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Pythagorean Theorem

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, ...

geometry/images/pyth1.js

... then the biggest square has the exact same area as the other two squares put together! (press Go).

It is the "Pythagorean 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

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.

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!

When we know two side lengths of a right triangle we can find the third side length.

Write it down as an equation:

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

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

Read about Squares and Square Roots to discover why √169 = 13

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

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.

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!

Does 8² + 15² = 16² ?

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

So, NO, it does not 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

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

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.

We also have a proof by adding up the areas.

Historical Note: while we call it Pythagorean Theorem, it was also known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived.

511,512,617,618, 1145, 1146, 1147, 2359, 2360, 2361

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