Unit 6 Lesson 1

The Pythagorean TheoremCCSS

G-SRT 4: Prove theorems about triangles.

Lesson Goals► Use the Pythagorean Th. to

find missing side lengths in a right triangle.

► Recognize a Pythagorean Triple.

ESLRs: Becoming Effective Communicators, Competent Learners and Complex Thinkers

the Pythagorean Theorem

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

ac

b

2 2 2c a b

proof

A C

BD

af

ec

bStatement Reason

1. is a rt.ABC 2. Draw CD AB3. ~ ~CAB DAC DCB 4. ,

c a c b

a e b f

2 25. ,a ce b cf 2 26. a b ce cf 2 27. ( )a b c e f

8. c e f

1. Given

2. Perpendicular Post.3. Right D Altitude Th

4. Def similar polygons

5. Cross-Products Prop.6. Add. Prop. of =7. Distributive Prop8. Segment Add. Post

c

2 2 29. a b c 9. Substitution

2 2 2

Given: is a rt.

Prove:

ABC

a b c

definition

Pythagorean Triples A set of three positive integers: a, b, c

that satisfy the equation

Examples

3, 4, and 5

5, 12, and 13

8, 15, and 17

2 2 2a b c

6a

2 36a

2180 144a

22 26 5 12a

A C

B

a

12

6 5

example

Find the missing side

Do the side lengths form a Pythagorean Triple?

You Try

Find the missing side

A C

B

7c

24

25c

2 625c

2 49 576c

2 2 27 24c

2 2 2c a b

Do the side lengths form a Pythagorean Triple?

You Try

Find the missing side

19.06 10x

2 360x

441 81 x

2 2 221 9 x

2 2 2c a b

Do the side lengths form a Pythagorean Triple?

39h

2 39h

2 25 64h

2 2 25 8h

Find the area of the triangle.

example

12A bh

12 10 39

25 39 m231 m

You TryFind the area of the triangle

2 11x

2 44x

2144 100x

2 2 212 10x

12A bh

2 11 10 220 11 m

266.3 m

x2 112 11

50 ft 50 ft100 ft

Two antennas are each supported by 100 foot cables. If the cables are attached to the antennas 50 feet fromthe ground, how far apart are the antennas?

d

100 ft

50 3x

2 7500x

2 2500 10,000x

2 2 250 100x

x

170 md 852d x

example

summary

• How is the Pythagorean Theorem useful?

• What is the difference between the Pythagorean Theorem in general and a Pythagorean Triple?

Today’s Assignment

p. 538: 8, 14, 18 – 28 e, 31 – 33, 37

Find the unknown side length. Simplify answers that are radicals. Tell whether the side lengths form a Pythagorean triple.

Find the area of the figure. Round decimal answers to the nearest tenth.

Find the value of x. Simplify answers that are radicals.

The variables r and s represent the lengths of the legs of a right triangle, and t represents the length of the hypotenuse. The values of r, s, and t form a Pythagorean triple. Find the unknown value.