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 \( AP : PB = 2 : 1 \) and \( AC = 18 \) cm, find the length of \( AQ \).

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

2. 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

3. 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 \(PB\).

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. 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

6. 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

7. 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.

8. 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 \(x\).

9. 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?

10. 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 _____.

11. 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

12. 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 \(PB:QC\).

13. 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

14. 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 \( PB:QC \).

15. 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 \(x\).

16. 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

17. 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

18. 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

19. In \(\triangle ABC\), \(AB = 9\) cm, \(BC = 6\) cm, and \(CA = 7.5\) cm. In \(\triangle DEF\), the corresponding side to \(BC\) is \(EF\), and \(EF = 8\) cm. Given that \(\triangle ABC \sim \triangle DEF\), determine the perimeter of \(\triangle DEF\).

20. 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.

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. 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 \(QR\).

23. Here’s the English translation of your math problem: In triangle \( \triangle ABC \), \( \angle ABC = 90^\circ \), \( BC = 24 \) cm, and \( E \) is the midpoint of \( AC \). If \( ED \perp BC \), then what is the length of \( BD \)?

(a) 6 cm (b) 8 cm (c) 9 cm (d) None of the above

24. In a circle centered at \(O\), two chords \(AB\) and \(CD\) have equal lengths. Given that \(\angle AOB = 60°\) and the radius of the circle is \(6\) cm, find the area of \(\triangle COD\).

(a) \(9\sqrt3\) square c.m (b) \(6\sqrt3\) square c.m (c) \(2\sqrt3\) square c.m (d) \(3\sqrt3\) square c.m

25. In a circle, triangle ABC is inscribed. The angle bisectors of \(\angle BAC\), \(\angle ABC\), and \(\angle ACB\) meet the circle at points X, Y, and Z respectively. Prove that in triangle ΔXYZ, \(\angle YXZ = 90^\circ - \frac{\angle BAC}{2}\).

26. In \(\triangle ABC\), \(\angle ABC = 90^\circ\) and \(BD \perp AC\); if \(AB = 6\) cm, \(BD = 3\) cm, and \(CD = 5.4\) cm, then calculate and write the length of side \(BC\).

27. 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.

28. 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