Mathematics (0580) | Circle theorems I

Geometric Mastery: Unlocking the Secrets of Circle Theorems I

Hello future mathematicians! Circles are everywhere—from the tires on your bike to the shape of the Earth seen from space. In IGCSE Mathematics, understanding how angles behave inside and around circles is crucial.

This chapter, Circle Theorems I, introduces fundamental rules that allow you to calculate unknown angles with confidence. These rules are powerful shortcuts! Don't worry if geometry seems tricky; we will break down each theorem into simple, memorable steps. Remember, you must always state the correct theorem (the geometrical reason) when calculating angles in the exam!

Part 1: Essential Circle Vocabulary (The Basics)

Before diving into the theorems, let's quickly review the key parts of a circle, which are essential for understanding where the angles are located:

• Centre: The middle point, usually labelled O.
• Radius: A line segment from the centre to the circumference.
• Diameter: A chord passing through the centre (equal to two radii).
• Circumference: The perimeter or distance around the circle.
• Chord: A straight line connecting two points on the circumference.
• Tangent: A straight line that touches the circle at exactly one point.
• Arc: A part of the circumference.
• Segment: The area enclosed by an arc and a chord.
• Sector: The area enclosed by two radii and an arc (like a slice of pizza).

Part 2: Core Circle Theorems (C5.6)

1. The Angle in a Semicircle Theorem

This is one of the easiest and most important rules to remember!

Theorem: The angle subtended by a diameter at any point on the circumference is a right angle (\(90^{\circ}\)).

Reason for Exam: Angle in a semicircle is \(90^{\circ}\).

Analogy: Imagine the diameter is the bottom edge of a ruler. If you place the corner of a set square anywhere on the circumference and keep the edges on the diameter ends, that corner will always be \(90^{\circ}\).

If AB is the diameter and C is any point on the circumference, then the angle \(ACB = 90^{\circ}\).

2. The Tangent and Radius Theorem

A tangent is a line that just kisses the circle at one point.

Theorem: The radius (or diameter) drawn to the point of contact with a tangent is perpendicular to the tangent.

Reason for Exam: Angle between tangent and radius is \(90^{\circ}\).

Step-by-Step Check:

1. Identify the tangent line.
2. Identify the single point where it touches the circle (the point of contact).
3. Draw a radius to that point.
4. The angle formed between the radius and the tangent is always \(90^{\circ}\).

Common Mistake to Avoid: This theorem only applies to the radius at the exact point of contact. If the radius is drawn to a different point, the angle is NOT \(90^{\circ}\).

Part 3: Extended Circle Theorems (E5.6)

These theorems relate angles within the circle itself, usually generated by arcs or chords.

3. Angle at the Centre and Circumference Theorem

This theorem links an angle measured from the centre to an angle measured from the circumference, both subtended by the same arc.

Theorem: The angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at any point on the circumference.

Reason for Exam: Angle at centre is twice angle at circumference.

Memory Aid (The Boss Rule): The angle at the Centre (the 'Boss' of the circle) is always twice as big as the angle at the Circumference (the 'Worker').

If arc AB subtends angle \(AOB\) at the centre and angle \(ACB\) at the circumference, then \(AOB = 2 \times ACB\).

Did You Know? This theorem is often combined with isosceles triangles (since radii are equal), making it possible to calculate multiple angles from just one known value.

4. Angles in the Same Segment Theorem

If you have a fixed chord (or arc), all the angles drawn from that chord to the circumference are identical.

Theorem: Angles subtended by the same arc (or chord) in the same segment of a circle are equal.

Reason for Exam: Angles in the same segment are equal.

Memory Aid (The Bowtie Rule): Draw a chord. If two triangles share this chord as their base and their third vertices both lie on the rest of the circumference, the angles at those third vertices are equal. They look like the ends of a bowtie or butterfly wings!

If points C and D are on the major arc subtended by chord AB, then angle \(ACB = ADB\).

5. Cyclic Quadrilateral Theorem

A cyclic quadrilateral is any four-sided shape (quadrilateral) where all four vertices lie exactly on the circumference of the circle.

Theorem: Opposite angles in a cyclic quadrilateral are supplementary (they sum to \(180^{\circ}\)).

Reason for Exam: Opposite angles of a cyclic quadrilateral sum to \(180^{\circ}\) (or supplementary angles in a cyclic quad).

If ABCD is a cyclic quadrilateral:

• Angle \(A + \text{Angle } C = 180^{\circ}\)
• Angle \(B + \text{Angle } D = 180^{\circ}\)

Accessibility Tip: If the question involves a quadrilateral, first check if all four corners touch the circumference. If they do, use the \(180^{\circ}\) rule! If even one corner is slightly off the circumference, the rule does not apply.

6. The Alternate Segment Theorem (Often the Trickiest!)

This theorem links an angle formed by a tangent and a chord to an angle inside the opposite triangle formed by that chord.

Theorem: The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

Reason for Exam: Alternate Segment Theorem.

Step-by-Step Trick (The Pointer/Tongue):

1. Identify the point where the tangent touches the circle (P).
2. Identify the chord starting at P (e.g., chord PQ).
3. Look at the angle formed between the tangent and the chord (e.g., angle \(TPQ\)).
4. This angle is equal to the angle in the triangle formed by the chord, but on the other side (the alternate segment).

Example: If the angle between the tangent and chord AB is \(65^{\circ}\), then the angle at the circumference (let's say angle C) subtended by chord AB is also \(65^{\circ}\).

Part 4: Synthesis and Problem Solving

Using Multiple Theorems

Most IGCSE exam questions require using two or three theorems together. Always calculate angles step-by-step, stating the reason for each new angle you find.

Step-by-Step Example Strategy

Suppose you are asked to find angle X in a diagram that includes a tangent, a chord, and a cyclic quadrilateral.

1. Examine the Diagram: Look for clear visual cues: Is there a diameter? A tangent? Are all points on the circumference?
2. Identify the Easiest Angle: Use the \(90^{\circ}\) rules first (Semicircle or Tangent-Radius). This gives you a foundation.
3. Apply Relationships: Use the Center/Circumference rule to relate angles if they share an arc.
4. Use \(180^{\circ}\) Rules: If you have a cyclic quadrilateral or angles on a straight line, use the supplementary rules.
5. Link Concepts: Use the Alternate Segment Theorem to bridge the gap between angles outside the circle (tangent) and angles inside the circle.

Common Pitfalls to Avoid

• Mixing up Center/Circumference: Always check which angle is bigger! The angle at the circumference is always half the angle at the center.
• Non-Cyclic Quadrilaterals: Do NOT assume opposite angles sum to \(180^{\circ}\) unless all four vertices are strictly on the circumference.
• Forgetting Reasons: In a calculation question, you might lose marks if you calculate correctly but fail to write down the geometric reason (e.g., "Angle at centre is twice angle at circumference").
• Isosceles Triangle Trap: Remember that any triangle formed by two radii is automatically an isosceles triangle, meaning two sides (the radii) are equal, and therefore the base angles are also equal. This is a common hidden step in circle theorem problems!

You’ve covered all the angle rules for Circle Theorems I! Practice makes perfect—try several examples where you have to apply multiple reasons in sequence. You've got this!