Question

ABC $~$DEF
If AB= 2.4 cm and DE= 1.6 cm, find the ratio of the areas of $△$ABC and $△$DEF.

Answer 1

$$\begin{gathered} \mathrm{Since}△\mathrm{ABC}~△\mathrm{DEF}, \\ \frac{\mathrm{A}(△\mathrm{ABC})}{\mathrm{A}(△\mathrm{DEF})}=\frac{{\mathrm{AB}}^2}{{\mathrm{DE}}^2}=\frac{{\mathrm{BC}}^2}{{\mathrm{EF}}^2}=\frac{{\mathrm{AC}}^2}{{\mathrm{DF}}^2} \\ \Rightarrow \frac{\mathrm{A}(△\mathrm{ABC})}{\mathrm{A}(△\mathrm{DEF})}=\frac{{\mathrm{AB}}^2}{{\mathrm{DE}}^2} \\ \mathrm{On}\mathrm{substituting}\mathrm{AB}=2.4\mathrm{cm}\mathrm{and}\mathrm{DE}=1.6\mathrm{cm},\mathrm{we}\mathrm{get}: \\ \frac{\mathrm{A}(△\mathrm{ABC})}{\mathrm{A}(△\mathrm{DEF})}=\frac{(2.4{)}^2}{(1.6{)}^2} \\ \Rightarrow \frac{\mathrm{A}(△\mathrm{ABC})}{\mathrm{A}(△\mathrm{DEF})}={\left(\frac{2.4}{1.6}\right)}^2={\left(\frac{3}{2}\right)}^2 \\ \Rightarrow \frac{\mathrm{A}(△\mathrm{ABC})}{\mathrm{A}(△\mathrm{DEF})}=\frac{9}{4} \\ \mathrm{Thus},\mathrm{A}\left(△\mathrm{ABC}\right):\mathrm{A}(△\mathrm{DEF})=9:4. \end{gathered}$$