Pythagorean 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, ...

geometry/images/pyth1.js

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


Pythagoras

It is the "Pythagorean Theorem" and can be written in one short equation:

a2 + b2 = c2

pythagoras 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?

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

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 = c2Put in what we know:52 + 122 = c2Calculate squares:25 + 144 = c225+144=169:169 = c2Swap sides:c2 = 169Square root of both sides:c = √169 Calculate:c = 13

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

Example: Solve this triangle.

right angled triangle 9 b 15

Start with:a2 + b2 = c2Put in what we know:92 + b2 = 152Calculate squares:81 + b2 = 225 Take 81 from both sides: 81 − 81 + b2 = 225 − 81Calculate: b2 = 144Square 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 = c2Put in what we know:12 + 12 = c2Calculate squares:1 + 1 = c21+1=2: 2 = c2Swap sides: c2 = 2Square root of both sides:c = √2Which is about:c = 1.4142...

Multiples of 3,4,5
There are practical uses!
Read Builder's Mathematics.

Is it a Right Angle?

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

Which shows us that a2 + b2 is equal to c2, 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
162
= 256

they are not equal

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  before

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.
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