Geometry

9.2 The Pythagorean Theorem

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 2

Goals

Prove the Pythagorean Theorem. Solve triangles using the theorem. Solve problems using the theorem.

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 3

This is ancient history.

The Egyptian Pyramid builders used it to make square corners.

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 4

Terminology

Leg

Leg

Hypotenuse

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 5

Hypotenuse = “stretched against”

35

4

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 6

Proof

Proofs of the Pythagorean Theorem are numerous – well over 300 known.

Discovered in many ancient cultures. We can use what we learned about the

altitude of right triangles to prove the Pythagorean Theorem, too.

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 7

From last lesson:

2 2

a cm b cn

a cm b cn

D

ab

mn

h

c

2 2

2 2 2

( )

a b cm cn

c m n

c c

a b c

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 8

Chinese Proof

a b

a

a

a

b

b

b

c

c

c

c

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 9

Chinese Proof

a b

a

a

a

b

b

b

c

c

c

c

Area of the square:

A = c2

Area of one triangle:

A = (½) ab

Area of 4 triangles:

A = 2ab

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 10

Chinese Proof

a b

a

a

a

b

b

b

c

c

c

c

Area of the square:

A = c2

Area of 4 triangles:

A = 2ab

Area Sum

c2 + 2ab

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 11

Chinese Proof

a b

a

a

a

b

b

b

c

c

c

c

Area Sum

c2 + 2ab

?a + b

?a + b

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 12

Chinese Proof

a b

a

a

a

b

b

b

c

c

c

c

Area Sum

c2 + 2ab

Area another way:

2

2 2

2 2

( )

( )( )

2

A a b

a b a b

a ab ab b

a ab b

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 13

Chinese Proof

a b

a

a

a

b

b

b

c

c

c

c

Area Sum

c2 + 2ab

or

a2 + 2ab + b2

These areas are equal.

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 14

Chinese Proof

a b

a

a

a

b

b

b

c

c

c

c

2 2 2

2 2 2

2 2a ab b c ab

a b c

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 15

President Garfield (1876)20th President of the United States

Area of Trapezoid = Sum of area of three triangles

21 1 1 12 2 2 2

2 2 21 12 2

2 2 2

2 2 2

2

2 2

a b a b ab ab c

a ab b ab c

a ab b ab c

a b c

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 16

The Pythagorean Theorem

a

b

c

a2 + b2 = c2

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Example 1 Solve.

5

6

c

2 2 25 6

25 36

61

61

7.81

c

c

c

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Example 2 Solve.

a

2

10

2 2 2

2

2

2 10

4 100

96

96 16 6

4 6 9.80

a

a

a

a

a

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 19

Example 3 Solve.

x

x20

2 2 2

2

2

20

2 400

200

200

10 2

14.14

x x

x

x

x

x

x

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 20

Solve these two triangles.

2 2 2

2

2

3 4

9 16

25

5

c

c

c

c

3

4

c5

12

c

2 2 2

2

2

5 12

25 144

169

13

c

c

c

c

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 21

Pythagorean Triples

3

4

55

12

13

3 – 4 – 5 and 5 – 12 – 13 are Pythagorean Triples.

Each side is an integer.

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Example

Is 10-10-20 a Pythagorean Triple? 102 + 102 = 202 ? 100 + 100 = 400 ? 200 = 400 ? False! Not a Pythagorean Triple.

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 23

Example

Is 20-21-29 a Pythagorean Triple? 202 + 212 = 292 ? 400 + 441 = 841 ? 841 = 841 True It is a Pythagorean Triple.

Generating Triples

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 24

Area of a Triangle

h

b

12A bh

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 25

Find the area.

12

15

12A bh

h

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 26

Find the area.

12

15h

2 2 2

2

2

12 15

144 225

81

9

h

h

h

h

A Pythagorean Triple

9

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 27

Find the area.

12

159

12 12 9

6(9)

54

A

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Problem

The distance between bases on a baseball diamond is 90 feet. A catcher throws the ball from home base to 2nd base. What is the distance?

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 29

Problem

90

90

2 2 2

2

2

90 90

8100 8100

16200

16200

127.3

c

c

c

c

c

c

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 30

Find the diagonal measure of the LCD screen to the nearest inch.

36.8 in.

20.7 in.

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 31

Find the diagonal measure of the LCD screen to the nearest inch.

36.8 in.

20.7 in.c

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 32

Find the diagonal measure of the LCD screen to the nearest inch.

2 2 2

2

2

36.8 20.7

1354.24 428.49

1782.73

42.22

c

c

c

c

36.8 in.

20.7 in.c

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 33

Find the diagonal measure of the LCD screen to the nearest inch.

2 2 2

2

2

36.8 20.7

1354.24 428.49

1782.73

42.22

c

c

c

c

36.8 in.

20.7 in.42 in.

About 42 inches

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 34

Summary

In a right triangle, the hypotenuse is the longest side.

The sum of the squares of the legs is equal to the square of the hypotenuse.

If the three sides are all integers, they form a Pythagorean Triple.

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 35

True or False? a b

c

a + b = c?http://www.youtube.com/watch?v=DUCZXn9RZ9s

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 36

False. It should have been…

The sum of the squares of the two legs of a

right triangle is equal to the

square of the remaining side. Oh joy! Rapture! I have a brain!

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 37

HomeworkHow to Generate

Pythagorean Triples

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 38

Generating Pythagorean Triples

Find two positive integers a & b which are relatively prime and a > b. That is, they have no factors in common other than 1.

Then the triples are: a2 + b2 , 2ab and a2 – b2.

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 39

Generating Pythagorean Triples

Example: Choose a = 4 and b = 3. a2 + b2 = 42 + 32 = 25. 2ab = 2(4)(3) = 24. a2 – b2 = 42 – 32 = 7. 7, 24, 25 is a Pythagorean Triple.

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 40

Generating Pythagorean Triples

7, 24, 25 is a Pythagorean Triple. Check: 72 + 242 = 252 ? 49 + 576 = 625 ? 625 = 625 That’s a triple!

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 41

Pythagorean Triples

a and b are relatively prime. a > b a2 + b2

2ab a2 – b2

April 21, 2023 Geometry 9.2 The Pythagorean Theorem 42

Try it.

Using a = 8 and b = 3, find the Pythagorean Triple.

Answer: 82 – 32 = 64 – 9 = 55 2(8)(3) = 48 82 + 32 = 73 552 + 482 = 732? 3025 + 2304 = 5329 ? 5329 = 5329 checks.

Area