Ch 4 · 4.7–4.8 Circle Theorems · IGCSE Mathematics 0580
Topic 4 of 4 in Geometry — IGCSE Mathematics 0580 — 38 lessons.
Introduction
Circles hide a surprising number of fixed angle relationships. Six theorems and three symmetry properties let you find almost any angle in a circle diagram, but the marks depend on quoting the right reason in the syllabus wording. (Syllabus sections 4.7 and 4.8.)
Key ideas — The angle theorems
🧑🏫 Teacher's explanation
Think of a circle not just as a shape, but as a perfectly balanced system of symmetry. Every point on the edge is exactly the same distance from the middle, which creates rigid rules for how lines and angles must behave. If you change one part of the circle, the angles elsewhere must adjust to maintain that perfect balance.
📖 Definitions
🔍 Worked example — Easy
A triangle ABC is inscribed in a circle where AC is the diameter. Calculate the size of ∠ ABC.
🔍 Worked example — Standard
In a circle with centre O, an angle at the centre ∠ AOB is 110^°. Find the angle ∠ ACB at the circumference, where C is a point on the major arc.
🔍 Worked example — Exam-style
A quadrilateral ABCD is inscribed in a circle. If ∠ ABC = 85^°, find the size of the opposite angle ∠ ADC.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Symmetry properties
🧑🏫 Teacher's explanation
Think of a circle as the most perfectly balanced shape possible. Because every point on the edge is exactly the same distance from the center, any line drawn through the circle (a chord) creates a mirror-image relationship with the center. If you draw a line that cuts a chord exactly in half at a 90^° angle, that line is essentially a 'symmetry axis' for that part of the circle, meaning it must point directly to the center.
📖 Definitions
🔍 Worked example — Easy
A chord of length 8 cm is drawn in a circle. A perpendicular line from the center of the circle bisects the chord. If the distance from the center to the chord is 3 cm, calculate the radius of the circle.
🔍 Worked example — Standard
Two tangents are drawn from an external point P to a circle with center O. The length of the tangent segment PA is 12 cm. Calculate the length of the tangent segment PB.
🔍 Worked example — Exam-style
A chord AB has length 24 cm. The perpendicular distance from the center O to the chord is 5 cm. Find the length of the radius OA.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Method
Mark known angles on the diagram, look for radii (isosceles triangles), diameters, tangents and cyclic quadrilaterals, and write one reason for each step.
🧑🏫 Teacher's explanation
Imagine you are looking at a circular clock face. If you draw a line from the 12 to the 6, you've drawn a diameter. If you then draw a line from the 12 to the 3, you've created a right angle at the edge of the circle. Circle theorems are simply the 'rules of the game' that tell you exactly how much space is left in those angles based on where the lines are drawn.
📖 Definitions
🔍 Worked example — Easy
A triangle is inscribed in a semicircle such that one side is the diameter. Calculate the angle at the circumference subtended by the diameter.
🔍 Worked example — Standard
In a circle, a cyclic quadrilateral ABCD has ∠ ABC = 85^°. Calculate ∠ ADC.
🔍 Worked example — Exam-style
A tangent XY touches a circle at point A. A radius OA is drawn. If ∠ XAO = 90^°, and a chord AB is drawn such that ∠ XAB = 40^°, find the angle ∠ ACB where C is a point on the major arc.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
A, B and P lie on a circle with centre O, and angle APB = 38^°. Angle AOB = 76^° (angle at the centre is twice the angle at the circumference). Triangle AOB is isosceles (OA = OB, radii), so angle OAB = tfrac12(180 - 76) = 52^°.
🔍 Worked example — Easy
A, B and P lie on a circle with centre O. Given that ∠ APB = 38^°, calculate ∠ AOB.
🔍 Worked example — Standard
ABCD is a cyclic quadrilateral. Given ∠ ABC = 104^° and ∠ BCD = 85^°, find ∠ ADC and ∠ DAB.
🔍 Worked example — Exam-style
Tangents from point T touch a circle with centre O at points A and B. Given ∠ ATB = 50^°, find ∠ AOB and ∠ TAB.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) angle ACB = 90^° (angle in a semicircle), so 55^°. (2) 4x + 20 = 180 ⇒ x = 40^°. (3) 58^° (alternate segment theorem). (4) the perpendicular from the centre bisects the chord: r = √(8^2 + 6^2) = 10 cm.