# Pythagoras Theorem: x2+y2 = z2. But Why ?

Pythagorean Theorem and its proof is considered as one of the first proofs of Mathematics.

The Theorem is stated as follows:

*In any right angled triangle, the sum of squares of two sides having the right angle is equal to the square of its hypotenuse*.

Please refer to the **fig a** which shows a right angled triangle with sides x and y having right angle and z as hypotenuse

So from **fig.a** and based on the above theorem

** x ^{2}+y^{2} = z^{2}.**

Ever wondered how the above statement is true. Let's prove this geometrically.

- Lets consider an arbitrary line of arbitrary length and consider an arbitrary point dividing the length in to x and y. Please refer to
**fig. b**. So the length of the line will become**x+y.** - Now, lets draw a square having length
**x+y**and please refer to**fig.c** - Now, the area of the square in
**fig.c**is**(x+y)**^{2}

**(x+y) ^{2} = x^{2}+y^{2}+2xy** ----------- (i)

The geometrical proof for the above equation can be found in one of my hubs named Why (a+b)^{2} = a^{2}+b^{2}+2ab ?

- Now, lets draw a square inside the square having length
**x+y**taking the arbitrary points on the square as the corners and let the length of the square be**z.**Please refer to**fig.d** - Considering
**fig. d**

Area of Outer Square of length (x+y) equals sum of areas of inner square of length z and area of 4 triangles.

Therefore

the area of inner square of

length z = **z ^{2}** --------------------------(ii)

the area of triangle with

sides x ,y and z = **(x*y)/2** -------------------(iii)

implies **(x+y) ^{2} = Area of inner square of length z + 4*(Area of triangles)**

implies **(x+y) ^{2 }= (ii) + (iii)**

implies **(x+y) ^{2} = z^{2 }+ 4 * (x*y) / 2** --------------(iv)

Substituting (i) in (iv) (i.e above equation)

** x ^{2}+y^{2}+2 xy = z^{2 }+ 2 * (x*y) ---> **

**x**

^{2}+y^{2}+2 xy = z^{2}+ 2 xyCancelling **2xy** from Left Hand Side and Right Hand Side from the above equation

Therefore we get

**x ^{2} + y^{2 } = z^{2}** which proves the

**Pythagoras Theorem**