International Mathematics (0607) | Circle theorems I

Welcome to Circle Theorems I: Unlocking the Geometry of Circles!

Hello! This chapter might seem daunting because of all the new rules, but don't worry. Circle theorems are powerful shortcuts that allow you to find missing angles in complex diagrams almost instantly, provided you know the rules!

We are going to break down the essential properties of circles (Geometry topic C5.6 and E5.6) so you can tackle any angle problem with confidence. Remember, in exams, you usually need to not just find the angle, but also state the correct geometric reason. Let's master these reasons!

Section 1: Essential Circle Vocabulary (Quick Review)

Before diving into the theorems, let's refresh some key terms you need to know:

• Centre (O): The middle point, equidistant from all points on the circumference.
• Radius (r): A line segment from the centre to the circumference.
• Diameter (d): A chord passing through the centre (\(d = 2r\)).
• Chord: A line segment connecting two points on the circumference.
• Tangent: A straight line that touches the circle at only one point.
• Arc: A part of the circumference (can be minor or major).
• Semicircle: Half of a circle, formed by the diameter.
• Segment: The area enclosed by an arc and a chord.
• Sector: The area enclosed by two radii and an arc.

Key Takeaway: Understanding these terms is the foundation for correctly applying the theorems.

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Section 2: The Core Angle Theorems (The Basics)

Theorem 1: The Angle between a Tangent and a Radius

This is one of the simplest, yet most useful, theorems.

Rule: The angle between a radius and a tangent at the point of contact is always \(90^\circ\).

Analogy: Imagine a bicycle wheel (the circle) meeting the ground (the tangent line). The spoke (radius) that touches the ground must be perfectly straight up, forming a right angle with the flat ground.

Reason to state in the exam: "Angle between tangent and radius = \(90^\circ\)."

Theorem 2: Angle in a Semicircle

If you draw a triangle inside a circle where one side is the diameter, you automatically know one angle!

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

• If a triangle is inscribed in a semicircle (with the diameter as its base), the angle opposite the diameter is \(90^\circ\).

Reason to state in the exam: "Angle in a semicircle = \(90^\circ\)."

Did you know? This is sometimes called Thales' Theorem, named after the ancient Greek philosopher and mathematician Thales of Miletus.

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Section 3: Angles from the Centre and Circumference

This theorem establishes a crucial relationship between the angle created at the circle's centre and the angle created at the circumference, both using the same arc.

Theorem 3: Angle at the Centre

Rule: The angle subtended by an arc at the centre is twice the angle subtended by the same arc at any point on the circumference.

• If the angle at the centre is \(2x\), the angle at the circumference is \(x\).
• Make sure both angles are created by the exact same pair of points (the same arc).

Memory Aid: Think of the centre as the "boss" or the "big angle" who has double the power (double the degrees) of the "worker" or "little angle" sitting on the edge.

Common Mistake to Avoid: Ensure the circumference angle you use is *not* in the major segment if the centre angle is in the minor segment (and vice versa). Check that both angles are subtended by the same arc.

Reason to state in the exam: "Angle at centre is twice angle at circumference."

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Section 4: Angles Formed by a Single Arc

Theorem 4: Angles in the Same Segment (The Butterfly Theorem)

If two or more angles are drawn from the same chord (or arc) and end up on the same part of the circumference, they must be equal.

Rule: Angles subtended by the same arc (or chord) in the same segment are equal.

Visualization: Imagine the chord as the base of a seesaw. If you pick up two angles using that base, no matter where they touch the circumference (in that segment), they will be identical. The resulting shape often looks like a butterfly or a bowtie.

Reason to state in the exam: "Angles in the same segment are equal."

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Section 5: Cyclic Quadrilaterals

Definition: Cyclic Quadrilateral

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

Theorem 5: Opposite Angles of a Cyclic Quadrilateral

Rule: The opposite angles of a cyclic quadrilateral sum to \(180^\circ\) (they are supplementary).

• If the vertices are A, B, C, D in order around the circle, then \(\angle A + \angle C = 180^\circ\) and \(\angle B + \angle D = 180^\circ\).

Analogy: Think of a rubber band stretched tightly around four pins on a circle. The opposing tension always pulls the opposite angles to total \(180^\circ\).

Reason to state in the exam: "Opposite angles of a cyclic quadrilateral sum to \(180^\circ\)."

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Section 6: The Tangent-Chord Relationship

This is often considered the trickiest theorem, so take your time and follow the specific steps needed to identify the angle pair.

Theorem 6: Alternate Segment Theorem

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

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

Step-by-Step Identification:

1. Identify the tangent line.
2. Identify the chord that touches the tangent (at point P).
3. Look at the angle formed outside the chord (the Tangent-Chord Angle, T).
4. The angle equal to T is the one made by the chord in the segment opposite the Tangent-Chord Angle. This angle is always part of the triangle drawn inside the circle.

Mnemonics: If you draw the two equal angles, they often form a shape resembling a person wearing a big hat, or sometimes an arrowhead pointing toward the tangent.

Reason to state in the exam: "Alternate segment theorem."

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Section 7: Circle Symmetry Properties (Circle Theorems II Summary)

The syllabus also requires knowledge of symmetry properties (E5.7). While not technically "angle theorems," these rules are vital for finding lengths and establishing right angles within circle geometry problems.

Property 7A: Perpendicular Bisector of a Chord

Rule: The perpendicular line drawn from the centre of the circle to a chord bisects (cuts exactly in half) the chord.
Conversely, the perpendicular bisector of a chord passes through the centre.

Key Use: When the centre is connected to the midpoint of a chord, a \(90^\circ\) angle is formed, allowing you to use Pythagoras' theorem or trigonometry.

Property 7B: Tangents from an External Point

Rule: The two tangents drawn to a circle from the same external point are equal in length.

Key Use: This often creates an isosceles triangle (and sometimes a kite shape) outside the circle, helping you find unknown lengths or angles.

Property 7C: Equal Chords

Rule: Equal chords are equidistant (the same distance) from the centre.

Key Use: If you know two chords have the same length, their shortest distance to the centre (which is the perpendicular distance) will be equal.