Bhaskara's First Proof

1)      Bhaskara’s First Proof of the Pythagorean Theorem

References: 

Loomis, Elisha S. (1968). The Pythagorean proposition. Washington, DC: National Council of Teachers of Mathematics.

Bogomolny, Alexander. (1996). The pythagorean theorem [Cut the knot]. Retrieved from http://www.cut-the-knot.org/pythagoras/

To complete Bhaskara’s Proof, you must draw the following diagram. The outside shape is a square with side of length c. Inside the large square are 4 right triangle with base length b and altitude length a. In the center is a small square with length a – b.

To prove the Pythagorean Theorem using Bhaskara’s method, we must find the area of the large square in two ways.

First, area of large square equals c².

Second, we find the area of the four congruent triangles with legs with length a and b, and hypotenuse with length c, and the area of the small square with length a – b.

The area of the 4 congruent triangles:

4 (½ ab) = 2ab

The area of the small interior square:

(a – b)²  = a² – 2ab + b²

When we sum the area of the four triangles and the small square, we find the total area of the large square equal to:

a² – 2ab + b² + 2ab = a² + b²

Since the area of the large square is equal to the sum of the areas of its parts:

c² = a² + b²