Question
If $a: b=c: d$, then prove that $\frac{a^2+c^2}{b^2+d^2}=\frac{a c}{b c}$
✓
Answer
$\frac{ a }{ b }=\frac{ c }{ d } \Rightarrow a =\frac{ bc }{ d }$
To prove,
$\frac{ a ^2+ c ^2}{ b ^2+ d ^2}=\frac{ ac }{ bd }$
LHS
$\frac{a^2+c^2}{b^2+d^2} $
$=\frac{\left(\frac{b c}{d}\right)^2+c^2}{b^2+d^2} $
$=\frac{\frac{b^2 c^2}{d^2}+c^2}{b^2+d^2} $
$=\frac{c^2\left(b^2+d^2\right)}{d^2\left(b^2+d^2\right)} $
$=\frac{c^2}{d^2}$
$\text { RHS } $
$ \frac{ ac }{ bd }$
$=\frac{\frac{ bc }{ d c}}{ bd } $
$ =\frac{ bc ^2}{ bd ^2} $
$ =\frac{ c ^2}{ d ^2} $
$ \text { LHS }=\text { RHS }$
To prove,
$\frac{ a ^2+ c ^2}{ b ^2+ d ^2}=\frac{ ac }{ bd }$
LHS
$\frac{a^2+c^2}{b^2+d^2} $
$=\frac{\left(\frac{b c}{d}\right)^2+c^2}{b^2+d^2} $
$=\frac{\frac{b^2 c^2}{d^2}+c^2}{b^2+d^2} $
$=\frac{c^2\left(b^2+d^2\right)}{d^2\left(b^2+d^2\right)} $
$=\frac{c^2}{d^2}$
$\text { RHS } $
$ \frac{ ac }{ bd }$
$=\frac{\frac{ bc }{ d c}}{ bd } $
$ =\frac{ bc ^2}{ bd ^2} $
$ =\frac{ c ^2}{ d ^2} $
$ \text { LHS }=\text { RHS }$
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