Pre-Algebra

6-3 The Pythagorean Theorem

6-3 The Pythagorean Theorem

Pre-Algebra

Warm Up

Problem of the Day

Lesson Presentation

Pre-Algebra

6-3 The Pythagorean Theorem

Warm UpGraph the figures with the given verticals and find the area.

1. (–1, –1), (–1, 3), (6, –1)

2. (2, 1), (8, 1), (6, –3)

3. (3, –2), (15, –2), (14, 6), (4, 6)

Pre-Algebra

6-3 The Pythagorean Theorem

14 units2

12 units2

88 units2

Pre-Algebra

6-3 The Pythagorean Theorem

Problem of the DayA side of a square A is 5 times the length of square B. How many times as great is the area of square A than the area of square B? 25

Pre-Algebra

6-3 The Pythagorean Theorem

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

6-3 The Pythagorean Theorem

Pre-Algebra

6-3 The Pythagorean Theorem

Pythagorean Theorem

leg

hypotenuse

Vocabulary

Pre-Algebra

6-3 The Pythagorean Theorem

a2 + b2 = c2

Pre-Algebra

6-3 The Pythagorean Theorem

4

5

c

6.40 c

A.

Pythagorean TheoremSubstitute for a and b.

a2 + b2 = c2

42 + 52 = c2

16 + 25 = c2

41 = c

Simplify powers. Solve for c; c = c2.

Additional Example 1A: Find the the Length of a Hypotenuse

Find the length of the hypotenuse.

41 = c2

Pre-Algebra

6-3 The Pythagorean Theorem

15 = c

B.

Pythagorean TheoremSubstitute for a and b.

a2 + b2 = c2

92 + 122 = c2

81 + 141 = c2

225 = cSimplify powers. Solve for c; c = c2.

Additional Example 1B: Find the the Length of a Hypotenuse

Find the length of the hypotenuse.

triangle with coordinates

(1, –2), (1, 7), and (13, –2)

Pre-Algebra

6-3 The Pythagorean Theorem

Try This: Example 1A

5

7

cA.

Find the length of the hypotenuse.

8.60 c

Pythagorean TheoremSubstitute for a and b.

a2 + b2 = c2

52 + 72 = c2

25 + 49 = c2

74 = cSimplify powers. Solve for c; c = c2.

Pre-Algebra

6-3 The Pythagorean Theorem

B. triangle with coordinates (–2, –2), (–2, 4), and (3, –2)

x

y

The points form a right triangle.

(–2, –2)

(–2, 4)

(3, –2)

Try This: Example 1B

Find the length of the hypotenuse.

7.81 c

Pythagorean Theorema2 + b2 = c2

62 + 52 = c2

36 + 25 = c2

61 = cSimplify powers. Solve for c; c = c2.

Substitute for a and b.

Pre-Algebra

6-3 The Pythagorean Theorem

Additional Example: 2 Finding the Length of a Leg in a Right Triangle

25

7

b

576 = 24b = 24

a2 + b2 = c2

72 + b2 = 252

49 + b2 = 625–49 –49

b2 = 576

Solve for the unknown side in the right triangle.

Pythagorean TheoremSubstitute for a and c. Simplify powers.

Pre-Algebra

6-3 The Pythagorean Theorem

Try This: Example 2

b 11.31

12

4

ba2 + b2 = c2

42 + b2 = 122

16 + b2 = 144–16 –16

b2 = 128

128 11.31

Solve for the unknown side in the right triangle.

Pythagorean TheoremSubstitute for a and c. Simplify powers.

Pre-Algebra

6-3 The Pythagorean Theorem

Additional Example 3: Using the Pythagorean Theorem to Find Area

a6 6

4 4

a2 + b2 = c2

a2 + 42 = 62

a2 + 16 = 36

a2 = 20a = 20 units ≈ 4.47 units

Find the square root of both sides.

Substitute for b and c.Pythagorean Theorem

A = hb = (8)( 20) = 4 20 units2 17.89 units212

12

Use the Pythagorean Theorem to find the height of the triangle. Then use the height to find the area of the triangle.

Pre-Algebra

6-3 The Pythagorean Theorem

a2 + b2 = c2

a2 + 22 = 52

a2 + 4 = 25

a2 = 21

a = 21 units ≈ 4.58 units

Find the square root of both sides.

Substitute for b and c.

Pythagorean Theorem

A = hb = (4)( 21) = 2 21 units2 4.58 units212

12

Try This: Example 3

Use the Pythagorean Theorem to find the height of the triangle. Then use the height to find the area of the triangle.

a5 5

2 2

Pre-Algebra

6-3 The Pythagorean Theorem

Lesson Quiz

1. Find the height of the triangle.

2. Find the length of side c to the nearest meter.

3. Find the area of the largest triangle.

4. One leg of a right triangle is 48 units long, and the hypotenuse is 50 units long. How long is the other leg?

8m

12m

60m2

14 units

h

c10 m

6 m 9 m

Use the figure for Problems 1-3.