MathematicsExam code: 0580

Circle Theorems

Geometry

PractiseFlashcards
  • 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

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

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