In the given figure, P is a point on AB such that AP : PB = 4 : 3. PQ is parallel to AC.

(i) Calculate the ratio PQ : AC, giving reason for your answer.

(ii) In triangle ARC, ∠ARC = 90° and in triangle PQS, ∠PSQ = 90°. Given QS = 6 cm, calculate the length of AR.

(i) Given,

AP : PB = 4 : 3

Let AP = 4x and PB = 3x.

From figure,

AB = AP + PB = 4x + 3x = 7x.

PB : AB = 3x : 7x = 3 : 7.

In △PQB and △ACB,

QP || AC

∠BPQ = ∠BAC (Corresponding angles are equal)

∠BQP = ∠BCA (Corresponding angles are equal)

△PQB ~ △ACB.

Since, corresponding sides of similar triangle are proportional to each other.

$\therefore \dfrac{PQ}{AC} = \dfrac{PB}{AB} = \dfrac{3}{7}$.

Hence, PQ : AC = 3 : 7.

(ii) In △ARC and △QSP,

∠ARC = ∠QSP = 90°

∠ACR = ∠SPQ (Alternate angles are equal)

∴ △ARC ~ △QSP [By AA]

Since, corresponding sides of similar triangle are proportional to each other.

$\Rightarrow \dfrac{AR}{QS} = \dfrac{AC}{PQ} \\[1em] \Rightarrow \dfrac{AR}{6} = \dfrac{7}{3} \\[1em] \Rightarrow AR = 6 \times \dfrac{7}{3} \\[1em] \Rightarrow AR = 14 \text{ cm}.$

Hence, AR = 14 cm.