Question

In two triangles ABC and DEF, ∠A = ∠D and the sum of the angles A and B is equal to the sum of the angles D and E. If BC = 6 cm and EF = 8 cm, then ar(∆ABC) : ar(∆DEF) = ___________.

Answer 1

In ∆ABC and ∆DEF,

∠A = ∠D (Given)

Also, ∠A + ∠B = ∠D + ∠E

⇒ ∠B = ∠E

∴ ∆ABC $~$ ∆DEF (AA Similarity)

$\Rightarrow \frac{\mathrm{ar}\left(∆\mathrm{ABC}\right)}{\mathrm{ar}\left(∆\mathrm{DEF}\right)}={\left(\frac{\mathrm{BC}}{\mathrm{EF}}\right)}^2$ (The ratio of areas of two similar triangles is equal to the ratio of the squares of any two corresponding sides)

$$\begin{gathered} \Rightarrow \frac{\mathrm{ar}\left(∆\mathrm{ABC}\right)}{\mathrm{ar}\left(∆\mathrm{DEF}\right)}={\left(\frac{6\mathrm{cm}}{8\mathrm{cm}}\right)}^2=\frac{9}{16} \\ \Rightarrow \mathrm{ar}\left(∆\mathrm{ABC}\right):\mathrm{ar}\left(∆\mathrm{DEF}\right)=9:16 \end{gathered}$$

In two triangles ABC and DEF, ∠A = ∠D and the sum of the angles A and B is equal to the sum of the angles D and E. If BC = 6 cm and EF = 8 cm, then ar(∆ABC) : ar(∆DEF) = ___9 : 16___.