9-8 The Pythagorean Theorem

Course 2

Warm UpWarm Up

Problem of the DayProblem of the Day

Lesson PresentationLesson Presentation

Warm UpEstimate each square root to the nearest whole number. Use a calculator to check the reasonableness of your answers.

1. 2. 3.

459

Course 2

9-8 The Pythagorean Theorem

√18√26

√86

4. √125 11

Problem of the Day

A shipping carton measures 12 in. by 15 in. by 16 in. What is the longest rod that can be shipped in it?25 in.

Course 2

9-8 The Pythagorean Theorem

Learn to use the Pythagorean Theorem to find the length of a side of a right triangle.

Course 2

9-8 The Pythagorean Theorem

Vocabulary

leghypotenusePythagorean Theorem

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Course 2

9-8 The Pythagorean Theorem

Course 2

9-8 The Pythagorean Theorem

Hypotenuse

Leg

Leg

In a right triangle, the two sides that form the right angle are called legs. The side opposite the right angle is called the hypotenuse.

One of the first people to recognize the relationship between the sides of a right triangle was the Greek mathematician Pythagoras. This special relationship is called the Pythagorean Theorem.

Course 2

9-8 The Pythagorean Theorem

PYTHAGOREAN THEOREM

In a right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse.

a2 + b2 = c2 a

b

c

You can use the Pythagorean Theorem to find the length of any side of a right triangle.

Use the Pythagorean Theorem to find the missing measure.

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

Course 2

9-8 The Pythagorean Theorem

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

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

Course 2

9-8 The Pythagorean Theorem

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

Use the Pythagorean Theorem to find the missing measure.

Course 2

9-8 The Pythagorean Theorem

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

Course 2

9-8 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

Course 2

9-8 The Pythagorean Theorem

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.

Additional Example 2 Continued

Course 2

9-8 The Pythagorean Theorem

• 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.

Additional Example 2 Continued

Course 2

9-8 The Pythagorean Theorem

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

Additional Example 2 Continued

Course 2

9-8 The Pythagorean Theorem

Solve33

a2 + b2 = c2

752 + 752 = c2

5,625 + 5,625 = c2

11,250 = c2

106.066012 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.

Additional Example 2 Continued

Course 2

9-8 The Pythagorean Theorem

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.

Check It Out: Example 2

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Course 2

9-8 The Pythagorean Theorem

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.

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.

Check It Out: Example 2 Continued

Course 2

9-8 The Pythagorean Theorem

• 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.

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

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.

Check It Out: Example 2 Continued

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Course 2

9-8 The Pythagorean Theorem

Solve33

a2 + b2 = c2

332 + 1002 = c2

1089 + 10,000 = c2

11,089 = c2

105.3043208 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.

Check It Out: Example 2 Continued

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Course 2

9-8 The Pythagorean Theorem

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.

Lesson Quiz: Part I

21 in.40 m

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16

29

Course 2

9-8 The Pythagorean Theorem

Use the Pythagorean Theorem to find each missing measure.

1. 2.

3. a = , b = 30, c = 34

4. a = 20, b = 21, c =

Lesson Quiz: Part II

5. 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

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Course 2

9-8 The Pythagorean Theorem