Question

$△$ABC with sides AB = 6 cm, BC = 12 cm and AC = 8 cm is enlarged to $△$PQR, such that its largest side is 18 cm. Find the ratio and hence, find the lengths of the remaining sides of $△$PQR.

Answer 1

We have triangle ABC with sides AB = 6 cm, BC = 12 cm and AC = 8 cm.

ΔABC is enlarged to ΔPQR, such that the largest side QR = 18 cm.

Thus, when ΔABC is enlarged to ΔPQR by increasing the corresponding sides, ΔABC ~ ΔPQR.
$\Rightarrow \frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{\mathrm{BC}}{\mathrm{QR}}=\frac{\mathrm{AC}}{\mathrm{PR}}$ …(1)
Since BC = 12 cm and QR = 18 cm, we get:
$$\begin{gathered} \frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{12}{18}=\frac{\mathrm{AC}}{\mathrm{PR}} \\ \Rightarrow \frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{2}{3}=\frac{\mathrm{AC}}{\mathrm{PR}}...\left(2\right) \end{gathered}$$

From the first two ratios of equation (2), we get:
$\frac{\mathrm{AB}}{\mathrm{PQ}}=\frac{2}{3}$
Since AB = 6 cm, we get:
$$\begin{gathered} \frac{6}{\mathrm{PQ}}=\frac{2}{3} \\ \Rightarrow \mathrm{PQ}=\frac{3\times 6}{2}=9\mathrm{cm} \end{gathered}$$

From the last two ratios of equation (2), we get:
$$\begin{gathered} \frac{2}{3}=\frac{\mathrm{AC}}{\mathrm{PR}} \\ \mathrm{Since}\mathrm{AC}=8\mathrm{cm},\mathrm{we}\mathrm{get}: \\ \frac{2}{3}=\frac{8}{\mathrm{PR}} \\ \Rightarrow \mathrm{PR}=\frac{3\times 8}{2}=12\mathrm{cm} \end{gathered}$$

Thus, the ratio is $\frac{2}{3}$ and the lengths of the remaining sides of Δ PQR are 9 cm and 12 cm.