a² + b² = c²

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.

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:

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

a² + b² = c²

5² + 12² = c²

25 + 144 = c²

169 = c²

c² = 169

c = √169

c = 13

++++++++++++++++++++++++++

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 this triangle have a Right Angle?

Does a² + b² = c² ?

  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