a

b

c

222 cba

Student Expectations

8th Grade:

• 8.3.7C Use pictures or models to demonstrate the Pythagorean theorem.

• 8.4.9A Use the Pythagorean theorem to solve real-life problems.

This is a right triangle:

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 theangle tells you it is aright angle.

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

Pythagorus 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 ?

Pythagorean Theorem

You can demonstrate the Pythagorean Theorem

geometrically…

1. Square both legs

3 ft

4 ft

4ft

3 ft

1 3

4 5 6

7 8 9

1

2

2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

2. Count the total squares

3 ft

4 ft

4ft

3 ft

1 3

4 5 6

7 8 9

1

2

2 3 4

5 6 7 8

9 10 11 12

13 14 15 16

9 + 16 = 25

3 ft

4 ft

4ft

3 ft

1 3

4 5 6

7 8 9

1

2

2 3 45 6 7 8

9 10 11 1213 14 15 16

9 + 16 = 25

3. Put that number of squares on the hypotenuse

2

4

6

8

10

12

14

16

18

20

22

24

1

3

5

7

9

11

13

15

17

19

21

23

25

3 ft

4 ft

4ft

3 ft

1 3

4 5 6

7 8 9

1

2

2 3 45 6 7 8

9 10 11 1213 14 15 16

9 + 16 = 25

4. Count the number of squares that touch the hypotenuse.

2

4

6

8

10

12

14

16

18

20

22

24

1

3

5

7

9

11

13

15

17

19

21

23

25

# = 5

3 ft

4 ft

4ft

3 ft

1 3

4 5 6

7 8 9

1

2

2 3 45 6 7 8

9 10 11 1213 14 15 16

9 + 16 = 25

5.That number is the length of the hypotenuse.

2

4

6

8

10

12

14

16

18

20

22

24

1

3

5

7

9

11

13

15

17

19

21

23

25

# = 5Length = 5

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

8

6

10222 1086

1006436

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

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

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

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

And the length of the hypotenuse

is usually labeled c.

a

b

c

The relationship Pythagorus 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

a

b

c

.222 cba

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

Suppose you drive directly west for 48 miles,

48

Then turn south and drive for 36 miles.

48

36

How far are you from where you 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 2c

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

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 b24b

12

b

15

= 9