Applying the Pythagorean Theorem and Its Converse

Warm Up

Lesson Presentation

Problem of the Day

Lesson Quizzes

Applying the Pythagorean Theorem and Its Converse

Warm UpFind the distance between each pair of points.

1. (8, 2) and (8, 7)

2. (–2, 4) and (5, 4)

3. (–1, –1) and (9, –1)

4. (–8, –4) and (–2, –4)

5 units

7 units

10 units

6 units

Applying the Pythagorean Theorem and Its Converse

Problem of the DayThe sum of the squares of two positive numbers is 100. One number is two more than the other. What are the numbers?

6 and 8

Applying the Pythagorean Theorem and Its Converse

Learn to use the Distance Formula and the Pythagorean Theorem and its converse to solve problems.

Applying the Pythagorean Theorem and Its Converse

7.615 c

Use the Pythagorean TheoremSimplify.

72 + 32 = c2

49 + 9 = c2

58 = c2

58 = c

Add.

Additional Example 1: Marketing Application

What is the diagonal length of the projector screen?

The diagonal length should be given as about 7.62 feet.

Applying the Pythagorean Theorem and Its Converse

14.42 c

Use the Pythagorean TheoremSimplify.

122 + 82 = c2

144 + 64 = c2

208 = c2

208 = c

Add.

Check It Out: Example 1

What is the diagonal length of the projector screen?

The diagonal length should be given as about 14.4 feet.

12 ft

8 ft

Applying the Pythagorean Theorem and Its Converse

Applying the Pythagorean Theorem and Its Converse

Use the Distance Formula.

J and K

Additional Example 2A: Finding Distance on the Coordinate Plane

Find the distances between the points to the nearest tenth.

Let A be (x1, y1)² and B be (x2, y2)².

Applying the Pythagorean Theorem and Its Converse

Substitute.

Subtract.

Additional Example 2A Continued

Find the distances between the points to the nearest tenth.

d = (0 – (–4))² + (–3 – 0 )²

d = (4)² + (–3)²

Simplify powers.

d = 16 + 9

The distance between A and B is 5 units.

Applying the Pythagorean Theorem and Its Converse

L and M

Additional Example 2B: Finding Distance on the Coordinate Plane

Find the distances between the points to the nearest tenth.

Use the Distance Formula.

Let A be (x1, y1)² and B be (x2, y2)².

Applying the Pythagorean Theorem and Its Converse

Substitute.

Subtract.

Additional Example 2B Continued

Find the distances between the points to the nearest tenth.

d = (5 – 4)² + (–3 – 0)²

d = (1)² + (–3)²

Simplify powers.

d = 1 + 9

The distance between L and M is about 3.2 units.

Applying the Pythagorean Theorem and Its Converse

J and L

Check It Out: Example 2A

Find the distances between the points to the nearest tenth.

Use the Distance Formula.

Let A be (x1, y1)² and B be (x2, y2)².

Applying the Pythagorean Theorem and Its Converse

Substitute.

Subtract.

Check It Out: Example 2A Continued

Find the distances between the points to the nearest tenth.

d = (4 – (–4))² + (0 – 0)²

d = (8)² + (0)²

Simplify powers.

d = 64 + 0

The distance between J and L is 8 units.

Applying the Pythagorean Theorem and Its Converse

K and M

Check It Out: Example 2B

Find the distances between the points to the nearest tenth.

Use the Distance Formula.

Let A be (x1, y1)² and B be (x2, y2)².

Applying the Pythagorean Theorem and Its Converse

Substitute.

Subtract.

Check It Out: Example 2B Continued

Find the distances between the points to the nearest tenth.

d = ((–4) – 0)² + (– 3 – (–3)²

d = (–4)² + (0)²

Simplify powers.

d = 16 + 0

The distance between K and M is 4 units.

Applying the Pythagorean Theorem and Its Converse

Additional Example 3A: Identifying a Right Triangle

a2 + b2 = c2

92 + 122 = 152

81 + 144 = 225

225 = 225 √

Substitute.

Compare a² + b² to c².

The side lengths form a right triangle.

Tell whether the given side lengths form a right triangle.

9, 12, 15

Simplify.

Add.

Applying the Pythagorean Theorem and Its Converse

Additional Example 3B: Identifying a Right Triangle

a2 + b2 = c2

82 + 102 = 132

63 + 100 = 169

163 ≠ 169 x

Substitute.

Compare a² + b² to c².

The side lengths do not form a right triangle.

Tell whether the given side lengths form a right triangle.

8, 10, 13

Simplify.

Add.

Applying the Pythagorean Theorem and Its Converse

Check It Out: Example 3A

a2 + b2 = c2

52 + 62 = 92

25 + 36 = 81

61 ≠ 81 x

Substitute.

Compare a² + b² to c².

The side lengths do not form a right triangle.

Tell whether the given side lengths form a right triangle.

5, 6, 9

Simplify.

Add.

Applying the Pythagorean Theorem and Its Converse

Check It Out: Example 3B

a2 + b2 = c2

82 + 152 = 172

64 + 225 = 289

289 = 289 √

Substitute.

Compare a² + b² to c².

The side lengths form a right triangle.

Tell whether the given side lengths form a right triangle.

8, 15, 17

Simplify.

Add.