1. A line parallel to side \(BC\) of triangle \(∆ABC\) intersects sides \(AB\) and \(AC\) at points \(P\) and \(Q\), respectively. Given that \(PB = AQ\), \(AP = 9\) units, and \(QC = 4\) units, find the length of \(PB\).

2. In triangle \( \triangle ABC \), a straight line parallel to side BC intersects sides AB and AC at points P and Q respectively. Given that \( AP = 18 \) cm, \( QC = 9 \) cm, and \( AQ = 2 \times PB \), find the length of \( PB \).

(a) 6 cm (b) 12 cm (c) 18 cm (d) 9 cm

3. In triangle \( \triangle ABC \), a straight line parallel to side BC intersects sides AB and AC at points P and Q respectively. Given that \( AP : PB = 2 : 1 \) and \( AC = 18 \) cm, find the length of \( AQ \).

(a) 12 cm (b) 9 cm (c) 6 cm (d) None of the above

4. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points P and Q respectively. If \( AB = 3 \times PB \) and \( BC = 18 \) cm, then what is the length of \( PQ \)?

(a) 10 cm (b) 9 cm (c) 12 cm (d) 8 cm

5. A straight line parallel to side BC of triangle ∆ABC intersects sides AB and AC at points D and E respectively. If AD : BD = 3 : 5, then what is the ratio of the area of triangle ∆ADE to the area of trapezium DBCE?

6. In \(\triangle\)ABC, a line parallel to side BC intersects AB and AC at points P and Q, respectively. If AP = 4 cm, QC = 9 cm, and PB = AQ, then what is the length of PB?

7. A straight line parallel to side BC of \(\triangle\)ABC intersects AB and AC at points P and Q, respectively. If AQ = 2AP, then what is the ratio PB:QC?

(a) 1:2 (b) 2:1 (c) 1:1 (d) None of these

8. In triangle ABC, a straight line parallel to side BC intersects sides AB and AC at points P and Q respectively. AP = QC, AB = 12 cm, AQ = 2 cm. Find the length of CQ.

(a) 4 cm (b) 6 cm (c) 9 cm (d) None of the above

9. In triangle ∆ABC, a straight line parallel to side BC intersects AB and AC at points P and Q respectively. Given that \(\frac{AQ}{QC} = \frac{3}{4}\) and AB = 21 cm, find the length of PB.

10. In triangle ∆ABC, a straight line parallel to side BC intersects sides AB and AC at points X and Y respectively. If AX = 2.4 cm, AY = 3.2 cm, and YC = 4.8 cm, find the length of AB.