Question

Why pythagore find theta

Answer 1

Let us take again our last diagram to indicate each sides by the following letters:



One has four right-angled triangles of which :
the side opposed by a,
the base is indicated by b,
and the hypotenuse by n.

Each hypotenuse n thus represent sides small square.

With sides opposite more the base (a + b) of each right-angled triangle represent each one of with sides great square.

 

Surface squares

Small square : n2.

Great square : (a + b)2.

 

Surface right-angled triangles

If one associates two right-angled triangles, one finds oneself with a rectangle of surface : a * b.

The total surface of the four right-angled triangles is : 2ab.

 

Difference in surface

The surface of the small square is equal to the surface of the great square minus the surface of the four right-angled triangles :

n2 = (a + b)2 - 2ab

from where

n2 = a2 + 2ab + b2 - 2ab

what gives

n2 = a2 + b2

Conclusion, the square of the hypotenuse is equal to the sum of the squares of the sides of the right angle.