Circle Theorems

9.A, B, C are points on a circle, centre O. Angle ABC = 50°. Find the obtuse angle AOC.

10.AB and DE are chords of a circle, centre O. M is midpoint of AB, N is midpoint of DE. AB = DE = 9 cm, OM = 5 cm. Find ON.

11.In a circle, points A, B, C, D lie on the circle with centre O, and FDE is a tangent. Using circle theorems, the marked angles satisfy y − x = 90°. If y = 200°, find x.

12.A, B and C are points on a circle, centre O. Angle AOB = 80°. What is angle ACB?

13.In a circle, a chord of length 10 cm is 12 cm from the centre. What is the radius of the circle?

14.A, B, C, D are points on a circle. Angle ABC = 70°. What is angle ADC?

15.A, B, C are points on a circle, centre O. Angle AOC = 140°. Find the obtuse angle ABC.

16.Tangents from an external point are equal in length.

17.In a circle, the angle at the centre is 80°. What is the angle at the circumference subtended by the same arc?

18.Which of the following statements are true? (Select all that apply)

19.Match each theorem to its description.

20.Arrange the following steps in order to prove that the angle in a semicircle is 90° using the angle at centre theorem.

21.Match each circle theorem to its correct description.

22.Complete the sentence.

23.A chord of length 10 cm is 12 cm from the centre of a circle. Find the radius of the circle in cm.

24.State the circle theorem that relates the angle between a tangent and a chord to the angle in the alternate segment.

25.In a circle, two tangents from an external point P touch the circle at Q and R. The distance from P to the centre O is 13 cm and the radius is 5 cm. Find the length of PQ in cm.

26.In a circle, two tangents from an external point P touch the circle at Q and R. The distance from P to the centre O is 13 cm and the radius is 5 cm. Find the length of PQ in cm.

27.P, Q and R are points on the circumference of a circle, centre O. PO is parallel to QR and angle POQ = 48°. Find angle OPR.

28.A, B and C are points on a circle with centre O. Which of the following statements are true? (Select all that apply)

29.In the diagram, TA is a tangent at A, and O is the centre. OA = 5 cm, TA = 12 cm. Find the length OT in cm.

30.In the diagram, TA is a tangent at A, and O is the centre. OA = 5 cm, TA = 12 cm. Find the length OT in cm.

31.ABCD is a cyclic quadrilateral, with all four vertices on a circle centred at O, as shown. Angle DAB = 95°. Find angle BCD, the angle opposite angle DAB.

32.WXYZ is a cyclic quadrilateral, with its vertices in that order around the circle. Angle ZWX = 3k° and angle XYZ = (2k + 10)°, where W and Y are opposite vertices. Complete the sentence.

33.O is the centre of a circle, and A, B, C, D lie on the circle in that order around its circumference, with B on the major arc AC. Angle AOC = (7x + 45)° and angle ABC = 5x°. Given that ABCD is a cyclic quadrilateral, find angle ADC, the angle opposite angle ABC.

34.ABCD is a cyclic quadrilateral. Side AB is extended beyond B to a point E, forming the exterior angle CBE. Angle CBE = 68°. Using the exterior angle property of a cyclic quadrilateral together with the opposite-angles property, find angle ABC.

35.The following steps prove that opposite angles of a cyclic quadrilateral sum to 180°, using the angle at centre theorem. Arrange them in the correct logical order.

36.A tangent touches a circle at point A. AB is a chord of the circle, and the angle between the tangent and the chord AB is 62°. C is a point on the circle in the alternate segment. Find angle ACB.

37.The alternate segment theorem states that the angle between a tangent and a chord equals the angle subtended by that chord in the segment on the SAME side of the chord as the tangent line.

38.PT is a tangent to a circle at point P. PQ is a chord, and the angle between the tangent PT and the chord PQ is 64°. R is a point on the circle in the alternate segment, and angle PQR = 55°. Find angle QPR.

39.A tangent line touches a circle at point X, with A and B two points on the tangent line either side of X (so A, X, B lie in a straight line). Chords XY and XZ are drawn from X into the circle. The angle between the tangent ray XA and the chord XY is 48°, and the angle between the tangent ray XB and the chord XZ is 55°. Find angle YXZ, the angle between the two chords.

40.O is the centre of a circle. A and B lie on the circle, and OA, OB are radii. Angle OAB = 35°. C lies on the major arc AB. Find angle ACB.

41.O is the centre of a circle. P and Q lie on the circle, and R lies on the minor arc PQ (the shorter arc between P and Q). The reflex angle POQ, measured through the major arc, is 250°. Find angle PRQ.

42.AB is a diameter of a circle with centre O, and C is a point on the circle. Angle ACB = 90°. Which of the following correctly explain why angle ACB = 90°? Select all that apply.

43.Match each theorem to its description.

44.A, B, P and Q lie on a circle, with P and Q both on the major arc AB. Angle APB = 43°. Find angle AQB.

45.O is the centre of a circle. A and B lie on the circle, and C and D both lie on the major arc AB. Angle AOB = 100°. Find angle ADB.

46.A chord AB of a circle has length 24 cm. M is the midpoint of AB, and the centre O is joined to M. Given that OM = 5 cm, find the radius of the circle.