Pythagoras' Theorem

pythagoras
Pythagoras

 

Over 2000 years ago there was an amazing discovery about triangles:

When a 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

pythagoras squares a^2 + b^2 = c^2

Note:

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.

triangle 3 4 5

Let's check if the areas are the same:

32 + 42 = 52

Calculating this becomes:

9 + 16 = 25

It works ... like Magic!

triangle 3 4 5 lego

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


Then we use algebra to find any missing value, as in these examples:

Example: Solve this triangle

right angled triangle 5 12 c

Start with:a2 + b2 = c2
Put in what we know:52 + 122 = c2
Calculate squares:25 + 144 = c2
25+144=169:169 = c2
Swap sides:c2 = 169
Square root of both sides:c = √169
Calculate:c = 13

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

Example: Solve this triangle.

right angled triangle 9 b 15

Start with:a2 + b2 = c2
Put in what we know:92 + b2 = 152
Calculate squares:81 + b2 = 225
Take 81 from both sides: 81 − 81 + b2 = 225 − 81
Calculate: b2 = 144
Square root of both sides:b = √144
Calculate:b = 12

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

Unit Square Diagonal

Start with:a2 + b2 = c2
Put in what we know:12 + 12 = c2
Calculate squares:1 + 1 = c2
1+1=2: 2 = c2
Swap sides: c2 = 2
Square root of both sides:c = √2
Which is about:c = 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 ?

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 ?

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:

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 !

 


Activity: Pythagoras' Theorem
Activity: A Walk in the Desert