Circle theorems I

1. Overview

Circle theorems are fundamental geometric principles that govern the relationships between angles, lines, and arcs within a circle. They provide the tools to calculate unknown angles and solve geometric problems involving circles. This revision note covers the essential circle theorems required for the IGCSE Cambridge Mathematics (0580) syllabus, including both Core and Extended curriculum topics. Understanding and applying these theorems is crucial for success in geometry questions.

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Key Definitions

• Tangent: A straight line that touches the circumference of a circle at exactly one point.
• Chord: A straight line segment joining two points on the circumference.
• Diameter: A chord that passes through the center of the circle.
• Radius: The distance from the center to any point on the circumference (half the diameter).
• Arc: A part of the circumference.
• Segment: An area of a circle bounded by a chord and an arc.
• Cyclic Quadrilateral: A four-sided shape where all four vertices (corners) lie on the circumference of a circle.
• Subtended: An angle "created" by an arc or chord at a specific point (either the center or the circumference).

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Core Content

Angle in a Semicircle

The angle subtended at the circumference by a diameter is always 90°. Any triangle drawn using the diameter as its base and a third point on the circumference will be a right-angled triangle.

Worked example 1 — Angle in a semicircle

Question: In a circle with center $O$, $AB$ is the diameter. Point $C$ lies on the circumference. If angle $CAB = 35^\circ$, find angle $CBA$.

1. Identify the diameter: $AB$ passes through $O$.
2. Apply theorem: Angle $ACB = 90^\circ$ (angle in a semicircle).
3. Sum of angles in a triangle: $180^\circ - 90^\circ - 35^\circ = 55^\circ$.
4. Final Answer: Angle $CBA = 55^\circ$.

Worked example 2 — Finding an angle using semicircle property

Question: A circle has diameter $PQ$. Point $R$ lies on the circumference such that $\angle RPQ = 62^\circ$. Calculate the size of $\angle RQP$.

1. $\angle PRQ = 90^\circ$ (Angle in a semicircle)
2. $\angle RPQ + \angle PRQ + \angle RQP = 180^\circ$ (Angles in a triangle)
3. $62^\circ + 90^\circ + \angle RQP = 180^\circ$ (Substitute known values)
4. $152^\circ + \angle RQP = 180^\circ$
5. $\angle RQP = 180^\circ - 152^\circ$ (Subtract $152^\circ$ from both sides)
6. $\angle RQP = 28^\circ$
7. Final Answer: $\angle RQP = 28^\circ$

Tangent and Radius

The angle between a tangent and the radius at the point of contact is 90°.

Worked example 3 — Tangent and radius

Question: A tangent $TP$ touches a circle at $P$. The radius of the circle is $OP$. A line connects the center $O$ to a point $T$ outside the circle. If angle $OTP = 20^\circ$, find angle $POT$.

1. Apply theorem: Angle $OPT = 90^\circ$ (tangent is perpendicular to radius).
2. Sum of angles in triangle $OPT = 180^\circ$.
3. Calculation: $180^\circ - (90^\circ + 20^\circ) = 70^\circ$.
4. Final Answer: Angle $POT = 70^\circ$.

Worked example 4 — Combining tangent/radius with isosceles triangle

Question: $AB$ is a tangent to a circle at point $B$. $O$ is the center of the circle, and $OB = 5$ cm. $OA = 13$ cm. Calculate the length of $AB$.

1. $\angle OBA = 90^\circ$ (Tangent meets radius at $90^\circ$)
2. Triangle $OBA$ is a right-angled triangle.
3. $OA^2 = OB^2 + AB^2$ (Pythagoras' Theorem)
4. $13^2 = 5^2 + AB^2$ (Substitute known values)
5. $169 = 25 + AB^2$
6. $AB^2 = 169 - 25$ (Subtract 25 from both sides)
7. $AB^2 = 144$
8. $AB = \sqrt{144}$ (Take the square root of both sides)
9. $AB = 12$ cm
10. Final Answer: $AB = 12$ cm

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Extended Content (Extended Curriculum Only)

Angle at the Centre

The angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at the circumference.

Worked example 5 — Angle at the centre

Question: Points $A$, $B$, and $C$ lie on a circle with center $O$. If angle $ABC = 48^\circ$, find the reflex angle $AOC$.

1. Angle at center is $2 \times$ angle at circumference.
2. Obtuse angle $AOC = 48^\circ \times 2 = 96^\circ$.
3. Reflex angle $AOC = 360^\circ - 96^\circ = 264^\circ$.
4. Final Answer: $264^\circ$.

Worked example 6 — Combining angle at centre with isosceles triangle

Question: Points $P$ and $Q$ lie on a circle with center $O$. $\angle OPQ = 24^\circ$. Find the angle subtended by the arc $PQ$ at the center of the circle, $\angle POQ$.

1. $OP = OQ$ (Both are radii of the circle)
2. Triangle $OPQ$ is isosceles.
3. $\angle OQP = \angle OPQ = 24^\circ$ (Base angles of an isosceles triangle are equal)
4. $\angle POQ + \angle OPQ + \angle OQP = 180^\circ$ (Angles in a triangle)
5. $\angle POQ + 24^\circ + 24^\circ = 180^\circ$
6. $\angle POQ + 48^\circ = 180^\circ$
7. $\angle POQ = 180^\circ - 48^\circ$ (Subtract $48^\circ$ from both sides)
8. $\angle POQ = 132^\circ$
9. Final Answer: $\angle POQ = 132^\circ$

Angles in the Same Segment

Angles subtended by the same arc (or chord) at the circumference are equal. This is often called the "Bow-tie" theorem.

Cyclic Quadrilaterals

The opposite angles in a cyclic quadrilateral sum to 180° (they are supplementary).

Worked example 7 — Cyclic quadrilateral

Question: $ABCD$ is a cyclic quadrilateral. If angle $DAB = 110^\circ$ and angle $ABC = 85^\circ$, find angles $BCD$ and $ADC$.

1. Opposite angles sum to $180^\circ$.
2. Angle $BCD = 180^\circ - 110^\circ = 70^\circ$.
3. Angle $ADC = 180^\circ - 85^\circ = 95^\circ$.
4. Final Answer: Angle $BCD = 70^\circ$ and Angle $ADC = 95^\circ$.

Alternate Segment Theorem

The angle between a tangent and a chord is equal to the angle in the alternate segment.

Worked example 8 — Alternate segment theorem

Question: A tangent $PT$ touches a circle at point $A$. Chord $AB$ is drawn. $\angle BAT = 52^\circ$. Point $C$ lies on the circumference such that $\angle ACB$ is subtended by chord $AB$. Find $\angle ACB$.

1. $\angle ACB = \angle BAT$ (Alternate Segment Theorem)
2. $\angle ACB = 52^\circ$
3. Final Answer: $\angle ACB = 52^\circ$

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Key Equations

• Angle in semicircle: $\angle = 90^\circ$
• Radius $\perp$ Tangent: $\angle = 90^\circ$
• Angle at centre: $\angle_{centre} = 2 \times \angle_{circumference}$
• Opposite angles of cyclic quad: $\angle_1 + \angle_2 = 180^\circ$
• Sum of angles around a point: $\sum \angle = 360^\circ$

Note: These formulas are not provided on the IGCSE formula sheet. You must memorize the theorems and the specific geometric language used to describe them.

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Common Mistakes to Avoid

• ❌ Wrong: Assuming that any four-sided shape inside a circle is a cyclic quadrilateral. ✓ Right: Verify that all four vertices of the quadrilateral lie exactly on the circumference before applying the rule that opposite angles sum to $180^\circ$.
• ❌ Wrong: Forgetting to double-check whether you need the reflex angle at the center. ✓ Right: Read the question carefully. If it asks for the reflex angle, remember to subtract the acute/obtuse angle you initially calculated from $360^\circ$.
• ❌ Wrong: Assuming that if a line looks like a tangent, it is definitely a tangent. ✓ Right: Only use the "tangent meets radius at $90^\circ$" theorem if the question explicitly states that the line is a tangent, or if you are asked to prove that it is a tangent.
• ❌ Wrong: Confusing the "angle at the center" theorem with the "angles in the same segment" theorem. ✓ Right: The "angle at the center" theorem relates an angle at the center to an angle at the circumference subtended by the same arc. "Angles in the same segment" relates two angles at the circumference subtended by the same arc.

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Exam Tips

• Command Words: If the question says "Give a reason for your answer," you must write the name of the theorem (e.g., "angles in the same segment are equal"). You will lose marks even if your calculation is correct.
• The Isosceles Trap: Look for triangles formed by two radii. These are always isosceles triangles, meaning the two angles at the circumference are equal. This is the most common "hidden" step in IGCSE circle questions.
• Calculator vs Non-Calculator: In non-calculator papers, angles are often multiples of $15, 30, 45,$ or $60$. If you get $37.42^\circ$ in a non-calculator section, re-check your subtraction!
• Multi-step logic: Often, you need to use "Angles on a straight line = $180^\circ$" or "Angles in a triangle = $180^\circ$" in combination with circle theorems. Work through the diagram step-by-step, labeling every angle you find.