Question

The corresponding altitudes of two similar triangles are 6 cm & 9 cm, respectively . Find the ratio of their areas

Answer 1

$$\begin{gathered} \text{Let}\mathrm{the}\mathrm{altitudes}\mathrm{of}\text{the}\mathrm{two}\mathrm{triangles}\text{be}\mathit{}{h}_1=6\mathrm{cm}\mathrm{and}{h}_2=9\mathrm{cm}. \\ \mathrm{We}\mathrm{know}\mathrm{that}\mathrm{if}{\mathrm{A}}_1\mathrm{and}{\mathrm{A}}_2\mathrm{are}\mathrm{the}\mathrm{areas}\mathrm{of}\mathrm{two}\mathrm{similar}\mathrm{triangles}\mathrm{and}{\mathrm{h}}_1\mathrm{and}{\mathrm{h}}_2 \\ \mathrm{are}\mathrm{their}\mathrm{corresponding}\mathrm{heights},\mathrm{then} \\ \frac{{\mathrm{A}}_1}{{\mathrm{A}}_2}=\frac{(h_1{)}^2}{(h_2{)}^2} \\ \Rightarrow \frac{{\mathrm{A}}_1}{{\mathrm{A}}_2}=\frac{(6{)}^2}{(9{)}^2}=\frac{36}{81}=\frac{4}{9} \\ \mathrm{Thus},\mathrm{the}\mathrm{ratio}\mathrm{of}\mathrm{the}\mathrm{areas}\mathrm{of}\mathrm{both}\mathrm{the}\mathrm{triangles}\mathrm{is}4:9. \end{gathered}$$