A circle theorem proof is a chain of steps, each with a reason. Radii are equal, so triangles with two radii are isosceles; angles in a triangle add to 180∘.
Join the centre to the points to make isosceles triangles.
Label the base angles with letters such as x and y.
Add up the angles and show the result in terms of those letters.
Watch out Give a reason for every step: missing reasons lose the proof marks.
See it worked
Worked exampleNon-calculator·4 marks
AB is a diameter of a circle with centre O. C is any point on the circle other than A or B. Join OC.
(a) Prove that angle ACB is a right angle without quoting the angle-in-a-semicircle theorem.
(b) Why may the two base-angle pairs be taken equal?
(a) Use OA=OC and OB=OC to establish both pairs of equal base angles.
Write the full triangle angle sum using x and y. x+y+(x+y)=180
Let OAC=OCA=x and OBC=OCB=y, using equal radii. Then triangle ABC has angles x, y and x+y. Its angle sum gives 2x+2y=180, so ACB=x+y=90 degrees.
(b) Each relevant triangle has two radii as equal sides, so it is isosceles.
Answer (a) Let OAC=OCA=x and OBC=OCB=y, using equal radii. Then triangle ABC has angles x, y and x+y. Its angle sum gives 2x+2y=180, so ACB=x+y=90 degrees.; (b) Each relevant triangle has two radii as equal sides, so it is isosceles.
Your turn
Question 1Non-calculator·5 marks
PT and PU are tangents from an external point P to a circle with centre O. T and U are the points of contact.
(a) Prove that PT = PU.
(b) If OP = 15 cm and the radius is 9 cm, find PT.
Write your answer, then show the working
(a) Identify the two right angles from the radius-tangent theorem.
Use the equal radii and common hypotenuse OP for RHS congruence.
Therefore OT=OU as radii, OP is common, and OT is perpendicular to PT while OU is perpendicular to PU. Right triangles OTP and OUP are congruent by RHS, so PT=PU.
(b) Apply Pythagoras in the radius-tangent right triangle. 152−92
Therefore 12 cm.
Answer (a) OT=OU as radii, OP is common, and OT is perpendicular to PT while OU is perpendicular to PU. Right triangles OTP and OUP are congruent by RHS, so PT=PU.; (b) 12 cm
Question 2Non-calculator·4 marks
TA is a tangent to a circle at A.AB is a chord and C is a point on the major arc AB (the alternate segment). T and C are on opposite sides of the line AB. The diameter through A meets the circle again at D.
Prove that angle TAB = angle ACB (the alternate segment theorem).
Write your answer, then show the working
Angle TAD=90∘: the tangent is perpendicular to the radius (diameter) at A.
So angle TAB=90∘− angle BAD.
Angle ABD=90∘: the angle in a semicircle (AD is a diameter).
In triangle ABD: angle ADB=180−90− angle BAD=90∘− angle BAD. So angle TAB = angle ADB.
Angle ADB = angle ACB: angles in the same segment (both stand on the chord AB).
A circle has centre O and radius 10 cm. P is outside the circle with OP = 26 cm. Tangents from P touch the circle at T and U. Work out the exact length of chord TU. You must show your working.
In past papers
13 official questions in 13 papers since 2017. The latest:
