Pythagoras' Theorem
Pythagoras' theorem | proving
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.
|
