International Mathematics (0607) | Circle theorems II

Study Notes: Geometry – Circle Theorems II (IGCSE 0607 Extended)

Welcome to Circle Theorems II! If Circle Theorems I (C5.6) was the warm-up, this section is where we dive into the full power of circle geometry. These theorems are essential for finding unknown angles and lengths in complex diagrams. The key to success here is not just knowing the rules, but being able to *name them* correctly when providing reasons for your answers.

Don't worry if the diagrams look intimidating. Just break them down, look for the center, tangents, and cyclic shapes, and the solution will reveal itself!

Quick Review: Essential Circle Vocabulary

Before we start, remember these crucial terms (from E5.1):

• Radius: A line segment from the center to the circumference.
• Chord: A line segment connecting two points on the circumference.
• Diameter: The longest chord; passes through the center (\(d = 2r\)).
• Tangent: A line that touches the circle at exactly one point.
• Arc: A part of the circumference.
• Segment: The area enclosed by a chord and an arc.

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Section 1: Angle Theorems – The Basics (Review of C5.6)

These two fundamental theorems connect the circle's properties to right angles:

1. Angle in a Semicircle (\(90^\circ\) Theorem)

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

Think of it this way: If a triangle is drawn inside a circle and its longest side (hypotenuse) is the diameter, the corner opposite the diameter must be \(90^\circ\).

Reason required: Angle in a semicircle = \(90^\circ\).

2. Angle between Tangent and Radius (\(90^\circ\) Theorem)

Theorem: A tangent is always perpendicular (\(90^\circ\)) to the radius (or diameter) at the point of contact.

Analogy: Imagine a tire (the circle) meeting the road (the tangent). The spoke (radius) going straight down to the contact point must be vertical (perpendicular) to the horizontal road.

Reason required: Angle between tangent and radius = \(90^\circ\).

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Section 2: Extended Angle Theorems (E5.6)

These theorems govern how angles relate to each other when they share the same arc.

3. Angle at Centre is Twice Angle at Circumference

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

If the angle at the center is \(\theta\), the angle at the circumference is \(\frac{1}{2}\theta\). If the angle at the circumference is \(\alpha\), the angle at the center is \(2\alpha\).

Common Mistake: Students sometimes confuse which angle is the center and which is the circumference. The angle at the center is always the larger one.

Reason required: Angle at centre is twice angle at circumference.

4. Angles in the Same Segment are Equal

Theorem: Angles subtended by the same arc (or chord) at the circumference are equal.

Analogy: If you have two different observers on the edge of a pond (circumference) looking at the same piece of shore (arc), they measure the same angle of sight.

Reason required: Angles in the same segment (or subtended by the same arc) are equal.

5. Properties of a Cyclic Quadrilateral

A Cyclic Quadrilateral is a four-sided shape where all four vertices (corners) lie on the circumference of the circle.

A. Opposite Angles

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

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

Reason required: Opposite angles of a cyclic quadrilateral sum to \(180^\circ\).

B. Exterior Angle

Did you know? An additional property is that the exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

6. The Alternate Segment Theorem (AST)

This is often the most challenging theorem, but extremely powerful. It connects a tangent, a chord, and an angle inside the circle.

Theorem: 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 Trick:

1. Identify the point where the chord meets the tangent.
2. Look at the angle formed by the tangent and the chord (e.g., \(\angle PQR\)).
3. Look at the segment *opposite* that angle. The angle subtended by the chord in that opposite segment is equal to the tangent angle.

Reason required: Alternate segment theorem.

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Section 3: Symmetry and Length Theorems (E5.7)

These theorems rely on symmetry and are often used to find unknown lengths or to prove geometric relationships.

7. Tangents from an External Point

Theorem: If two tangents are drawn to a circle from the same external point, they are equal in length.

If point P is outside the circle, and tangents meet the circle at A and B, then the length of PA must equal the length of PB.

Analogy: This theorem creates a perfect kite shape (where the radii meet the tangents at \(90^\circ\)). Since the two tangent lengths are equal, you often end up with congruent triangles and can use Pythagoras' theorem.

Reason required: Tangents from an external point are equal in length.

8. Perpendicular Bisector of a Chord

Theorem: The perpendicular bisector of a chord passes through the centre of the circle.

This is crucial for constructions or finding the centre of a circle given a chord.

Reason required: Perpendicular bisector of a chord passes through the centre.

9. Equal Chords are Equidistant

Theorem: In the same circle (or in congruent circles), equal chords are equidistant (equal distance) from the centre.

If chord AB = chord CD, then the perpendicular distance from the centre O to AB is equal to the perpendicular distance from the center O to CD.

Remember: When finding the distance from the centre to a chord, that line segment must be perpendicular to the chord. This line segment also bisects the chord (cuts it into two equal halves). This allows you to use Pythagoras' theorem heavily!

Reason required: Equal chords are equidistant from the centre.

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Practice Strategies and Common Pitfalls

Step-by-Step Problem Solving

When approaching a circle geometry problem, follow this mental checklist:

1. Identify Key Features: Does the diagram show the Centre (O)? Are there any Diameters? Are there any Tangents?
2. Look for Isosceles Triangles: Any triangle formed by two radii (Radius-Radius-Chord) must be an Isosceles Triangle. This means the base angles are equal. This is a common starting point!
3. Check for \(90^\circ\) Angles: Use Theorem 1 (Semicircle) and Theorem 2 (Tangent/Radius).
4. Scan the Circumference: Look for pairs of angles subtended by the same arc (Theorem 4) or cyclic quadrilaterals (Theorem 5).
5. Incorporate Lengths: If lengths are involved, check Theorem 7 (External Tangents) or Theorem 9 (Equal Chords).

⚠️ Common Mistake to Avoid

Assuming that a line passing through a chord is the perpendicular bisector just because it looks like it! You must be told that the line is perpendicular OR that it bisects the chord OR that it passes through the centre to use the related theorems.