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Pythagoras' Theorem

Pythagoras

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

When the 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

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.

Example: A "3,4,5" triangle has a right angle in it. pythagoras theorem 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:

abc triangle	 	a2 + b2 = c2

Now you can use algebra to find any missing value, as in the following examples:

Example: Solve this triangle. right angled triangle 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. right angled triangle 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? Unit Square Diagonal 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? 10 24 26 triangle 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! 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? 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:

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 sqaure 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.

before	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 !

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