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Pythagoras' Theorem

Pythagoras' Theorem

  

Pythagoras' theorem states that the square of the hypotenuse will equal the sum of the squares of the other two sides. (It doesn't matter which way around a and b are.)

a2 + b2 = c2

Two easy to remember right triangles which fulfill this condition with whole numbers are the 3, 4, 5 triangle and the 5, 12, 13 triangle.

32 + 42 = 52
9 + 16 = 25

52 + 122 = 132
25 + 144 = 169

These three numbers are together called Pythagorean triples.

Proving Pythagoras' theorem

One way of proving Pythagoras' theorem is by algebra.

We can arrange four triangle so that they bound a square area (light blue). We know it's a square because the angles in a triangle add to 180° and so do angles in a straight line.

The area of a triangle is given by 1/2 base x height, and the area of a square is given by (length)2. So the area of the whole arrangement in the diagram will be given by the area of 4 triangles plus the large blue square.

4 x (1/2 a x b ) + c2

But the arrangement is a square - there is a right angle at each corner and each side has a length of a + b. So the total area can also be given by:

( a + b )2

Of course, these areas are the same, so we equate the two then simplify.

( a + b )2 = 4( 1/2 ab ) + c2

( a + b )( a + b ) = 4/2 ab + c2

a2 + ab + ba + b2 = 2ab + c2

a2 + 2ab + b2 = 2ab + c2

a2 + b2 = c2

  

It's also possible to prove by rearrangement.

What happens if you double the size of one of the examples given, say 3, 4, 5 becomes 6, 8, 10? Will the triangle still be a right angle triangle? Do the numbers still work out for Pythagoras' theorem?

For information see the Pythagoreas Triples page.