Question

Verify whether the following measures represent the sides of a right angled triangle.
$m^2-n^2,2mn,m^2+n^2$

Answer 1

Let $△ABC$ be a triangle where $AB=(m^2-n^2)$, $BC=2mn$ and $AC=(m^2+n^2)$

Now consider $A{B}^2+B{C}^2$

$A{B}^2+B{C}^2=(m^2-n^2{)}^2+(2mn{)}^2=(m^2{)}^2+(n^2{)}^2-(2\times m^2\times n^2)+4m^2{n}^2=m^4+n^4-2m^2{n}^2+4m^2{n}^2$

$=m^4+n^4+2m^2{n}^2=(m^2{)}^2+(n^2{)}^2+(2\times m^2\times n^2)=(m^2+n^2{)}^2=A{C}^2(\because (a-b{)}^2=a^2+b^2-2ab,(a+b{)}^2=a^2+b^2+2ab)$

Since $A{B}^2+B{C}^2=A{C}^2$, therefore, $△ABC$ is a right angled triangle.

Hence, $(m^2-n^2)$, $2mn$ and $(m^2+n^2)$ represent the sides of a right angled triangle.