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.**

Also Read:

**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.

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