Start with the lower point and count how much you rise and run to get to the

other point!

Remember slope of the line.

63

run

3

6= =

1

2

rise

Notice segments formed by the rise and run form a

right triangle.This takes us to the….

a

b

c

222 cba

8.1 The Pythagorean Theorem

& Its Converse

(PAGES 491 – 495)

OBJECTIVE: To use the Pythagorean Theorem and its converse to solve problems related to right triangles

STANDARD ADDRESSED: Use the triangle angle sum theorem and/or the Pythagorean Theorem and its converse, to solve simple triangle problems and justify results.

What is the Pythagorean Theorem used for?

• to find the length of a missing side in a right triangle

5

4

Cb

156

10

a 9

We call it a right triangle because it contains a right angle.

The measure of a right angle is 90o

90o

The little square

90o

in the angle tells you it is aright angle.

The two sides which come together in a right angle are called legs.

The lengths of the legs are usually called a and b.

a

b

The side across from the right angle

a

b

is called the

HYPOTENUSE

And the length of the hypotenuse

is usually labeled c.

a

b

c

HypotenuseLega c

Where is the Hypotenuse?

Always opposite of the 90 degree angle and it is the longest side of a

right triangle.

b

Leg

OPPO

SITE

Click here to view interactive example from Mathopenref.com

About 2,500 years ago, a Greek mathematician named Pythagoras discovered a special relationship between the sides of right triangles.

Pythagoras realized that if you have a right triangle,

3

4

5

and you square the lengths of the two sides that make up the right angle,

24233

4

5

and add them together,

3

4

5

2423 22 43

22 43

you get the same number you would get by squaring the other side.

222 543 3

4

5

Is that correct?

222 543 ?

25169 ?

Click here to explore an interactive example

It is. And it is true for any right triangle.

8

6

10222 1086

1006436

The relationship Pythagoras discovered is now called The Pythagorean Theorem:

a

b

c

The Pythagorean Theorem says, given the right triangle with legs a and b and hypotenuse c,

a

b

c

then .222 cba

a

b

c

Pythagorean Theorem Formula

Page 491a2 + b2= c2

a and b represent the

legs

c represents the hypotenuse a

b

c

You can use The Pythagorean Theorem to solve many kinds of problems.

Suppose a boat travels directly west for 48 miles,

48

Then turn south and continue for 36 miles.

48

36

How far are is the boat from where it started?

48

36?

482

Using The Pythagorean Theorem,

48

36c

362+ = c2

Why? Can you see that we have a right triangle?

48

36c

482 362+ = c2

Which side is the hypotenuse? Which sides are the legs?

48

36c

482 362+ = c2

22 3648

Then all we need to do is calculate:

12962304

3600 2cTo get c by itself and solve. Remember to apply the square root to both sides of the equation (square root removes the square)

c

c

60

3600 2

And you end up 60 miles from where you started.

48

3660

So, since c2 is 3600, c is 60.So, since c2 is 3600, c is

Find the length of a diagonal of the rectangle:

15"

8"?

Find the length of a diagonal of the rectangle:

15"

8"?

b = 8

a = 15

c

222 cba 222 815 c 264225 c 2892 c 17c

b = 8

a = 15

c

SO

c

c

17

289 2

Find the length of a diagonal of the rectangle:

15"

8"17

Practice using The Pythagorean Theorem

to solve these right triangles:

5

12

c = 13

10

b

26

10

b

26

= 24

(a)

(c)

222 cba 222 2610 b

676100 2 b1006762 b

5762 b 24b5762 b

12

b

15

= 9

Pythagorean Theorem

Pythagorean Triple: A set of nonzero whole numbers, a, b, and, c

that makes a2+b2=c2 a true statement.

Some common Pythagorean Triples are:

3, 4, 5

5, 12, 13

8, 15, 17

7,24,25

Pythagorean Triples

3 4 5

6 8 10

9 12 15

12 16 20

Pythagorean Triples

3 4 5

6 8 10

9 12 15

12 16 20

5 12 13

10 24 26

15 36 39

7 24 25

14 48 50

21 72 75

915

12

Applying the Pythagorean Theorem

Find the value of the missing length. Do the lengths of sides of

ABC form a Pythagorean Triple?

208

x20

21

x

Converse of the Pythagorean Theorem

Theorem 8-2: Converse of the Pythagorean Theorem – If the square of the lengths of one side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.

If a2+b2=c2, then the triangle is a right triangle.

If , complete each statement.

Applying the Converse of the Pythagorean Theorem

Are the following triangles right triangles? EXPLAIN

8584

1321

20

28

Classifying Triangles

Theorem 8-3: If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is obtuse.

If c2 > a2+b2, then the triangle is obtuse

A

B

C

a

b

c

Page # 494

Classifying Triangles

Theorem 8-4: If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is acute.

If c2 < a2+b2, then the triangle is acute

A

B

C

a

b

c

Page # 494

Classifying Triangles

If c2 = a2+b2, then the triangle is a right triangle

Given a triangle with sides 6, 11, and 14 how can you classify the triangle?

Given a triangle with sides 7, 8, and 9 how can you classify the triangle?

If c2 > a2+b2, then the triangle is obtuseIf c2 < a2+b2, then the triangle is acute

Application

The Parks Department rents paddle boats at docks near each entrance to the park. To the nearest meter how far is it to paddle from one dock to the other?

cmeters

c

c

c

c

430

185000

185000

62500122500

250350

2

2

222

430 m

8.1 The Pythagorean Theorem and its Converse

Section 8.1; Pg. 495

Complete lesson check #1 – 6 before leaving class. Let Mr. Miller check.

Complete # 7 – 32

DUE MONDAY Feb 13