Question

derive the Pythagorean theorem (a²+b²=c²) which is one of the most important theorems in mathematics

Answer 1

Pythagorean Theorem states that: The Square of the hypotenuse (the side opposite to the right angle of a triangle) is equal to the sum of the squares of the other two sides. The Pythagorean Theorem is written as shown below:

a²+b²=C²

In the Pythagorean equation stated above, c is the length of the hypotenuse while the length of the other two sides (height and base) of the triangle is represented by b and a.

                            

 This theorem is named after the Greek mathematician Pythagoras (ca. 570 BC–ca. 495 BC), who by tradition is credited with its proof

The theorem can be proofed by algebra as shown below:

We consider four triangles inside a square. The height of the triangle is b, the base is a and the hypotenuse is C. the length of the triangle is (a+b) as shown in the diagram: 

        The area of the whole square can be given by:

A = (a+b) (a+b)

The area of the pieces can be given by:

i. The area of the inside (tilted) square: C²

ii. The area of the triangle:  0.5ab

The area of all the triangles is, therefore:  0.5ab X 4 = 2ab

The sum of the areas of the triangles and the tilted square is given by:

        A = 2ab + C²

The area of the large square is equal to the area of the tilted square and the 4 triangles. This can be represented as:

(a+b)(a+b) = 2ab +C²

We expand the expression on the left-hand side to get: a²+2ab+b²

We subtract 2ab from the two sides to get a²+b²=C² which is the Pythagorean

The derivation of the Pythagorean theorem is clearly shown in the explanation below