# Pythagorean Theorem proof explained

Have you learnt the Pythagorean Theorem proof? Which is considered as an important theorem in mathematics. This theorem works only for right angled triangle. So you need to remember that for other triangles we can’t apply Pythagorean Theorem.

Now we will see what is Pythagorean Theorem statement. This Theorem statement is given as “hypotenuse length (side) square is equals to sum of squares of lengths of remaining 2 sides for any right angular triangle”.

Let us consider a right angled triangle DEF right angled at E. In this triangle the side opposite to right angle is considered as hypotenuse.

Therefore, according to Pythagorean Theorem statement we can write

${ DF }^{ 2 }$ = ${ DE }^{ 2 }$ + ${ EF }^{ 2 }$

## Steps to follow for Pythagorean Theorem proof:

Consider four right angled triangles as shown in below diagram.

Join all the four right angled triangles in such a way that it looks like below given diagram.

So now if you observe the above diagram it is looking a square with side a + b.

By observing the above diagram, we can say that

Area of a square with side “a + b” is equal to sum of area of four right angled triangles and area of a square with side “c”.

${ \left( a+b \right) }^{ 2 }$ = 4 x$\frac { 1 }{ 2 }$ x a x b + ${ c }^{ 2 }$

${ a }^{ 2 }$ + ${ b }^{ 2 }$ + 2 x a x b = 2 x a x b + ${ c }^{ 2 }$

${ a }^{ 2 }$ + ${ b }^{ 2 }$ = ${ c }^{ 2 }$

Therefore finally we can say that hypotenuse square is equals to sum of squares of other two sides for any right angled triangle.

Hence Pythagorean Theorem proved.

Pythagorean triplets

Example:

Find the length of PQ from the diagram.

Solution:

We know that according to Pythagorean Theorem

${ Hypotenuse }^{ 2 }$ = ${ side }^{ 2 }$ + ${ side }^{ 2 }$

${ 29 }^{ 2 }$ = ${ 21 }^{ 2 }$ + ${ PQ }^{ 2 }$

841 = 441 + ${ PQ }^{ 2 }$

841-441 = ${ PQ }^{ 2 }$

400 = ${ PQ }^{ 2 }$

${ 20 }^{ 2 }$ = ${ PQ }^{ 2 }$

PQ = 20 cm.