If a b =c d Show that a + b : c + d =a^2+b^2: c^2+d^2. - Vidyadip

Question

If $\frac{a}{b}=\frac{c}{d}$ Show that a + b : c + d =$\sqrt{a^2+b^2}: \sqrt{c^2+d^2}$.

✓

Answer

$\text { Let } \frac{a}{b}=\frac{c}{d}=k $
$ \Rightarrow a = bk \text { and } c = dk$
$ \text { L.H.S. } $
$ =\frac{a+b}{c+d}=\frac{b k+b}{d k+d} $
$ =\frac{b(k+1)}{d(k+1)}=\frac{b}{d}$
R.H.S.
$=\frac{\sqrt{a^2+b^2}}{\sqrt{c^2+d^2}}=\frac{\sqrt{b^2 k^2+b^2}}{\sqrt{d^2 k^2+d^2}}$
$ =\frac{b\left(\sqrt{k^2+1}\right)}{d\left(\sqrt{k^2+1}\right)}=\frac{b}{d}$
L.H.S. = R.H.S.
Hence proved.

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