Circle theorems worksheet — GCSE Higher

1.PT is a tangent to a circle with centre O, touching the circle at T. C is a point on the circle such that C lies inside angle OTP (the right angle between the radius OT and the tangent PT), and OC is a radius. Angle TOC = 130°. Work out the size of angle PTC, the angle between the tangent PT and the chord TC.

2.From an external point P, two tangents PA and PB touch a circle with centre O at A and B. Angle AOB = 112°. Work out the size of angle APB.

3.A, B and C are points on a circle with centre O. B is on the major arc AC. Angle AOC = 116°. Work out the size of angle ABC.

4.A, B and C are points on a circle with centre O. B is on the minor arc AC. Angle ABC = 100°. Work out the size of the non-reflex angle AOC.

5.To prove that the angle at the centre is twice the angle at the circumference, a student draws radii OA and OC, where A and C are points on the circle and O is the centre. The student's first step is: 'Since OA = OC, both being radii of the circle, triangle OAC is isosceles, so angle OAC = angle OCA.' Is this first step correct?

6.A, B, C and D are points on a circle, with B and D both on the same major arc AC. E lies on the straight line through A and B, beyond B, so that angle CBE = 145°. Work out the size of angle ADC.

7.ABCD is a cyclic quadrilateral, with its vertices in that order around the circle. Angle DAB = 108°. Work out the size of angle BCD.

8.A, B, C and D are points on a circle, and ABCD is a cyclic quadrilateral with the vertices in that order around the circle. Angle ABC = 108° and angle ADC = 72°. Which circle theorem is the reason that angle ABC + angle ADC = 180°?

9.A skateboard ramp is designed so that a straight rail RT is tangent to a circular curve at the point T, and TC is a chord of the curve from T to another point C on the curve. A second rail continues straight through T on the other side, so that the two rails together form a straight line, and the angle between that second rail and the chord TC is 109°. A support point D is placed on the curve, on the major arc TC. Work out the size of angle TDC, the angle subtended by the chord TC at D.

10.Points A, B, C and D lie on a circle with centre O, in that order around the circle, and AC is a diameter. Which of the following circle facts does NOT apply to this diagram?

11.From an external point P, two tangents PA and PB touch a circle with centre O at the points A and B. Angle APB = 50°. Work out the size of angle OAB.

12.A circle with centre O has radius 17 cm. A chord AB is drawn so that the perpendicular distance from O to AB is 8 cm. Work out the length of the chord AB.

13.PT is a tangent to a circle with centre O, touching the circle at T. OT is a radius. Angle OPT = 27°, where P is a point outside the circle. Work out the size of angle POT.

14.A, B and C are points on a circle with centre O, and AC is a diameter of the circle. Work out the size of angle ABC.

15.From an external point P, two tangents PA and PB touch a circle with centre O at A and B. Angle APB = 40°. Work out the size of angle AOB.