Pythagorean Theory

Belinda OramHampton Middle School

Graphics From MS Clipart and www.PBS.org

Pythagoras

• Pythagoras lived in the 500's BC, and was one of the first Greek mathematical thinkers.

The Egyptians knew that a triangle with sides 3, 4, and 5 make a 90o angle.

                                                                      

       

They used this knowledge to build perfect corners in their buildings and pyramids. It is believed that they only knew about the 3, 4, 5 triangle and not the general theorem that applies to all right triangles.

So why is it called the Pythagorean Theorem?

• Even though the theorem was known long before his time, Pythagoras generalized it and made it popular.

• It was Pythagoras who is attributed with its first geometrical demonstration.

The Theorem• The square described upon the

hypotenuse of a right-angled triangle is equal to the sum of the squares described upon the other two sides

WHAT?????

The Theorem (In simple terms)

• When the two shorter sides in a right triangle are squared and then added, the sum equals the square of the longest side or hypotenuse.

Theorem Proof

32 = 9

42 = 16

a2 + b2 = c2

32 + 42 = 52

9 + 16 = 25

52 = 25

http://www.pbs.org/wgbh/nova/proof/puzzle/images/example.gif

Used with permission from PBS.org

Finding the Unknown Side

First, sketch the diagram into your notebook.

6 cm

8 cm

c

6 cm

8 cm

c

Then copy out the formula.

a2 + b2 = c2

Identify the hypotenuse:“c” in the formula.

Always the side opposite the right angle

Make the substitutions into the formula.

62 + 82 = c2

Note: it doesn’t

matter which of the two sides is

a or b.

Do the calculations

62 + 82 = c2

36 + 64 = c2

100 = c2

c2100

10 = c

To find the value of c:

You have to do the opposite function of squaring a number.

The opposite function is √ (square root).

BEDMAS tells us to do the

exponents first.

Then add.

6 cm

8 cm

c

a2 + 81 = 225

a

9 cm

15 cm

Solving if one of the sides is missing.

1442 a

Always start by writing the formula.

a2 + b2 = c2

Make the substitutions.

a2 + 92 = 152

Isolate the variableGet rid of the +81!

-81

-81

Cancel

Rewrite the equation

a2 = 144

To isolate the variable, do the opposite function. Take the square root of each side.

12a

Calculate and rewrite

C is always the hypotenuse

The side opposite the right angle

Using the Pythagorean Theorem, solve for the missing side.

11 cm

26 cm

x

Senteo QuestionTo set the properties right click and select Senteo Question Object->Properties...

Solve for ϰ

x23 cm

28 cm

Senteo QuestionTo set the properties right click and select Senteo Question Object->Properties...

Check your understanding

Practical Applications

• The Pythagorean Theorem is used in everyday life all the time.

• Carpenters use it to check to see if a structure is square.

• If you know two sides of any right angle triangle, you can calculate the length of the unknown side.

Baseball

• A baseball diamond consists of four bases each 90 ft apart situated at 90º angles from each other.

• If the catcher tries to throw out a runner at second base, how far does he throw the ball?

90ft

90ft90ft

90ft

Senteo QuestionTo set the properties right click and select Senteo Question Object->Properties...

Around the house

• You have to fix a window that is 12 ft high from the ground. You have a bush at the bottom of the window, so you will need to place the ladder back 3 ft from the base of the house. How long will the ladder need to be?

Senteo QuestionTo set the properties right click and select Senteo Question Object->Properties...

Note: Ladders can only be bought in full feet, not decimals.

3, 4, 5 Rule • Carpenters use the “3, 4, 5

Rule” to make sure that a deck is square.

• They measure a spot on one side and mark the 3 ft line.

• They measure a spot on the other side and mark the 4 ft line.

• If the measurement between the two marks equals 5 ft, they know the deck is “square”.

• Prove this by using the Pythagorean Theorem.

x

x