Question

The areas of two similar triangles are 81 cm² and 49 cm² respectively. Find the ratio of their corresponding heights. What is the ratio of their corresponding medians?

Answer 1

$$\begin{gathered} \mathrm{We}\mathrm{have}\mathrm{the}\mathrm{areas}\mathrm{of}\text{the}\mathrm{two}\mathrm{triangles}\mathrm{as}{\mathrm{A}}_1=81{\mathrm{cm}}^2\mathrm{and}{\mathrm{A}}_2=49{\mathrm{cm}}^2. \\ \mathrm{We}\mathrm{know}\mathrm{that}\mathrm{if}{\mathrm{A}}_1\mathrm{and}{\mathrm{A}}_2\mathrm{are}\mathrm{the}\mathrm{areas}\mathrm{of}\mathrm{two}\mathrm{similar}\mathrm{triangles}\mathrm{and}{h}_1\mathrm{and}{h}_2\mathrm{are} \\ \mathrm{their}\mathrm{corrsponding}\mathrm{heights}and{m}_1\mathrm{and}{m}_2\mathrm{their}\mathrm{corresponding}\mathrm{medians},\text{then} \\ \frac{{\mathrm{A}}_1}{{\mathrm{A}}_2}=\frac{{\left(h_1\right)}^2}{(h_2{)}^2}=\frac{{\left(m_1\right)}^2}{{\left(m_2\right)}^2} \\ \Rightarrow \frac{81}{49}=\frac{{\left(h_1\right)}^2}{(h_2{)}^2}=\frac{{\left(m_1\right)}^2}{{\left(m_2\right)}^2} \\ \Rightarrow \frac{(9{)}^2}{(7{)}^2}=\frac{{\left(h_1\right)}^2}{(h_2{)}^2}=\frac{{\left(m_1\right)}^2}{{\left(m_2\right)}^2} \\ \mathrm{On}\mathrm{taking}\mathrm{square}\mathrm{root}\mathrm{of}\mathrm{both}\mathrm{sides},\mathrm{we}\mathrm{get}: \\ \frac{9}{7}=\frac{h_1}{h_2}=\frac{m_1}{m_2} \\ \Rightarrow \frac{h_1}{h_2}=\frac{m_1}{m_2}=\frac{9}{7} \\ \mathrm{Thus},\mathrm{the}\mathrm{ratio}\mathrm{of}\mathrm{the}\mathrm{corresponding}\mathrm{heights}\mathrm{is}9:7\mathrm{and}\text{that of the}\mathrm{corresponding}\mathrm{medians}\mathrm{is}9:7. \end{gathered}$$