1. 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

2. 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

3. 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

4. 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.

5. In triangle \(\triangle ABC\), the incircle touches the sides AB, BC, and CA at points D, E, and F respectively. Given: AD = 12 cm, BE = 5 cm, and CF = 4 cm. Find the lengths of AB, BC, and CA.

6. In \( \triangle ABC \), points \( D \) and \( E \) lie on sides \( AB \) and \( AC \) respectively, such that \( DE \parallel BC \) and \( AD:DB = 3:1 \). If \( EA = 33 \) cm, then the length of \( AC \) is _____.

7. In \(\triangle ABC\), points \(D\) and \(E\) lie on sides \(AB\) and \(AC\) respectively, such that \(DE \parallel BC\) and \(AD : DB = 3 : 1\). If \(EA = 3.3\) cm, determine the length of \(AC\).

(a) 1.1 cm (b) 4 cm (c) 4.4 cm (d) 5.5 cm

8. 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 \(PB:QC\).

9. 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 \(PB\).

10. A line parallel to side \(BC\) of triangle \(∆ABC\) 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 side \(AB\).

(a) 3.6 cm (b) 6 cm (c) 6.4 cm (d) 7.2 cm

11. 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

12. In \(\triangle ABC\), the center of the incircle is \(O\), and the incircle touches the sides \(AB\), \(BC\), and \(CA\) at points \(P\), \(Q\), and \(R\) respectively. Given that \(AP = 4\) cm, \(BP = 6\) cm, \(AC = 12\) cm, and \(BC = x\) cm, determine the value of \(x\).

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

14. Points \(X\) and \(Y\) are taken on sides \(PQ\) and \(PR\) of triangle \(∆PQR\), respectively. Given: \(PQ = 8\) units, \(YR = 12\) units, \(PY = 4\) units, and the length of \(PY\) is 2 units less than the length of \(XQ\). Determine, with reasoning, whether segment \(XY\) is parallel to side \(QR\).

15. In trapezium \(ABCD\), where \(AB ∥ DC\), points \(P\) and \(Q\) lie on sides \(AD\) and \(BC\) respectively such that \(PQ ∥ DC\). Given: \(PD = 18\) cm, \(BQ = 35\) cm, and \(QC = 15\) cm Find the length of side \(AD\).

(a) 60 cm (b) 30 cm (c) 12 cm (d) 15 cm

16. Points \(D\) and \(E\) lie on the sides \(AB\) and \(AC\) of triangle \(∆ABC\) such that \(DE ∥ BC\) and \(AD : DB = 3 : 1\). If \(EA = 3.3\) cm, find the length of side \(AC\).

(a) 1.1 cm (b) 4 cm (c) 4.4 cm (d) 5.5 cm

17. 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

18. 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

19. 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?

20. On the sides AC and BC of \(\triangle\)ABC, two points L and M are positioned respectively such that \(LM \parallel AB\), and \(AL = (x - 2)\) units, \(AC = 2x + 3\) units, \(BM = (x - 3)\) units, and \(BC = 2x\) units. Then, find the value of \(x\).

21. A line parallel to side \(BC\) of triangle \(∆ABC\) intersects sides \(AB\) and \(AC\) at points \(P\) and \(Q\), respectively. The length of \(PB\) is twice the length of \(AP\), and the length of \(QC\) is 3 units more than the length of \(AQ\). Find the length of side \(AC\).

22. In triangle \(∆PQR\), points \(X\) and \(Y\) are taken on sides \(PQ\) and \(PR\), respectively. Given: \(PX = 2\) units, \(XQ = 3.5\) units, \(YR = 7\) units, and \(PY = 4.25\) units. Determine whether line segment \(XY\) is parallel to side \(QR\), with justification.

23. Triangles \(∆ABC\) and \(∆DBC\) are drawn on the same base \(BC\) and on the same side of it. Let \(E\) be any point on side \(BC\). From point \(E\), lines parallel to \(AB\) and \(BD\) are drawn, intersecting sides \(AC\) and \(DC\) at points \(F\) and \(G\) respectively. Prove that: \(AD ∥ FG\).

24. On triangle \( \triangle ABC \), points P and Q are such that \( \angle ABC = \angle APQ \). Given that \( AP = 3.6 \) cm, \( QC = 1.6 \) cm, and \( AQ = 4.8 \) cm, find the length of \( PB \).

(a) 1.2 cm (b) 2.4 cm (c) 6 cm (d) None of the above

25. In triangle \( \triangle ABC \), AD is a median. Point E divides AD in the ratio 1:2. The extended line BE intersects AC at point F. If \( AC = 10 \) cm, find the length of \( AF \).

(a) 5 cm (b) 4 cm (c) 2 cm (d) None of the above

26. In triangle \(\triangle ABC\), AB = AC. Points E and F are the midpoints of sides AB and AC respectively. AD is perpendicular to BC, and AD = 4 cm. If EF = 3 cm, then what is the length of BD?

(a) 4 cm (b) 3 cm (c) 6 cm (d) 7 cm

27. A straight line intersects sides AB and AC of triangle ABC at points D and E respectively such that \(\frac{AD}{DB} = \frac{AE}{EC}\). If \(\angle ADE = \angle ACB\), prove that triangle \( \triangle ABC \) is isosceles.

28. 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 \(AD\).

29. In a right-angled triangle, if the ratio of the perpendicular (opposite side) to the hypotenuse with respect to a positive acute angle \(\theta\) is \(12 : 13\), then determine the ratio of the perpendicular to the base and the ratio of the hypotenuse to the base, and verify that \( \sec^2\theta = 1 + \tan^2\theta \).