Welcome to Circle Theorems II: Mastering the Extended Geometry!

Hello! If you’ve made it to Circle Theorems II, congratulations! You’ve already covered the basics, and now we are diving into the powerful properties that make solving complex circle geometry problems fun (yes, really!).

This chapter focuses on the theorems needed for the Extended syllabus (E5.6 and E5.7). These rules let you find almost any missing angle or length inside a circle. Don't worry if these look tricky at first; we'll break them down step-by-step. Mastering these is crucial, especially since Cambridge expects you to quote the correct geometrical reasons for your answers!

Section 1: Core Foundation (Quick Review)

Let's quickly refresh the absolute basics, as you’ll need these constants for the harder theorems.

1. Angle in a Semicircle

If you draw a triangle inside a circle where one side is the diameter, the angle opposite the diameter (the angle on the circumference) is always a right angle.

  • Property: The angle subtended by a diameter at any point on the circumference is \(90^\circ\).
  • Reason (Required): Angle in a semicircle is \(90^\circ\).

2. Tangent and Radius Property

A tangent is a straight line that touches the circle at exactly one point. The radius drawn to this point of contact is special.

  • Property: The radius (or diameter) is perpendicular to the tangent at the point of contact.
  • Reason (Required): Angle between tangent and radius is \(90^\circ\).
Quick Review Key Takeaway

The rules \(90^\circ\) and right angles are always your first tools when dealing with diameters and tangents!

Section 2: Angles Subtended by Arcs and Chords (E5.6)

These theorems relate the size of an angle to the arc or chord that “creates” it. Think of the arc as the “stage” and the angles as the “viewers.”

3. Angle at the Centre vs. Angle at the Circumference

When an arc (or chord) creates an angle at the centre (O) and another angle at the circumference (P), there is a fixed relationship.

  • Property: The angle subtended by an arc at the centre is twice the angle subtended by the same arc at any point on the remaining part of the circumference.
  • Formula: Angle at centre \( = 2 \times \) Angle at circumference.
  • Reason (Required): Angle at centre is twice angle at circumference.

Analogy: Imagine a piece of pizza (the arc). If you measure the angle at the very centre point, it will be twice as wide as the angle measured from the crust (the circumference).

4. Angles in the Same Segment

If two angles are formed by the same arc/chord and both have their vertices on the circumference, they must be equal.

  • Property: Angles subtended by the same arc (or chord) in the same segment of a circle are equal.
  • Reason (Required): Angles in the same segment are equal.

Common Mistake to Avoid: This rule only works if the angles are "looking at" the exact same arc. Be careful not to confuse this with the Centre/Circumference rule.

5. Cyclic Quadrilaterals

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

  • Property: The opposite angles of a cyclic quadrilateral sum to \(180^\circ\) (they are supplementary).
  • Formula: \(A + C = 180^\circ\) and \(B + D = 180^\circ\)
  • Reason (Required): Opposite angles of a cyclic quadrilateral sum to \(180^\circ\).
Did you know?

You can sometimes prove that a quadrilateral is cyclic if you can show that one pair of opposite angles sums to \(180^\circ\). This often comes up in proving questions!

Quick Review Key Takeaway

These three theorems (Centre/Circumference, Same Segment, Cyclic Quad) are the foundation of angle solving in circles. Always look for the arc that generates the angle!

Section 3: The Alternate Segment Theorem (E5.6)

This is often considered the trickiest theorem, but once you spot the shape, it’s straightforward!

6. Alternate Segment Theorem (The “Pigtail” Rule)

This theorem connects an angle formed by a tangent and a chord to an angle deep inside the circle.

  • Setup: You need a tangent touching the circle and a chord starting from the point of contact.
  • The Angle: Measure the angle between the tangent and the chord (e.g., angle BAT).
  • The Alternate Segment: This is the angle in the triangle created by the chord, in the segment opposite (alternate) to the angle you measured.
  • Property: The angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.
  • Reason (Required): Alternate segment theorem.

Step-by-Step Spotting Guide:

  1. Find the tangent line (L).
  2. Find the point of contact (T).
  3. Identify a chord (TC) starting at T.
  4. Look at the angle formed by L and TC (e.g., Angle ATC).
  5. The equal angle is formed by the remaining two ends of the chord (A and C) and any point (P) on the circumference in the *other* segment (Angle APC).
Quick Review Key Takeaway

The Alternate Segment Theorem is your go-to when a tangent and a chord meet at the same point. The angle outside equals the angle inside, opposite the chord.

Section 4: Symmetry and Chord Properties (E5.7)

These theorems deal with the relationship between chords, the centre, and the radius, relying heavily on symmetry.

7. Perpendicular Bisector of a Chord

This is a fundamental symmetry property relating the centre of the circle to any chord.

  • Property 7a: The perpendicular from the centre to a chord bisects (cuts in half) the chord.
  • Property 7b: The perpendicular bisector of a chord always passes through the centre of the circle (O).
  • Reason (Required): Perpendicular from centre to chord bisects chord, OR Perpendicular bisector of a chord passes through the centre.

Why this matters: If you draw a radius (or line from the centre) to the midpoint of a chord, you automatically create a \(90^\circ\) angle, forming a right-angled triangle. This allows you to use Pythagoras' Theorem (\(a^2 + b^2 = c^2\)) to find unknown lengths!

8. Equal Chords

If you have multiple chords in the same circle, symmetry tells us something about their distance from the centre.

  • Property 8a: Equal chords are equidistant (the same distance) from the centre.
  • Property 8b: Chords equidistant from the centre are equal in length.
  • Reason (Required): Equal chords are equidistant from the centre.

9. Tangents from an External Point

If you pick any point outside the circle and draw two tangents from that point, they stop being equal exactly where they touch the circle.

  • Property: Tangents drawn to a circle from an external point are equal in length.
  • Reason (Required): Tangents from an external point are equal in length.

Bonus consequence: Because the radii to the points of contact are \(90^\circ\) (Theorem 2), drawing radii from the centre (O) to the points of contact (A and B) and a line from the external point (P) back to O creates two congruent right-angled triangles (OAP and OBP). This is often used to calculate angles or solve problems using trigonometry.

Quick Review Key Takeaway

Symmetry theorems are often about lengths and involve Pythagoras’ theorem. Look for chords and radii to form right-angled triangles.

Final Checklist: Essential Reasons for Exam Success

In all geometry questions, especially those involving circles, you must state the geometric rule used to justify your calculation. Here is a simplified list of acceptable reasons based on the IGCSE curriculum:

  1. Angle in a semicircle is \(90^\circ\).
  2. Angle between tangent and radius is \(90^\circ\).
  3. Angle at centre is twice angle at circumference.
  4. Angles in the same segment are equal.
  5. Opposite angles of a cyclic quadrilateral sum to \(180^\circ\).
  6. Alternate segment theorem.
  7. Tangents from an external point are equal in length.
  8. Perpendicular from centre to chord bisects chord.

Remember: You may also need to use standard angle rules like “Angles on a straight line sum to \(180^\circ\),” “Vertically opposite angles are equal,” or “Angle sum of a triangle is \(180^\circ\).” Always keep these basic geometric facts ready!

A Word of Encouragement

Circle theorems are about patterns! The more diagrams you practice, the faster you will recognize which rule to apply. If you get stuck, look for the ‘special lines’: the diameter, the centre, and the tangents. These are almost always clues to a specific theorem!

Keep practising, and you will nail this topic!