A b c. About 2,500 years ago, a Greek mathematician named Pythagoras discovered a special...

a

b

c

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

Pythagoras was not only a mathematician, but a philosopher and religious leader, as well.

He was responsible for many important developments in math,

astronomy, and

music.

Pythagoras (~560-480 B.C.)

His students formed a secret society called the Pythagoreans.

As well as studying math, they were a political and religious organization.

Members could be identified by a five pointed star they wore on their clothes.

The Secret Brotherhood

They had to follow some unusual rules. They were not allowed to wear wool, drink wine or pick up anything they had dropped!

The Secret Brotherhood

The initiation into the secret society asked that for the first 5 years into the brotherhood, they would not speak. Just listen.

A right angled triangle

A Pythagorean Puzzle

Draw a square on each side.


cb

a

Measure the length of each side


Work out the area of each square.


a

b

C²

b²

a²

c


c²

b²

a²

Proof

a

a

a2

bb

cc

b2

c2

Let’s look at it this way…


1


1

2


1

2


1

2

3


1

2

3


1

23

4 A Pythagorean Puzzle

1

23

4


1

23

45


1

23

4

5

What does this tell you about the areas of the three squares?

The red square and the yellow square together cover the green square exactly.

The square on the longest side is equal in area to the sum of the squares on the other two sides.


1

23

4

5

Put the pieces back where they came from.


1

23

45



1

23

4

5



1

2

3

4

5



1

23

4

5



1

23

4

5



This is the Pythagorean Theorem.


c²

b²

a²

c²=a²+b²

It only works with right-angled triangles.

hypotenuse

The longest side, which is always opposite the right-angle, has a special name:

This is the name of Pythagoras’ most famous discovery.

Pythagorean Theorem

c

b

a

c²=a²+b²

Pythagorean Theorem

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

This is true for any right triangle.

8

6

10222 1086

1006436

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

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"?


15"

8"?

b = 8

a = 15

c

222 cba 222 815 c 264225 c 2892 c 17c

b = 8

a = 15

c


15"

8"17

Check It Out! Example 2

A rectangular field has a length of 100 yards and a width of 33 yards. About how far is it from one corner of the field to the opposite corner of the field? Round your answer to the nearest tenth.

Check It Out! Example 2 Continued

11 Understand the Problem

Rewrite the question as a statement.

• Find the distance from one corner of the field to the opposite corner of the field.

• The segment between the two corners is the hypotenuse.

• The sides of the fields are legs, and they are 33 yards long and 100 yards long.

List the important information:

• Drawing a segment from one corner of the field to the opposite corner of the field divides the field into two right triangles.


22 Make a Plan

You can use the Pythagorean Theorem towrite an equation.


Solve33

a2 + b2 = c2

332 + 1002 = c2

1089 + 10,000 = c2

11,089 = c2

105.304 c

The distance from one corner of the field to the opposite corner is about 105.3 yards.

Use the Pythagorean Theorem.

Substitute for the known variables.

Evaluate the powers.

Add.

Take the square roots of both sides.

105.3 c Round.

Ladder Problem

A ladder leans against a second-story window of a house. If the ladder is 25 meters long, and the base of the ladder is 7 meters from the house, how high is the window?

Ladder ProblemSolution

First draw a diagram that shows the sides of the right triangle.

Label the sides: Ladder is 25 m

Distance from house is 7 m

Use a2 + b2 = c2 to solve for the missing side. Distance from house: 7 meters

Ladder ProblemSolution

72 + b2 = 252

49 + b2 = 625

b2 = 576

b = 24 m

How did you do?

A = 7 m

Baseball Problem

A baseball “diamond” is really a square.

You can use the Pythagorean theorem to find distances around a baseball diamond.

Baseball Problem

The distance between

consecutive bases is 90

feet. How far does a

catcher have to throw

the ball from home

plate to second base?

Baseball Problem

To use the Pythagorean theorem to solve for x, find the right angle.

Which side is the hypotenuse?

Which sides are the legs?

Now use: aa22 + b + b22 = c = c22

Baseball ProblemSolution

The hypotenuse is the distance from home to second, or side x in the picture.

The legs are from home to first and from first to second.

Solution:

x2 = 902 + 902 = 16,200

x = 127.28 ft

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A b c. About 2,500 years ago, a Greek mathematician named Pythagoras discovered a special...

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