# CBSE Class 10 Answered

**Pythagoras Theorem**

Dear Student,

However its unclear in the question, as to which 'given proof ' is referred to, still, We are giving the following two proofs of Pythagoras Theorem:

The Pythagoras Theorem says, that for a right triangle, with hypotenuse 'c', and legs 'a' and 'b',

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

PROOF#1. In order to prove the theorem, we construct squares on each of the sides of the triangle:

If we can show that the *area* of the green square plus the *area* of the red square is equal to the *area* of the blue square, then we have proven the Pythagorean Theorem.

Now let's draw a line extending the 'a' side and a line extending the 'b' side. Then we draw 2 lines perpindicular to these lines so that the blue square is surrounded by a larger square each side having a length of (a+b). Therefore, the area of the larger square equals (a+b)² which equals a² + 2ab + b².

Note that the blue square is surrounded by 4 right triangles, the area of each being ½ (a•b) and so the area of all 4 triangles totals 2•a•b.

So, the area of the blue square = area of the surrounding square minus the area of the 4 triangles.

**c² = a² + b².**PROOF#2:

The simplest proof is an algebraic proof using similar triangles ABC, CBX, and ACX (in the diagram):

Since corresponding parts of similar triangles are proportional,

a/x=c/a

=> a²=cx.

And b/(c-x)=c/b

=> b²=c²-cx

=> c²=cx+b².

Substituting a² for cx, we get,

__c²=a²+b²__

Hence Proved.

Regards Topperlearning.