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.

$1) \dfrac{a}{b} = \dfrac{c}{d}\ ==\textgreater ac=bd\\\\ 2)\dfrac{a+b}{c+d} = \dfrac{a}{c}\\\\ Indeed \\\\ \dfrac{a}{b} = \dfrac{c}{d}\\ ==\textgreater\ ad=bc\\ ==\textgreater\ ad+ac=bc+ac\\ ==\textgreater\ a(c+d)=c(a+b)\\ ==\textgreater\ \dfrac{a+b}{c+d}= \dfrac{a}{c}$


$\dfrac{ \sqrt{a^2+b^2}}{ \sqrt{c^2+d^2}} = \dfrac{a+b}{c+d} = \dfrac{a}{c}\\ ==\textgreater\ \dfrac{a^2+b^2}{c^2+d^2} = \frac{a^2}{c^2}\\ ==\textgreater\ a^2c^2+b^2c^2=a^2c^2+a^2d^2\\ ==\textgreater\ (bc)^2-(ad)^2=0\\ ==\textgreater\ (bc+ad) (bc-ad)=0\\ ==\textgreater\ bc=-ad\ (exclude)\ or\ bc=ad$