1. An equilateral triangle where each side is 6 cm. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the radius of the circumcircle. [Only drawing symbols required]

2. From a point outside a circle, a tangent of length 8 cm is drawn to the circle. If the radius of the circle is 6 cm, what is the distance from the center of the circle to the external point?

(a) 10 cm (b) 8 cm (c) 6 cm (d) 14 cm

3. 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°

4. 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°

5. In triangle ABC, the circumcenter is O. If \(\angle\)BAC = 85° and \(\angle\)BCA = 75°, then what is the measure of \(\angle\)OAC?

(a) 70° (b) 40° (c) 110° (d) 140°

6. In triangle ABC, the incenter is O. If \(\angle\)BOC = 120°, then what is the measure of \(\angle\)BAC?

(a) 60° (b) 90° (c) 45° (d) 120°

7. AB is a diameter of a circle with center O. CD is a chord whose length is equal to the radius of the circle. When AC and BD are extended, they intersect at point P. What is the measure of ∠APB?

8. 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 \(\angle PTA\)?

9. In triangle \(\triangle ABC\), O is the circumcenter. If \(\angle OAB = 35^\circ\), then what is the measure of \(\angle ACB\)?

(a) 45° (b) 50° (c) 55° (d) 60°

10. In a circle with center O, triangle ABC is inscribed. If \(\angle\)BAC = 85° and \(\angle\)BCA = 75°, then find the measure of \(\angle\)AOC.

11. In triangle ABC, if O is the orthocenter and ∠BOC = 120°, then what is the measure of ∠BAC?

(a) 80° (b) 60° (c) 90° (d) 75°

12. In triangle ABC, the circumcenter is \(O\). If \(\angle\)BAC = 85° and \(\angle\)BCA = 75°, then what is the measure of \(\angle\)AOC?

13. In triangle ABC, the incenter is O, and \(\angle\)BOC = 120°. What is the measure of \(\angle\)BAC?

14. A right-angled triangle where the hypotenuse is 12 cm and one of the other sides is 5 cm. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the radius of the circumcircle. [Only drawing symbols required]

15. Draw a triangle where one side is 6.7 cm and the two adjacent angles are 75° and 55°. — Draw the triangle and then draw its circumcircle. Mark the position of the circumcenter and measure and write the length of the circumradius (i.e., the radius of the circumcircle). [Only drawing symbols required]

16. "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

17. What is the radius of the incircle of an equilateral triangle whose each side is \(2a\)?

(a) \(\sqrt{3a}\) cm (b) \(\cfrac{a}{\sqrt3}\) cm (c) \(\cfrac{a}{\sqrt2}\) cm (d) \(\sqrt{2a}\) cm

18. Two circles intersect at points A and B, and the circumference of each passes through the center of the other. If a straight line drawn through point A intersects the two circles again at points C and D respectively, then what type of triangle is BCD?

(a) right-angled isosceles (b) scalene (c) isosceles (d) equilateral

19. If the circumradius of an equilateral triangle is 4 cm, what is its inradius?

(a) 2 cm (b) 4 cm (c) 1 cm (d) 3 cm

20. In the cyclic quadrilateral PQRS, the side PS is a diameter of the circle. If \(\angle\)PQR = 120°, then what is the measure of \(\angle\)SPR?

(a) 90° (b) 30° (c) 60° (d) 120°

21. In triangle ABC, AB = AC. A line drawn from point C intersects the extended line BA at point D. If AC = AD, what is the measure of \(\angle\)BCD in radians?

(a) \(\cfrac{π}{2}\) (b) \(π\) (c) \(\cfrac{π}{4}\) (d) \(\cfrac{π}{3}\)

22. In the cyclic quadrilateral PORS, the side PS is a diameter of the circle. If \(\angle\)PQR = 128°, then what is the measure of \(\angle\)SPR?

(a) 30° (b) 38° (c) 60° (d) None of the above

23. O is the circumcenter of triangle ABC. If \(\angle\)BAC = 85° and \(\angle\)BCA = 70°, then what is the measure of \(\angle\)OAC?

(a) \(65^o\) (b) \(42\cfrac{1}{2}^o\) (c) \(50^o\) (d) \(25^o\)

24. In triangle ABC, the incenter is I. When the internal bisector of ∠A (i.e., AI) is extended, it intersects the circumcircle at point P. If PB = 15 cm, then what is the length of PI?

(a) 5 cm (b) 15 cm (c) 10 cm (d) 20 cm

25. 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

26. If the orthocenter, centroid, circumcenter, and incenter of a triangle all lie at the same point, then what type of triangle is it?

(a) isosceles (b) scalene (c) equilateral (d) right-angled isosceles

27. 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°

28. From point A, which is located 26 cm away from the center O of a circle, a tangent is drawn to the circle with a length of 10 cm. Find the radius of the circle.

29. If the sum of a central angle and an inscribed angle of a circle is 180°, then what is the measure of the central angle?

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?