There are literally dozens of proofs for the Pythagorean Theorem. The
proof shown here is probably the clearest and easiest to understand.
The Pythagorean Theorem states that for any right triangle the
square of the hypotenuse equals the sum of the squares of the other
2 sides.
If we draw a right triangle having sides 'a' 'b' and 'c' (with 'c' being the
hypotenuse)
In order to prove the theorem, we construct squares on each of the
sides of the triangle.
Now, for the important concepts of this proof:
1) The Pythagoren Theorem concerns
squaring each side of a right triangle
Now let's draw a line extending the 'a' side and a line extending
the 'b' side. These have been drawn in gray and are labelled with the
accompanying lengths. Then we draw 2 lines perpindicular to these lines
so that the blue square is surrounded by a larger square each side having
a length of (a+b). Therefore, the area of the larger square equals (a+b)²
which equals a² + 2ab + b².
2) When we square the length of one side of the triangle,
that is exactly the same as determining the area of the
attached square because the area of a square is the length of its
side squared.
(For example, the area of the green square, would be a²).
3) If we can show that the area of the green square plus
the area of the red square is equal to the area of the blue
square, then we have proven the Pythagorean Theorem.
So, the area of the blue square = area of the surrounding square minus the area of the 4 triangles.
We have just proven the Pythagorean Theorem.