Question

Sides of two similar triangles are in the ratio 4 9. What will be the ratio of area of these triangles?

Answer 1

The ratio of the area of similar triangles :

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

The area ratio of two similar triangles (say $\left(\mathrm{\Delta ABC}\right)\mathrm{and}\left(\mathrm{\Delta DEF}\right)$ ) is equal to the square of their side ratios.

i.e.

$$\begin{gathered} \frac{\mathrm{Area}\left(\mathrm{\Delta ABC}\right)}{\mathrm{Area}\left(\mathrm{\Delta DEF}\right)}=\frac{{\mathrm{AB}}^2}{{\mathrm{DE}}^2} \\ \frac{\mathrm{Area}\left(\mathrm{\Delta ABC}\right)}{\mathrm{Area}\left(\mathrm{\Delta DEF}\right)}={\left(\frac{4}{9}\right)}^2 \\ \frac{\mathrm{Area}\left(\mathrm{\Delta ABC}\right)}{\mathrm{Area}\left(\mathrm{\Delta DEF}\right)}=\frac{16}{81} \end{gathered}$$

Therefore, if the sides of two similar triangles are in the ratio 4: 9, then the areas of these triangles are in the ratio 16: 81.

Hence, the area of triangles in the ratio is 16: 81.