Pythagoras Theorem

Over 2000 years ago there was an amazing discovery about triangles: When the triangle has a right angle (90°) ... ... and squares are made on each of the three sides, then ...
... the biggest square has the exact same area as the other two squares put together!

It is called "Pythagoras' Theorem" and can be written in one short equation:

a² + b² = c²

Note:

• c is the longest side of the triangle
• a and b are the other two sides

Definition

The longest side of the triangle is called the "hypotenuse", so the formal definition is:

In a right angled triangle:
the square of the hypotenuse is equal to
the sum of the squares of the other two sides.

Sure ... ?

Let's see if it really works using an example.

Example: A "3,4,5" triangle has a right angle in it.

Let's check if the areas are the same: 32 + 42 = 52 Calculating this becomes: 9 + 16 = 25 It works ... like Magic!

Why Is This Useful?

If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right angled triangles!)

How Do I Use it?

Write it down as an equation:

a2 + b2 = c2

Now you can use algebra to find any missing value, as in the following examples:

Example: Solve this triangle.

a2 + b2 = c2 52 + 122 = c2 25 + 144 = c2 169 = c2 c2 = 169 c = √169 c = 13

You can also read about Squares and Square Roots to find out why √169 = 13

Example: Solve this triangle.

a2 + b2 = c2 92 + b2 = 152 81 + b2 = 225 Take 81 from both sides: b2 = 144 b = √144 b = 12

Example: What is the diagonal distance across a square of size 1?

a2 + b2 = c2 12 + 12 = c2 1 + 1 = c2 2 = c2 c2 = 2 c = √2 = 1,4142...

It works the other way around, too: when the three sides of a triangle make a² + b² = c², then the triangle is right angled.

Example: Does this triangle have a Right Angle?

Does a2 + b2 = c2 ? a2 + b2 = 102 + 242 = 100 + 576 = 676 c2 = 262 = 676 They are equal, so ... Yes, it does have a Right Angle!

Example: Does an 8, 15, 16 triangle have a Right Angle?

Does 8² + 15² = 16² ?

• 8² + 15² = 64 + 225 = 289,
• but 16² = 256

So, NO, it does not have a Right Angle

Example: Does this triangle have a Right Angle?

Does a2 + b2 = c2 ? Does (√3)2 + (√5)2 = (√8)2 ? Does 3 + 5 = 8 ? Yes, it does! So this is a right-angled triangle

And You Can Prove The Theorem Yourself !

Get paper pen and scissors, then using the following animation as a guide:

Draw a right angled triangle on the paper, leaving plenty of space. Draw a square along the hypotenuse (the longest side) Draw the same sized square on the other side of the hypotenuse Draw lines as shown on the animation, like this: Cut out the shapes Arrange them so that you can prove that the big square has the same area as the two squares on the other sides

Another, Amazingly Simple, Proof

Here is one of the oldest proofs that the square on the long side has the same area as the other squares.
Watch the animation, and pay attention when the triangles start sliding around.
You may want to watch the animation a few times to understand what is happening.
The purple triangle is the important one.

becomes

We also have a proof by adding up the areas.

What is the Pythagorean Theorem?

You can learn all about the Pythagorean Theorem, but here is a quick summary:

The Pythagorean Theorem states that, in a right triangle, the square of a (a2) plus the square of b (b2) is equal to the square of c (c2): a2 + b2 = c2

Proof of the Pythagorean Theorem using Algebra

We can show that a² + b² = c² using Algebra
Take a look at this diagram ... it has that "abc" triangle in it (four of them actually):

Area of Whole Square

It is a big square, with each side having a length of a+b, so the total area is:

A = (a+b)(a+b)

Area of The Pieces

Now let's add up the areas of all the smaller pieces:

First, the smaller (tilted) square has an area of		A = c2
And there are four triangles, each one has an area of		A =½ab
So all four of them combined is		A = 4(½ab) = 2ab
So, adding up the tilted square and the 4 triangles gives:		A = c2+2ab

Both Areas Must Be Equal

The area of the large square is equal to the area of the tilted square and the 4 triangles. This can be written as:

(a+b)(a+b) = c²+2ab

NOW, let us rearrange this to see if we can get the pythagoras theorem:

Start with:		(a+b)(a+b)	=	c2 + 2ab
Expand (a+b)(a+b):		a2 + 2ab + b2	=	c2 + 2ab
Subtract "2ab" from both sides:		a2 + b2	=	c2
		DONE!

Now we can see why the Pythagorean Theorem works ... and it is actually a proof of the Pythagorean Theorem.

This proof came from China over 2000 years ago!

There are many more proofs of the Pythagorean theorem, but this one works nicely.

source: http://www.mathsisfun.com/geometry/pythagorean-theorem-proof.html