Question

In Fig. 7.248, PQ || BC and PQ :BC = 1 : 3. If ar(∆ABC) = 144 cm², then ar(∆APQ) = ____________.

Answer 1

In ∆APQ and ∆ABC,

∠APQ = ∠B (Corresponding angles)

∠A = ∠A (Common)

∴ ∆APQ $~$ ∆ABC (AA Similarity)

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

$$\begin{gathered} \Rightarrow \frac{\mathrm{ar}\left(∆\mathrm{APQ}\right)}{144{\mathrm{cm}}^2}={\left(\frac{1}{3}\right)}^2 \\ \Rightarrow \mathrm{ar}\left(∆\mathrm{APQ}\right)=\frac{144}{9}=16{\mathrm{cm}}^2 \end{gathered}$$

In Fig. 7.248, PQ || BC and PQ : BC = 1 : 3. If ar(∆ABC) = 144 cm², then ar(∆APQ) = ___16 cm²____.