0580

Circle Theorems

Geometry

  • Most questions need two or three facts chained together, so work outwards from whatever is given
  • Fill in every angle you can and label each one with its reason as you go
A circle with five points A to E on it joined by several chords, with angles of 12, 14 and 73 degrees marked at E and the angle theta at B to be found, which needs more than one theorem chained together
Source: Circles & segments by Save My Exams
  • A pair of tangents drawn from one outside point, together with the two radii, boxes in a quadrilateral whose angles total 360°, and two of those are the right angles where the tangents touch
Two tangents drawn from an outside point T touching a circle of centre O at R and S, the two tangent lengths TR and TS marked equal, and the radii OR and OS drawn in to complete the quadrilateral ORTS
Source: Circles & chords by Save My Exams
Exam tip

Write the theorem out in full words. Mark schemes are generous about wording — they accept "tangent and radius are perpendicular" and allow "sum" for "add" — but they reject abbreviations outright, naming "AST", "alt segment theorem" and "alternative segment theorem" as unacceptable. Where an answer takes two steps, give both reasons. Draw in the radius whenever you see a tangent, and mark equal radii to expose the isosceles triangles.

Worked example

Two tangents and a quadrilateral

Two tangents from an external point T touch a circle, centre O, at R and S. Angle RTS is 25°. Find the angle θ at the centre, giving reasons.

Solution:

  • OR and OS are radii drawn to the points where the tangents touch
  • A tangent meets a radius at 90°, so angle ORT = angle OST = 90°
  • ORTS is a quadrilateral, and the angles of a quadrilateral add to 360°
Two tangents from a point T touching a circle of centre O at R and S, with the right angles at R and S marked and the 25 degree angle at T shown, so the four angles of ORTS add to 360 and give the angle theta at the centre
Source: Circles & chords by Save My Exams
  • So θ = 360 − 90 − 90 − 25
  • θ = 155°
  • Check: 155 + 90 + 90 + 25 = 360