Pythagoras' Theorem
Years ago, a man named Pythagoras found an amazing fact about triangles:
... the biggest square had the exact same area as the other two squares put together! |
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It is called "Pythagoras' Theorem" and can be written in one short equation: a2 + b2 = c2
Note:
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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.
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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:
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a2 + b2 = c2 |
Now you can use algebra to find any missing value, as in the following examples:
Example: Solve this triangle.
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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.
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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?
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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?
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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 ?
- 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?
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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:
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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. | |||||
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We also have a proof by adding up the areas.
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Historical Note: while we call it Pythagoras' Theorem, it was also known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived ! |