The ratio between the sides of two identical triangles is 4: 9. Find the ratio of their respective areas.

Let $∆\mathrm{ABC}~∆\mathrm{DEF}$

And $\frac{\mathrm{AB}}{\mathrm{DE}}=\frac{\mathrm{AC}}{\mathrm{DF}}=\frac{\mathrm{BC}}{\mathrm{EF}}=\frac{4}{9}$

Due to the fact that the ratio of these triangles' areas will be equal to the square of their respective side ratios,

$\therefore \frac{Area(∆\mathrm{ABC})}{Area(∆\mathrm{DEF})}=\frac{{\mathrm{AB}}^2}{{\mathrm{DE}}^2}$

$\therefore \frac{Area(∆\mathrm{ABC})}{Area(∆\mathrm{DEF})}=\frac{4^2}{9^2} = \frac{16}{81}$