pythagorean theory belinda oram hampton middle school graphics from ms clipart and
TRANSCRIPT
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