Question

If a/b=c/d,show that (a+b):(c+d)=(((a^2+b^2))^1/2):(((c^2+d^2))^1/2)

Answer 1

Hello,

We suppose that b≠0 and d≠0.


`1) (a)/(b) = (c)/(d)\ ==\textgreater ac=bd\\\\ 2)(a+b)/(c+d) = (a)/(c)\\\\ Indeed \\\\ (a)/(b) = (c)/(d)\\ ==\textgreater\ ad=bc\\ ==\textgreater\ ad+ac=bc+ac\\ ==\textgreater\ a(c+d)=c(a+b)\\ ==\textgreater\ (a+b)/(c+d)= (a)/(c)`



`( √(a^2+b^2))/( √(c^2+d^2)) = (a+b)/(c+d) = (a)/(c)\\ ==\textgreater\ (a^2+b^2)/(c^2+d^2) = (a^2)/(c^2)\\ ==\textgreater\ a^2c^2+b^2c^2=a^2c^2+a^2d^2\\ ==\textgreater\ (bc)^2-(ad)^2=0\\ ==\textgreater\ (bc+ad) (bc-ad)=0\\ ==\textgreater\ bc=-ad\ (exclude)\ or\ bc=ad`