# Pythagoras Theorem

Read up about the History of the man who was the first to proof this Theorem:

Want to see more portraits of him? Go to: http://www-history.mcs.st-andrews.ac.uk/history/PictDisplay/Pythagoras.html

So, how did he discover the theorem?

Well, legend has it that he got his inspiration from looking at floor tiles~
(yup, just like how Decartes got his inspiration of the Coordinate System by looking at a fly~)

Here is one such tile:

See anything?

Ok, probably not...

Look again!

Got it? Hmm......

Ok, to get the idea, go to the next section...

Concept of Pythagoras Theorem

So, what exactly is this theorem that Pythagoras discovered?

Basically, the theorem states that

"For any Right-Angled Triangle, the square of the length of the longest side (called the hypotenuse) is equal to the sum of squares of the length of the other 2 sides".

Important points to note:
(1) The triangle MUST have an angle that is 90 deg.
(2) a^2 in the above equation always refers to the LONGEST side.

Proofs of the theorem:
To show that the theorem is true, think of it in this way:

In the above diagram, notice that the sides of the white triangle are also the lengths of 3 different squares. We can make use of the lengths to define the area of the 3 squares as a^2, b^2 and c^2.

To show that Pythagoras Theorem is true, all we need to do is to prove that the area of a = area of b + area of c.
Meaning, if we can show that a^2 = b^2 + c^2, we have proven pythagoras theorem.

There are many different ways to make the 2 smaller squares fit into the big square. Go to this website and try the interactive applets for yourself:
http://www.ies.co.jp/math/java/geo/pythagoras.html

To start, click on "Define" to show the red dots. Then move the pieces by clicking onto the red dot.
To reset, click on "Init".

Here's an animated version: http://www.davis-inc.com/pythagor/index.shtml

While there are many many proofs of Pythagoras Theorem, some basic and some advanced, understanding them all is beyond your scope. Still, to see 84 of them, go to: http://www.cut-the-knot.org/pythagoras/index.shtml

Fun Trival: One American President discovered a proof of Pythagoras Theorem. Do you know who he is?

"In 1876, Garfield discovered a novel proof of the Pythagorean Theorem using a trapezoid while serving as a member of the House of Representatives.]"
[Extracted From http://en.wikipedia.org/wiki/James_A._Garfield]

Pythagorean Triplets:

Pythagoras triplets are integers that form the three sides of a right triangle.
Example:
(3, 4, 5) --> 5^2 = 3^2 + 4^2
(5, 12, 13) --> 13^2 = 5^2 + 12^2

There are more of them! Can you think of some yourself?

There is a rule to generate these triplets. Can you find it? You may share your findings on the discussion page.

Examples on how to solve problems using Pythagoras Theorem:

1) You have allocated a space of 48 inches wide by 35 inches high for a TV. Will a 65-inch TV fit into the space? [65-inch refers to the length of the diagonal of the TV)

Solution: Use Pythagoras Theorem to find the length of the diagonal of the space you have.

You have only 60 inches available. Hence, a 65-inch TV will not fit into the space.

2) In the diagram below, find the lengths of AB and CD.

To find AB:

To find CD:

Practice:

Complete Textbook 2B Chapter 7, Ex 7.1 (Do in Exercise Book)

ConcepTest:

Attempt the ConcepTest by Sunday 25/4 6 pm.

Diagram for Q2:

The solution to the above problem is given in the following slides, as shared by one of your classmates in the JiTT exercise:

### Worksheets and Solutions:

Q1 - Q6 are basic questions.Q7 - Q14 are more challenging.

Solution for Q1 - Q6
Solution for Q7 - Q14: