Question

If a : b =c : d, then prove that ${\displaystyle \frac{{\text{a}}^2+{\text{c}}^2}{{\text{b}}^2+{\text{d}}^2}=\frac{\text{ac}}{\text{bc}}}$

Answer 1

${\displaystyle \frac{\text{a}}{\text{b}}=\frac{\text{c}}{\text{d}}\Rightarrow \text{a}=\frac{\text{bc}}{\text{d}}}$

To prove, 

${\displaystyle \frac{{\text{a}}^2+{\text{c}}^2}{{\text{b}}^2+{\text{d}}^2}=\frac{\text{ac}}{\text{bd}}}$

LHS

${\displaystyle \frac{{\text{a}}^2+{\text{c}}^2}{{\text{b}}^2+{\text{d}}^2}}$

${\displaystyle =\frac{{\left(\frac{\text{bc}}{\text{d}}\right)}^2+{\text{c}}^2}{{\text{b}}^2+{\text{d}}^2}}$

${\displaystyle =\frac{\frac{{\text{b}}^2{\text{c}}^2}{{\text{d}}^2}+{\text{c}}^2}{{\text{b}}^2+{\text{d}}^2}}$

${\displaystyle =\frac{{\text{c}}^2({\text{b}}^2+{\text{d}}^2)}{{\text{d}}^2({\text{b}}^2+{\text{d}}^2)}}$

${\displaystyle =\frac{{\text{c}}^2}{{\text{d}}^2}}$

RHS

${\displaystyle \frac{\text{ac}}{\text{bd}}}$

${\displaystyle =\frac{\frac{\text{bc}}{\text{d}}\text{c}}{\text{bd}}}$

${\displaystyle =\frac{{\text{bc}}^2}{{\text{bd}}^2}}$

${\displaystyle =\frac{{\text{c}}^2}{{\text{d}}^2}}$

LHS = RHS