1. 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?

2. In trapezium ABCD, AD is parallel to BC. A straight line parallel to BC intersects AB and DC at points P and Q respectively. If \( AP : PB = 2 : 1 \), then what is the ratio \( DQ : QC \)?

(a) 1:1 (b) 1:2 (c) 1:4 (d) 2:1

3. In triangle \( \triangle ABC \), a line parallel to side BC intersects sides AB and AC at points D and E respectively. If \( AB = 20 \) cm and \( BD = 14 \) cm, then what is the ratio \( DE : BC \)?

(a) 7:10 (b) 5:17 (c) 3:10 (d) 7:17

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. In triangle ∆ABC, a straight line parallel to side BC intersects sides AB and AC at points P and Q respectively. Given that PB = AQ, AP = 9 units, and QC = 16 units, what is the length of PB?

(a) 12 cm (b) 6 cm (c) 8 cm (d) 10 cm

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. In triangle ABC, a straight line parallel to side BC intersects AB and AC at points D and E respectively. If AE = 2AD, then calculate and write the value of the ratio DB : EC.

8. In triangle \( \triangle ABC \), AD is a median. A straight line parallel to side BC intersects sides AB, AD, and AC at points P, O, and Q respectively. What is the ratio \( PO : OQ \)?

(a) 1:2 (b) 2:3 (c) 1:1 (d) None of the above

9. In triangle ABC, a straight line parallel to side BC intersects AB and AC at points D and E respectively. If AE = 2AD, then find the ratio DB : EC.

10. A line parallel to side BC of triangle ABC intersects sides AB and AC at points D and E, respectively. If AD is half of AE, then the ratio BD:EC will be –

(a) 1:2 (b) 2:1 (c) 2:3 (d) 1:3

11. A straight line parallel to the side BC of triangle ABC intersects AB and AC at points P and Q, respectively. If AQ = 2AP, determine the ratio PB:QC.

12. 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

13. 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

14. 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

15. 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.

16. 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.

17. In triangle \(ABC\), a straight line parallel to side \(BC\) intersects sides \(AB\) and \(AC\) at points \(P\) and \(Q\), respectively. If \(AQ = 2AP\), determine the ratio \(PB:QC\).

18. We have drawn a median \(AD\) of triangle \( \triangle ABC \). If a straight line parallel to \(BC\) intersects sides \(AB\) and \(AC\) at points \(P\) and \(Q\) respectively, then prove that the line segment \(PQ\) is bisected by the median \(AD\).