Question

The sides of two similar triangles are in the ratio $4:9$. Find the ratio of areas of these triangles.

Answer 1

Finding the ratio of areas of $△ABC$ and $△DEF$:

Consider $△ABC$and $△DEF$ are two similar triangles, such that,

$\Delta ABC~\Delta DEF$

We know that, when two triangles are similar, their corresponding sides are in proportion

Hence, $\frac{AB}{DE}=\frac{AC}{DF}=\frac{BC}{EF}$

Given that the ratio of sides of $△ABC$ and $△DEF$ be $4:9$.

Therefore,$\frac{AB}{DE}=\frac{AC}{DF}=\frac{BC}{EF}=\frac{4}{9}.....\left(i\right)$

We know that the ratio of area of similar triangles is equal to the square of the ratio of the corresponding sides

$$\begin{gathered} \frac{\mathrm{Area}\mathrm{of}\Delta ABC}{\mathrm{Area}\mathrm{of}\Delta DEF}=\frac{A{B}^2}{D{E}^2} \\ \Rightarrow =\frac{4^2}{9^2} \\ \Rightarrow \frac{\mathrm{Area}\mathrm{of}\Delta ABC}{\mathrm{Area}\mathrm{of}\Delta DEF}=\frac{16}{81} \end{gathered}$$(from equation $\left(i\right)$)

Hence, the areas of $△ABC$ and $△DEF$ are in the ratio $16:81$.