Pythagoras Theorem (Baudhayan Theorem)

Pythagoras theorem says that in a right angle Triangle, the square of the hypotenuse i.e. the side opposite to the right angle is equal to the sum of the square of the other two sides of the Triangle.

If one angle is90^\circ , then {a^2} + {b^2} = {c^2}

 

Proof of Pythagoras Theorem

Statement: As per Pythagoras theorem“In a right-angled triangle, the sum of squares of two sides of a right triangle is equal to the square of the hypotenuse of the triangle.”

Proof –

Consider the right triangle, right-angled at B.

Construction-

Draw BD  \bot AC

Now,

So, AD/AB = AB/AC

or AD. AC = A{B^2} ……………(i)

Also,

So, CD/BC = BC/AC

or, CD. AC = B{C^2} ……………(ii)

Adding (i) and (ii),

  1. AC + CD. AC =  A{B^2} + B{C^2}

AC(AD + DC) = A{B^2} + B{C^2}

AC(AC) = A{B^2} + B{C^2}

A{C^2} = A{B^2} + B{C^2}

Hence, proved.

Example

In the given right angle Triangle, Find the hypotenuse.

Solution

AB and BC are the two sides of the right angle Triangle.

BC = 12 cm and AB = 5 cm

From Pythagoras Theorem, we have:

 C{A^2} = A{B^2} + B{C^2}

 = {(5)^2} + {(12)^2}

= 25+144

So, A{C^2} = 169

AC = 13 cm

Converse of Pythagoras Theorem

In a Triangle, if the sum of the square of the two sides is equal to the square of the third side then the given Triangle is a right angle Triangle.

If {a^2} + {b^2} = {c^2} then one angle is90^\circ .

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