Pythagoras' Theorem


Pythagoras

 

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!

 

Pythagoras

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.

pythagoras theorem

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:

abc triangle   a2 + b2 = c2


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

Example: Solve this triangle.

right angled 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.

right angled 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?

Unit Square Diagonal

 

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?

10 24 26 triangle  

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?

Triangle with roots  

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 sqaure
  • 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.

before  becomes  after

 

We also have a proof by adding up the areas.

history Historical Note: while we call it Pythagoras' Theorem, it was also known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived !