1. In the adjacent figure, BX and CY are the bisectors of \(\angle\)ABC and \(\angle\)ACB respectively. Given that AB = AC and BY = 4 cm, find the length of AX.

2. In the adjacent figure, a circle is centered at point O. From an external point C, two tangents are drawn to the circle, touching it at points P and Q respectively. Another tangent is drawn at point R on the circle, which intersects the tangents CP and CQ at points A and B respectively. If CP = 11 cm and BC = 7 cm, then find the length of BR.

3. 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

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 \( \triangle ABC \), 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

6. 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

7. AB is the diameter of a circle, and PQ is a chord that stands perpendicular to AB at point O. Given: OA = 8 cm, OB = 2 cm, OP = 4 cm Find: The length of OQ.

(a) 6 cm (b) 4 cm (c) 5 cm (d) None of the above

8. In triangle ABC, AB = AC and E, F are the midpoints of AB and AC respectively. AD is perpendicular to BC. Given: AD = \(2\sqrt{5}\) cm and EF = 4 cm Find: The length of AB.

(a) 7 cm (b) 4 cm (c) 6 cm (d) 5 cm

9. 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

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.

11. In the adjacent figure, O is the center of the circle and AB is the diameter. Given that \(\angle\)AOD = 140° and \(\angle\)CAB = 50°, find the measure of \(\angle\)BED.

12. 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.

13. 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

14. In the trapezium ABCD, AB || DC, and P and Q are two points on the sides AD and BC such that PQ || DC. If PD = 18 cm, BQ = 35 cm, and QC = 15 cm, find the length of AD.

15. In the adjacent figure, point O is located inside triangle PQR such that ∠POR = 90°, OP = 6 cm and OR = 8 cm. If PR = 24 cm and ∠QPR = 90°, then write the length of side QR.

16. In the adjacent figure, ∠ABC = 90° and BD is perpendicular to AC; if AB = 30 cm, BD = 24 cm, and AD = 18 cm, then write the length of BC.

17. In the adjacent figure, ∠ABC = 90° and BD is perpendicular to AC; if BD = 8 cm and AD = 4 cm, then write the length of CD.

18. 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 \(PB\).

19. 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

20. In the adjacent figure, triangle ABC is inscribed in a circle and touches the circle at points P, Q, and R. If AP = 4 cm, BP = 6 cm, AC = 12 cm, and BC = x cm, then find the value of x.

21. A line parallel to side BC of triangle ABC intersects AB and AC at points P and Q respectively. Given: AP = 4 cm, QC = 9 cm, and PB = AQ. Find the length of PB.

22. Triangle ABC is inscribed in a circle, and the circle touches the sides at points P, Q, and R respectively. If AP = 4 cm, BP = 6 cm, and AC = 12 cm, then find the length of BC.