In the given figure, Δ ABC ∼ Δ DEF. Find the lengths of the sides of both the triangles (Each side is in cm).

Given, Δ ABC ∼ Δ DEF

We know that,

Corresponding sides of similar triangles are proportional.

$\Rightarrow \dfrac{\text{AB}}{\text{DE}} = \dfrac{\text{BC}}{\text{EF}}\\[1em] \Rightarrow \dfrac{(2x + 1)}{18} = \dfrac{2(x + 1)}{3(x + 1)}\\[1em] \Rightarrow \dfrac{(2x + 1)}{18} = \dfrac{2}{3}\\[1em] \Rightarrow (2x + 1) \times 3 = 18 \times 2\\[1em] \Rightarrow 6x + 3 = 36\\[1em] \Rightarrow 6x = 36 - 3\\[1em] \Rightarrow 6x = 33\\[1em] \Rightarrow x = \dfrac{33}{6}\\[1em] \Rightarrow x = \dfrac{11}{2}$

Now, in Δ ABC, AB = 2x + 1 = 2 × $\dfrac{11}{2}$ + 1 = 11 + 1 = 12 cm

BC = 2(x + 1) = $2\Big(\dfrac{11}{2} + 1\Big) = 2 × \Big(\dfrac{11 + 2}{2}\Big) = 2 × \dfrac{13}{2} = 13$ cm

AC = 4x = 4 × $\Big(\dfrac{11}{2}\Big)$ = 22 cm

Now, in Δ DEF, DE = 18 cm

EF = 3(x + 1) = $3\Big(\dfrac{11}{2} + 1\Big) = 3 × \Big(\dfrac{11 + 2}{2}\Big) = 3 × \dfrac{13}{2} = \dfrac{39}{2}$ cm

DF = 6x = 6 × $\Big(\dfrac{11}{2}\Big)$ = 33 cm

Hence, AB = 12 cm, BC = 13 cm, AC = 22 cm, DE = 18 cm, EF = $\dfrac{39}{2}$ cm and DF = 33 cm.