/sci/ - Science & Math

We are given a right triangle with sides a,b known and c to be solved for
Create a square with side length of a+b
Partition the square as shown in the picture
since each triangle has 2 congruent sides and a congruent angle between the two sides they are all congruent to each other and to our given triangle. So they each have a hypotenuse equal to c
since the sum of the angles in a triangle is 180 degrees and a right angle is 90 degrees, a right triangle has acute angles that sum to 90 degrees
the sum of angles at each point of the figure inside of our square must be 180 degrees
two of those angles are the acute angles of our right triangle so the remaining angle is a right angle
So the figure inside of our constructed square is itself a square with side length c and area c^2
each triangle has area a*b/2, in total the triangles has area 2ab
the area of the larger square is (a+b)^2=a^2+2ab+b^2
the larger square is decomposable to the 4 triangles and the smaller square so we have that the area of the larger square is equal to sum of the triangles and the smaller square areas
(a+b)^2=2ab+c^2
a^2+2ab+b^2=2ab+c^2
a^2+b^2=c^2