.

Substitute for the known variables.

Evaluate the powers.

Add.

Take the square roots of both sides.

106.1 c Round.

4-9 The Pythagorean TheoremAdditional Example 2 Continued

Look Back44

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.

4-9 The Pythagorean TheoremCheck 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.

4-9 The Pythagorean TheoremCheck It Out! Example 2 Continued

11 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.

• 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.

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.

4-9 The Pythagorean TheoremCheck It Out! Example 2 Continued

22 Make a PlanYou can use the Pythagorean Theorem towrite an equation.

4-9 The Pythagorean TheoremCheck It Out! Example 2 Continued

Solve33

a2 + b2 = c2

332 + 1002 = c2

1089 + 10,000 = c2

11,089 = c2

105.304 c

The distance from one corner of the field to the opposite corner is about 105.3 yards.

Use the Pythagorean Theorem.

Substitute for the known variables.

Evaluate the powers.

Add.

Take the square roots of both sides.

105.3 c Round.

4-9 The Pythagorean TheoremCheck It Out! Example 2 Continued

Look Back44

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.

4-9 The Pythagorean Theorem

Tell whether the given side lengths form a right triangle.

Additional Example 3A: Identifying a Right Triangle

a2 + b2 = c2

122 + 352 = 372

144 + 1225 = 1369

The side lengths form a right triangle.

Compare a2 to b2 to c2.

Substitute the longest side length for c.Simplify the powers.

Add.1369 = 1369

A. 12, 35, 37

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4-9 The Pythagorean Theorem

Tell whether the given side lengths form a right triangle.

Additional Example 3B: Identifying a Right Triangle

a2 + b2 = c2

82 + 122 = 162

64 + 144 = 256

The side lengths do not form a right triangle.

Compare a2 to b2 to c2.

Substitute the longest side length for c.Simplify the powers.

Add.208 ≠ 256

B. 8, 12, 16

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4-9 The Pythagorean Theorem

Tell whether the given side lengths form a right triangle.

Check It Out! Example 3A

a2 + b2 = c2

102 + 152 = 202

100 + 225 = 400

The side lengths do not form a right triangle.

Compare a2 to b2 to c2.

Substitute the longest side length for c.Simplify the powers.

Add.325 ≠ 400

A. 10, 15, 20

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?

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4-9 The Pythagorean Theorem

Tell whether the given side lengths form a right triangle.

Check It Out! Example 3B

a2 + b2 = c2

82 + 152 = 172

64 + 225 = 289

The side lengths form a right triangle.

Compare a2 to b2 to c2.

Substitute the longest side length for c.Simplify the powers.

Add.289 = 289

B. 8, 15, 17

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4-9 The Pythagorean TheoremLesson Quiz

21 in.40 m

Use the Pythagorean Theorem to find each missing measure.

1. 2.

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

no yes5. 33, 56, 65 4. 2.5, 3, 4.5

Tell whether the given side lengths form a right triangle.