The Pythagorean

Theorem

A tool for right

triangle problems

only

When do I need to use

the Pythagorean

Theorem ?

When I know: length of 2 sides

and

Need to know: length of 3rd side

Pythagorean Theorem

In a right triangle, the square of the length

of the hypotenuse is equal o the sum of

the squares of the lengths of the legs.

What is the Pythagorean

Theorem?

a2 b2 c2

Where a and b are legs and c is the hypotenuse

leg2 + leg2 = hyp2

Sides are always positive numbers

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

63

2

2

2

2

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

63

2

2

2

2

= 5

= 28

= 98

= 4n2

= 9/5

= 1/2

= 24/9

Pythagorean Triples 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.

Using a2 + b2 = c2

Looking for length of

the hypotenuse a2 + b2 = c2

152 + 202 = c2

225 + 400 = c2

625 = c2

625 c2

25 c

Using a2 + b2 = c2

Looking for length of

A leg

a2 + b2 = c2

62 + b2 = 102

36 + b2 = 100

-36 -36

b2 = 64

b = 8

The Converse of Theorem 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 Theorems If c2 > a2 + b2,

then the triangle is obtuse.

If c2 < a2 + b2,

then the triangle is acute.

Applying the

Pythagorean Theorem

Practice #1. A 15 foot ladder leans up

against a building. The foot of the

ladder is 5 feet from the base of the

building. How high up the wall, to the

nearest foot does the ladder reach?

Draw a picture:

Solving the problem

a2 + b2 = c2 Write formula

x2 + 52 = 152 Substitute in

x2 + 25 = 225 Solve for x

- 25 -25

x2 = 200

x = 14.142135

x 200

The ladder reaches 14 feet up the wall.

Practice Problem #2.

Tim rode 8 miles due north, then 3 miles

due east. How far, to the nearest mile, is

Tim from where he started?

Draw a picture:

Solve the problem

a2 + b2 = c2

82 + 32 = c2

64 + 9 = c2

73 = c2

73 c

C = 8.5440037

Practice Problem #3.

As seen in the

accompanying diagram, a

person can travel from

NYC to Buffalo by going

north 170 miles to Albany

and then west 280 miles to

Buffalo.

A) If a highway is built to

connect NYC and Buffalo,

how many miles would be

saved on the trip?

Albany

New York City

170 miles

280 miles

???

Buffalo

Find length of new highway

Buf Albany

New York City

170 miles

280 miles

???

a2 + b2 = c2

2802+1702=c2

107300 m= c2

107300 c2

327.566= c

Did I answer question?

How many miles would be saved?

Old Distance: 280 + 170 = 450

New Distance: 327.566

Saved Miles: 122.4 or 122 miles