Question

If a, b, c, d are in proportion then show that $\frac{a^2}{b^2}=\frac{ac}{bd}$.

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ \frac{a}{b}=\frac{c}{d} \\ \mathrm{Let}: \\ \frac{a}{b}=\frac{c}{d}=k \\ a=bk\mathrm{and}c=dk \\ \mathrm{Now}, \\ \mathrm{LHS}=\frac{a^2}{b^2}={\left(\frac{a}{b}\right)}^2=k^2 \\ \mathrm{RHS}=\frac{ac}{bd} \\ =\frac{a}{b}\times \frac{c}{d}\left(\mathrm{Substituting}\mathrm{the}\mathrm{values}\mathrm{of}\frac{a}{b}\mathrm{and}\frac{c}{d}\right) \\ =k\times k \\ =k^2 \\ \mathrm{LHS}=\mathrm{RHS} \\ \therefore \frac{a^2}{b^2}=\frac{ac}{bd} \end{gathered}$$