The Math Factory

1. In triangle ∆ABO, the circumcenter is C; If \(\angle\)CAB=50° then what is \(\angle\)AOB =? (a) 50° (b) 100° (c) 40° (d) 80°
2. 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
3. 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°
4. 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°
5. In a circle centered at point O, AB is a diameter. If chord AC creates a 60° angle at the center, then the measure of \(\angle\)OCB will be: (a) 20° (b) 30° (c) 40° (d) 50°
6. If \(O\) is the circumcenter of the equilateral triangle \(ABC\), the value of \(\angle BOC\) will be - (a) 100° (b) 120° (c) 140° (d) 60°
7. If \(O\) is the circumcenter of \(∆ABC\) and \(\angle\)OAB = 50°, then \(\angle\)ACB = (a) 50° (b) 100° (c) 40° (d) 80°
8. In a circle centered at O, PQ is a diameter; R is a point on the circle, and PR = RQ. Then, the measure of \(\angle\)RPQ is: ____? (a) 30° (b) 90° (c) 60° (d) 45°
9. 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°
10. 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°
11. 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°
12. The equilateral triangle XYZ is inscribed in a circle. If O is the center of the circle, what is the measure of \(\angle\)YOZ? (a) 60° (b) 30° (c) 90° (d) 120°
13. G is the centroid of equilateral triangle ABC; if AB = 10 cm, then what is the length of AG? (a) \(10\sqrt3\) cm (b) \(\cfrac{10}{3}\) cm (c) \(10\) cm (d) \(\cfrac{10\sqrt3}{3}\) cm
14. 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°
15. AB is a diameter of a circle with center O, and P is a point on the circle. If ∠POA = 120°, then the measure of ∠PBO is — (a) 30° (b) 45° (c) 60° (d) 90°
16. An angle subtended by a segment smaller than a semicircle is an obtuse angle. True / False
17. 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?
18. In triangle ∆ABC, O is the circumcenter and ∠BAC = 50°. Find the measure of ∠OBC.
19. In triangle \( \triangle ABC \), if \( \angle ABC = 90^\circ \), \( AB = 6 \) cm, and \( BC = 8 \) cm, then what is the length of the circumradius of triangle \( \triangle ABC \)?
20. Prove that the central angle subtended by an arc of a circle is twice any inscribed angle subtended by the same arc.
21. ABC is an equilateral triangle inscribed in a circle. If P is any point on the arc BC opposite to vertex A, then prove that \(AP = BP + CP\).
22. Given: In triangle △ABC, O is the circumcenter and OD ⊥ BC. Prove that: ∠BOD = ∠BAC Let’s break it down in English: **Given:** In triangle △ABC, O is the circumcenter (the point where the perpendicular bisectors of the sides meet), and OD is perpendicular to side BC. **To Prove:** The angle ∠BOD formed at the center between points B and D is equal to the angle ∠BAC at the vertex A. This is a classic geometry result based on the properties of a circle and triangle. Would you like me to walk you through the full proof in English as well?
23. Prove that the angle subtended at the center of any circle by an arc is double the angle subtended by the same arc at any point on the remaining part of the circle.
24. Given: In triangle ABC, O is the circumcenter and OD ⊥ BC. Prove: \(\angle\)BOD = \(\angle\)BAC.
25. ABCD is a cyclic quadrilateral where AB = DC. Prove that: AC = BD.
26. Prove that the angle subtended by an arc at the center of a circle is double the angle subtended by the same arc at any point on the remaining part of the circumference.
27. In triangle ABC, the circumcenter is O. It is given that \(\angle\)BAC = 85° and \(\angle\)BCA = 55°. Find the value of \(\angle\)OAC.
28. Prove that all angles subtended by the same arc at the circumference of a circle are equal.
29. The central angle subtended by an arc of a circle is twice the inscribed angle subtended by the same arc—prove it.
30. Prove that the angle subtended at the center by an arc of a circle is twice the angle subtended by the same arc at any point on the circle.
31. AB and CD are two chords of a circle with center O. When extended, they intersect at point P. Prove that \(\angle AOC - \angle BOD = 2\angle BPC\).
32. In triangle ABC, \(\angle\)ABC is a right angle, and AB = 5 cm, BC = 12 cm. What is the radius of the circumcircle of triangle ABC?
33. 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.
34. 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.
35. The angle in a segment smaller than a semicircle is _____.
36. Prove that all angles subtended by the same arc of a circle are equal.
37. "Angle in a semicircle is a right angle — prove it."
38. The angle in a circular segment larger than a semicircle is an obtuse angle.
39. AB and CD are two chords of a circle with center O. OM and ON are perpendiculars drawn from the center to the chords such that OM = ON. Prove that AB = CD.
40. Prove that the central angle subtended by an arc of a circle is twice the inscribed angle subtended by the same arc.
41. 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.
42. The measure of an angle subtended by a semicircle is _____.
43. Two circles intersect each other at points P and Q. If PA and PB are the diameters of the respective circles, then prove that the points A, Q, and B lie on a straight line.
44. Determine the value of \( x \), where \( O \) is the center of the circle and \( AB \) is the diameter of the circle.
45. Prove that the central angle subtended by the same arc is twice the inscribed angle.
46. Prove that the central angle subtended by an arc of a circle is twice the inscribed angle subtended by the same arc.
47. In triangle ABC, AB = AC. Let E be any point on the extension of BC. The circumcircle of triangle ABC intersects line AE at point D. Prove that \[ \angle ACD = \angle AEC \]
48. In a circle, two chords AB and AC are perpendicular to each other. If AB = 4 cm and AC = 3 cm, find the radius of the circle.
49. AOB is a diameter of a circle. C is a point on the circle. If ∠OBC = 60°, then find the value of ∠OCA.
50. Prove that the central angle subtended by an arc of a circle is twice any inscribed angle subtended by the same arc.
51. The central angle formed by an arc of a circle is twice any inscribed angle formed by the same arc—prove it.