Question

If the areas of two similar triangles are in the ratio $25:64$, write the ratio of their corresponding sides.

Answer 1

Calculating the ratio of the corresponding sides:

Let the two triangles be $∆ABC$ and $∆PQR$.

It is given that $∆ABC~∆PQR$ ,

$\frac{\mathrm{Area}\mathrm{of}∆ABC}{\mathrm{Area}\mathrm{of}△PQR}=\frac{25}{64}$

By the similarity criterion, if two triangles are similar then the ratio of their area will be square of the ratio of their corresponding sides.

$$\begin{gathered} \frac{\mathrm{Area}\mathrm{of}∆ABC}{\mathrm{Area}\mathrm{of}△PQR}={\left(\frac{AB}{PQ}\right)}^2 \\ \Rightarrow \left(\frac{AB}{PQ}\right)=\sqrt{\frac{\mathrm{Area}\mathrm{of}∆ABC}{\mathrm{Area}\mathrm{of}△PQR}} \\ \Rightarrow \frac{AB}{PQ}=\sqrt{\frac{25}{64}}\left(\mathrm{given}\right) \\ \Rightarrow \frac{AB}{PQ}=\frac{5}{8} \end{gathered}$$

Hence the ratio of sides of the given two similar triangles is $5:8$.