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Ratio and Proportion — Mathematics STD 10 — Question

ICSE BoardEnglish MediumSTD 10MathematicsRatio and Proportion4 Marks
Question
If $\frac{a}{c}=\frac{c}{d}=\frac{c}{f}$ prove that:
$
\frac{a^2}{b^2}+\frac{c^2}{d^2}+\frac{e^2}{f^2}=\frac{ ac }{ bd }+\frac{ ce }{ df }+\frac{ ae }{ df }
$
✓

Answer


$\begin{aligned} & \frac{a}{c}=\frac{c}{d}=\frac{c}{f}= k \text { (say) } \\ & \therefore a = bk _1 c = dk _{ r } e = fk \\ & \text { L.H.S. }=\frac{a^2}{b^2}+\frac{c^2}{d^2}+\frac{e^2}{f^2} \\ & =\frac{b^2 k^2}{b^2}+\frac{d^2 k^2}{d^2}+\frac{f^2 k^2}{f^2} \\ & = k ^2+ k ^2+ k ^2 \\ & =3 k ^2 \\ & \text { R.H.S. }=\frac{ ac }{ bd }+\frac{ ce }{ df }+\frac{ ae }{ bf } \\ & =\frac{ bk \cdot dk }{ b \cdot d }+\frac{ dk \cdot fk }{ d \cdot f }+\frac{ bk \cdot fk }{ b \cdot f } \\ & =k^2+k^2+k^2 \\ & =3 k ^2 \\ & \therefore \text { L.H.S. }=\text { R.H.S. } \\ & \end{aligned}$

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