1. 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°

2. 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 \(\angle PTA\)?

3. In a circle with center \(O\), \(AB\) is the diameter, and \(P\) is a point on the circle. If \(\angle AOP = 104°\), find the value of \(\angle BPO\).

(a) 54° (b) 72° (c) 36° (d) 27°

4. 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

5. 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 _____.

6. AB is a diameter of a circle with center O. P is any point on the circle. Tangents are drawn at points A and B, which are intersected by the tangent drawn from point P at points Q and R respectively. If the radius of the circle is \(r\), prove that \(PQ \cdot PR = r^2\).

7. From an external point \(P\), two tangents \(PS\) and \(PT\) are drawn to a circle with center \(O\). \(QS\) is a chord of the circle that is parallel to \(PT\). If \(\angle SPT = 80^\circ\), then what is the measure of \(\angle QST\)?

8. 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

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. 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

11. 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°

12. In a semicircle with a radius of 4 cm, AB is the diameter and ∠ACB is an angle inscribed in the semicircle. If BC = \(2\sqrt{7}\) cm, find the length of AC.

13. In a circle with center O, AB and CD are two equal chords. If \(\angle\)AOB = 60° and CD = 10 cm, then what is the length of OD?

14. 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 \(x\).

15. If a circle has a radius of 5 cm and a tangent is drawn from an external point \(P\) to the circle with a length of 12 cm, what is the distance from the center to point \(P\)?

16. If the coordinates of the center of a circle are \((0, 0)\) and the coordinates of a point on the circle are \((3, 4)\), then the length of the radius of the circle is:

(a) 5 units (b) 4 units (c) 3 units (d) None of the above

17. 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

18. 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°

19. Here’s the English translation: *If the mean of a statistical distribution is 4.1, \(∑f_i.x_i = 132 + 5k\), and \(∑f_i = 20\), then what is the value of \(k\)?* Would you like help solving it too? I’d be glad to walk through it with you.

20. AOB is a diameter of the circle with center O, and C is a point on the circle. If \(\angle\)OBC = 60°, then find the measure of \(\angle\)OCA.

21. In triangle ABC, point O is located inside the triangle such that OA = OB and \(\angle\)AOB = 2\(\angle\)ACB. If a circle is drawn with center O and radius equal to OA, then point C will lie on the circle.

22. A circle is drawn with diameter \(AB\) and center \(O\). From a point \(P\) on the circle, a perpendicular is drawn to the diameter \(AB\), intersecting it at point \(N\). Prove that: \[ PB^2 = AB \cdot BN \]

23. Diameter PQ of a circle with radius 4 cm and angle ∠PRQ is an angle in the semicircle. Given QR = \(2\sqrt{7}\) cm, find the length of PR = _____

24. In a circle with center \( O \), \( AB \) and \( CD \) are two equal-length chords. \( E \) is the midpoint of \( CD \), and \( \angle AOB = 70^\circ \). The value of angle \( \angle COE \) is:

(a) 70° (b) 110° (c) 35° (d) 55°

25. In a circle with center \(O\), \(AB\) is a diameter. \(P\) is any point on the circumference. If \(\angle POA = 120°\), then the measurement of \(\angle PBO\) is –

(a) 30° (b) 60° (c) 90° (d) 120°

26. 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

27. 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

28. 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°

29. If the mean of a statistical distribution is 4.1, \(∑f_i x_i = 132 + 5k\), and \(∑f_i = 20\), then what is the value of \(k\)?

30. AB and CD are two chords of a circle with center O, and both have equal lengths. If \(\angle\)AOB = 60°, then what is the measure of \(\angle\)COD?