The Pythagorean Theorem

GeometryMrs. Pam MillerJanuary 2010

The Pythagorean Theorem

• Greek Mathematician, Pythagoras, proved this theorem.

• Applies to right triangles.• Many different proofs exist,

including one by President Garfield.

For any right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

The Theorem

hypotenuseleg

legleg2 + leg2 = hyp2

The Abbreviated Version

ca

b

We often see the Pythagorean Theorem stated as:

a2 + b2 = c2

Some Examples

Find the value of x.

( Remember that the length of a segment must be a positive number.)

a)

3

7

x

leg2 + leg2 = hyp2

32 + 72 = x2

9 + 49 = x2

58 = x2

√58 = √x2

√58 = x

Examples (cont’d.)

b)8

10x

leg2 + leg2 = hyp2

82 + x2 = 102

64 + x2 = 100

64 - 64 + x2 = 100 - 64X2 = 36

√x2 = √36

X = 6

Practice with Radicals

Work with the Pythagorean Theorem often requires us to work with radicals.

Simplify each expression:

A) (√3) 2

√3 × √3

√9

3

B) ( 3 √ 11 ) 2

3 √ 11 × 3 √ 11

9 × √ 121

9 × 11

99

Your Turn

Simplify each expression:

A. (√ 5) 2

B. (2 √ 7) 2

C. (7 √ 2 ) 2

D. (2n) 2

E. 2

5

3

F.

G.2

632

2

22

The Answers

Simplify each expression:

A. (√ 5) 2

B. (2 √ 7) 2

C. (7 √ 2 ) 2

D. (2n) 2

E. 2

5

3

F.

G.2

632

2

22

= 5

= 28

= 98

= 4n2

= 9/5

= 1/2

= 24/9

Pythagorean Triples

3, 4, 5

6, 8, 10

9, 12, 15

12, 16, 20

15, 20, 25

5, 12, 13

10, 24, 26

8, 15, 17 7, 24, 25

In every triple, the largest # is the length of the hypotenuse and the 2 smaller numbers are the lengths of the legs of the right triangle.

The Converse

If the square of 1 side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle.

c

b

a

More simply…

if c2 = a2 + b2,

then the triangle is a

right triangle.

Pythagorean Inequalities

c

b

a

If c2 > a2 + b2,

then the triangle is obtuse.

If c2 < a2 + b2,

then the triangle is acute.

Practice

The sides for 3 triangles are given.

Decide if each triangle is acute, right or obtuse.

A) 14, 7, 9

142 ___ 72 + 92

196 ___ 49 + 81

196 > 130

It’s obtuse!

B) 2.5, 6, 6.56.52 ___ 2.52 +

62

42.25 ___ 6.25 + 3642.25 = 42.25

It’s right!

C) 2, 3, 3.5

3.52 ___ 22 + 32

12.25 ___ 4 + 9

12.25 < 13

It’s acute!

Check It Out

A triangle has sides with the following lengths: 9, 40, & 41. Is this a right triangle?

Does 412 = 92 + 402 ?

Does 1681 = 81 + 1600?

YES !! So, the triangle is a right triangle.

More Practice

A right triangle has one leg with a length of 48 and a hypotenuse with a length of 80.

What is the length of the other leg? 64

Here’s how:

482 + x2 = 802 x2 = 4096

2304 + x2 = 6400 √ x2 = √ 4096

x2 = 6400 - 2304 x = 64

Practice (cont’d)

A triangle has side lengths of 7, 10, & 12.

Is the triangle a right triangle?

Use the Converse of the Pythagorean Theorem!

Here’s How: Does c2 = a2 + b2 ?

Does 122 = 72 + 102 ?Does 144 = 49 + 100?

NO !

Practice (cont’d)

A triangle has side lengths of 8, 15, and 18.

Is the triangle right, acute, or obtuse?

Here’s How: 182 ____ 82 + 152

Remember:

If c2 < a2 + b2, you have an acute triangle.

If c2 = a2 + b2, you have a right triangle.

If c2 > a2 + b2, you have an obtuse triangle.

324 ____ 64 + 225

324 > 289

It’s obtuse!

Other Applications

Find the area of the figure.

Leave your answer in radical form. 8

8

8

Area of a triangle = 1/2 bh

8

4

hUse the Pythagorean Theorem to find “h”:42 + h2 = 82

16 + h2 = 64

h2 = 64 - 16

h2 = 48

√ h2 = √ 48

h = 4 √ 3 More….

Find the Area of the Triangle

Area = 1/2 bh

8

4

4 √ 3 Area = 1/2 (8) (4 √ 3)

Area = 16 √ 3

8

8

8

Last Problem!

Find the area of the square. Leave your answer in radical form.

6Use the Pythagorean Theorem to find the length of the sides (s).

s2 + s2 = 62

2s2 = 36

s2 = 18

√ s2 = √ 18

s = 3 √ 2

So the area of the square is :

3 √ 2 × 3 √ 2= 9 √ 4 = 9 × 2= 18

s

s

What Have You Learned?

• Pythagorean Theorem • Converse of the Pythagorean

Theorem• Pythagorean Triples• Pythagorean Inequalities• Applications