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!
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.
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:
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 Builder's Mathematics to see practical uses for this.
Also read about Squares and Square Roots to find out 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
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.
becomes |
We also have a proof by adding up the areas.
Historical Note: while we call it Pythagoras' 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
Activity: Pythagoras' Theorem
Activity: A Walk in the Desert
Copyright © 2022 Rod Pierce
The Pythagorean theorem states that in any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the legs of the right triangle. This same relationship is often used in the construction industry and is referred to as the 3-4-5 Rule.
The right triangle below has one leg with a length of three, another leg with a length of four and a hypotenuse with a length of five.
Given the lengths of any two sides of a right triangle, the length of the third side can be calculated using the Pythagorean theorem. In the example above, there are three possible unknowns. Each case is outlined below.
There are many ways to prove the Pythagorean theorem. One such proof is given here.
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