Proof of the Circle Theorems
Proof of the circle theorems means using known facts, such as radii being equal and angle sums of 180 degrees, to derive a circle theorem from scratch rather than just applying it to find a missing angle. GCSE Higher papers occasionally ask for a full proof, most often of the angle at the centre theorem or the angle in a semicircle theorem.
Method
- Mark all radii on the diagram, and note that any two radii are equal in length, so a triangle formed using two radii and a chord is isosceles.
- Label the base angles of each isosceles triangle with a letter, such as x or y, since the base angles of an isosceles triangle are equal.
- Use the fact that angles in a triangle sum to 180 degrees to find the third angle of each isosceles triangle in terms of your letters.
- Use the exterior angle rule, or angles on a straight line or around a point, to link the angles together.
- Combine the relationships algebraically to reach the statement you are proving, writing each step of reasoning clearly.
- Finish with a concluding sentence that states exactly what has been proved, referencing the original claim.
Worked example
O is the centre of a circle. A and B are points on the circumference, and C is a point on the major arc AB. Prove that angle AOB = 2 x angle ACB.
- Draw the radius from O through C, extending it to meet the circumference at a point D on the far side, so that it splits angle AOB into two parts.
- Since OA = OC (both radii), triangle OAC is isosceles, so let angle OAC = angle OCA = x.
- The exterior angle of triangle OAC at O is angle AOD, so angle AOD = angle OAC + angle OCA = 2x, using the exterior angle rule.
- Similarly, since OB = OC, triangle OBC is isosceles; let angle OBC = angle OCB = y, so angle BOD = 2y.
- Angle AOB = angle AOD + angle BOD = 2x + 2y = 2(x + y).
- Final answer: since angle ACB = angle OCA + angle OCB = x + y, angle AOB = 2(x + y) = 2 x angle ACB, as required.
Practice questions
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Exam-style questions
Written in the style of a GCSE Maths exam paper, with a full mark scheme.
O is the centre of a circle. A and B are points on the circumference such that OA = OB. Angle OAB = 38 degrees. (a) State the size of angle OBA, giving a reason. (b) Calculate the size of angle AOB.
A circle has centre O and radius 13 cm. A chord AB has length 24 cm. M is the midpoint of AB, and OM is perpendicular to AB. (a) Calculate the length of OM, giving a reason for your method. (b) Hence calculate the size of angle AOB, giving your answer correct to 1 decimal place.
O is the centre of a circle. P, Q, R and S are points on the circumference such that PQRS is a cyclic quadrilateral (in that order), with Q and S on opposite arcs formed by chord PR. The reflex angle POR = 236 degrees. Calculate the size of angle PQR, giving a reason for your answer.
Free printable worksheet
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