Given that ΔABC ∼ ΔPQR.

(i) If ar(ΔABC) = 49 cm² and ar(ΔPQR) = 25 cm² and AB = 5.6 cm, find the length of PQ.

(ii) If ar(ΔABC) = 28 cm² and ar(ΔPQR) = 63 cm² and PR = 8.4 cm, find the length of AC.

(iii) If BC = 4 cm, QR = 5 cm and ar(ΔABC) = 32 cm² determine ar(ΔPQR).

(i) Given,

ΔABC ∼ ΔPQR

ar(ΔABC) = 49 cm²

ar(ΔPQR) = 25 cm²

AB = 5.6 cm

We know that,

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

$\therefore \dfrac{\text{ar(ΔABC)}}{\text{ar(ΔPQR)}} = \Big(\dfrac{AB}{PQ}\Big)^2 \\[1em] \Rightarrow \dfrac{49}{25} = \Big(\dfrac{5.6}{PQ}\Big)^2 \\[1em] \Rightarrow \sqrt{\dfrac{49}{25}} = \dfrac{5.6}{PQ} \\[1em] \Rightarrow \dfrac{7}{5} = \dfrac{5.6}{PQ} \\[1em] \Rightarrow PQ = \dfrac{5 \times 5.6}{7}\\[1em] \Rightarrow PQ = \dfrac{28}{7}\\[1em] \Rightarrow PQ = 4 \text{ cm.}$

Hence, PQ = 4 cm.

(ii) Given,

ΔABC ∼ ΔPQR

ar(ΔABC) = 28 cm²

ar(ΔPQR) = 63 cm²

PR = 8.4 cm

We know that,

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

$\therefore \dfrac{\text{ar(ΔABC)}}{\text{ar(ΔPQR)}} = \Big(\dfrac{AC}{PR}\Big)^2 \\[1em] \Rightarrow \dfrac{28}{63} = \Big(\dfrac{AC}{8.4}\Big)^2 \\[1em] \Rightarrow \sqrt{\dfrac{4}{9}} = \dfrac{AC}{8.4} \\[1em] \Rightarrow \dfrac{2}{3} = \dfrac{AC}{8.4} \\[1em] \Rightarrow AC = \dfrac{2 \times 8.4}{3}\\[1em] \Rightarrow AC = \dfrac{16.8}{3}\\[1em] \Rightarrow AC = 5.6 \text{ cm.}$

Hence, AC = 5.6 cm.

(iii) Given,

ΔABC ∼ ΔPQR

BC = 4 cm

QR = 5 cm

ar(ΔABC) = 32 cm²

We know that,

The ratio of the areas of two similar triangles is equal to the square of the ratio of their corresponding sides.

$\therefore \dfrac{\text{ar(ΔABC)}}{\text{ar(ΔPQR)}} = \Big(\dfrac{BC}{QR}\Big)^2 \\[1em] \Rightarrow \dfrac{32}{\text{ar(ΔPQR)}} = \Big(\dfrac{4}{5}\Big)^2 \\[1em] \Rightarrow \dfrac{32}{\text{ar(ΔPQR)}} = \dfrac{16}{25} \\[1em] \Rightarrow \text{ar(ΔPQR)} = \dfrac{32 \times 25}{16} \\[1em] \Rightarrow \text{ar(ΔPQR)} = 25 \times 2 \\[1em] \Rightarrow \text{ar(ΔPQR)} = 50 \text{ cm}^2.$

Hence, ar(ΔPQR) = 50 cm².