1. In the adjacent figure, O is the center of the circle and BOA is the diameter. A tangent is drawn at point P on the circle, which intersects the extended line BA at point T. If \(\angle PBO = 30^\circ\), then what is the measure of \(\angle PTA\)?

2. In a circle centered at O, AB is a diameter, and C is any point on the circle. Given that \(\angle\)OAC = 45°, determine the measure of \(\angle\)OCB.

(a) 90° (b) 45° (c) 30° (d) 60°

3. In the adjacent figure, O is the center of the circle, and BOA is the diameter. A tangent is drawn at point P on the circle, which intersects the extension of BA at point T. If ∠PBO = 30°, then find the measure of ∠PTA.

4. 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.

5. Two tangents are drawn to a circle from points A and B on the circumference, and they intersect at point C. Another point P lies on the circumference, on the side opposite to where point C is located with respect to the center. If \(\angle\)APB = 35°, then what is the measure of \(\angle\)ACB?

(a) 145° (b) 55° (c) 110° (d) None of the above

6. O is the center of a circle. PQ is a diameter, and R is a point on the circumference. If \(\angle\)PQR = 40°, then what is the measure of \(\angle\)POR?

(a) 80° (b) 40° (c) 20° (d) 100°

7. AB is a chord of a circle centered at O, and PT is a tangent to the circle at point A. If ∠ AOB = 120°, then what is the measure of ∠ BAT?

(a) 60° (b) 30° (c) 90° (d) 45°

8. If two tangents are drawn from points P and Q on a circle and intersect at point A such that ∠PAQ = 60°, then what is the measure of ∠APQ?

9. AB is a diameter of a circle centered at O. P is any point on the circumference. If \(\angle\)POA = 120°, then find the measure of \(\angle\)PBO.

10. In a circle, AB and AC are two chords. Tangents are drawn at points B and C, and they intersect at point P. If \(\angle\)BAC = 54°, then what is the measure of \(\angle\)BPC?

11. AB is the diameter of a circle centered at O. C is any point on the circle. Given ∠BAC = 50° and CD is perpendicular to AB, Find the value of ∠BCD.

12. Two tangents are drawn to a circle from points A and B on its circumference, and they intersect at point P. If \(\angle\)APB = 68°, then what is the measure of \(\angle\)PAB?

13. AOB is the diameter of a circle centered at O. C is a point on the circumference. If AC = 3 cm and BC = 4 cm, then the length of AB will be _____.

14. In the adjacent figure, O is the center of the circle, \(\angle AOD = 40^\circ\), and \(\angle ACB = 35^\circ\). Calculate and write the measures of \(\angle BCO\) and \(\angle BOD\), and provide reasoning to support your answers.

15. In the adjacent figure, AB is the diameter of a circle centered at O. Point C is any point on the circle. Given that ∠BAC = 50° and CD is perpendicular to AB, find the value of ∠BCD.

16. Draw an equilateral triangle ABC with each side measuring 5 cm. Then, draw the circumcircle of that triangle. At point A on the circle, draw a tangent. On the tangent, take a point P such that AP = 5 cm. From point P, draw another tangent to the circle, and write down which point on the circle this second tangent touches.

17. From an external point A of a circle, two tangents AB and AC are drawn, touching the circle at points B and C respectively. A tangent is drawn at point X, which lies on the arc BC, and it intersects AB and AC at points D and E respectively. Prove that the perimeter of triangle ∆ADE = 2 × AB.

18. Rumela has drawn a circle centered at point O, in which QR is a chord. Tangents are drawn at points Q and R, and they intersect at point P. If QM is a diameter of the circle, prove that ∠QPR = 2∠RQM.

19. Two circles intersect at points A and B, and the center of one of the circles is point O. A straight line from point A intersects the circle centered at O at point R and the other circle at point P. By joining points P to B and R to B, prove that PR = PB.

20. In the circle with center \( O \), if \( AOB \) is a diameter and point \( C \) is on the circle such that \( AC = 3 \) cm and \( BC = 4 \) cm, what is the length of \( AB \)?

(a) 3 cm (b) 4 cm (c) 5 cm (d) 8 cm

21. Two circles touch each other externally at point C. AB is a common tangent to both circles and touches the circles at points A and B, respectively. Find the measure of \(\angle\)ACB.

(a) 60° (b) 45° (c) 30° (d) 90°

22. Two circles touch each other externally at point \(C\). \(AB\) is a common tangent to the two circles, touching them at points \(A\) and \(B\), respectively. The measurement of \(\angle ACB\) is –

(a) 60° (b) 45° (c) 30° (d) 90°

23. In a circle with center \(O\), \(\bar{AB}\) is a diameter. On the opposite side of the circumference from the diameter \(\bar{AB}\), there are two points \(C\) and \(D\) such that \(\angle AOC = 130°\) and \(\angle BDC = x°\). Find the value of \(x\).

(a) 25° (b) 50° (c) 60° (d) 65°

24. 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

25. "AB is a diameter of a circle with center O. P is any point on the circumference. If ∠ POA = 120°, then what is the measure of ∠ PBO?"

(a) 90\(^o\) (b) 60\(^o\) (c) 75\(^o\) (d) None of the above

26. In triangle ABC, the circumcenter is O; points A and B, C lie on opposite sides of the center. If \(\angle BOC = 120^\circ\), then what is the measure of \(\angle BAC\)?

(a) 50° (b) 60° (c) 70° (d) 80°

27. AB and AC are two tangents drawn from point A to a circle with center O. The line OA intersects the chord BC (which joins the points of contact) at point M. If AM = 8 cm and BC = 12 cm, then what is the length of each tangent?

(a) 8 cm (b) 10 cm (c) 12 cm (d) 16 cm

28. In a circle centered at O, chords AB and CD intersect at point P. The line segment OP is the bisector of ∠APC. Prove that AB = CD.

29. Point P is any point inside a circle centered at O. If the radius of the circle is 5 cm and OP = 3 cm, then find the minimum length of the chord passing through point P.

30. In a circle with center O, PQ and PR are two chords. Tangents drawn at points Q and P intersect at point S. If ∠QSR = 70°, then what is the measure of ∠QPR?