A tangent is a straight line that touches a circle at exactly one point. This lesson covers the three essential theorems you need to solve geometry problems involving tangents.

1. The Three Core Theorems of Tangents

Theorem 1: Tangent and Radius are Perpendicular

• What it means: The tangent to a circle is always perpendicular (forms a 90° angle) to the radius at the point where it touches the circle (the point of contact).

• Why it's important: This is the most fundamental tangent rule. Whenever you see a radius meeting a tangent, you can immediately mark a 90° angle. This often allows you to use Pythagoras' Theorem or trigonometry.

Theorem 2: Tangents from an External Point

• What it means: If you draw two tangents to a circle from a single point outside the circle, then:

  1. The lengths of the two tangents (from the external point to the point of contact) are equal.

  2. The line joining the external point to the centre of the circle bisects the angle between the two tangents.

  3. The two tangents subtend (create) equal angles at the centre of the circle.

• Key takeaway: This setup creates two congruent right-angled triangles, which is a very common starting point for proofs.

Theorem 3: The Alternate Segment Theorem

• What it means: The angle between a tangent and a chord drawn through the point of contact is equal to the angle in the alternate (opposite) segment.

• How to spot it:

  1. Find a tangent and a chord that meet at the same point on the circle.

  2. The angle between them is equal to any angle formed by that chord at the circumference in the other part of the circle.

• This is a powerful shortcut for relating angles inside the circle to the tangent outside.

2. Strategy for Solving Problems (Riders)

1. Identify Tangents and Radii: Immediately mark any 90° angles where a radius meets a tangent. This is your most reliable starting point.

2. Look for Tangents from an External Point: If you see this pattern, you know two lengths are equal (creating an isosceles triangle) and some angles are bisected.

3. Check for the Alternate Segment Theorem: If there's a triangle inside the circle with one vertex at the point of contact, you can almost certainly use this theorem to relate the angles.

4. Combine with Other Geometry Rules: Tangent problems are almost never solved with just one rule. You will need to use other theorems you know:

   • Cyclic Quadrilateral theorems.

   • Angle at the centre is twice the angle at the circumference.

   • Angles in the same segment.

   • Properties of parallel lines.

   • Sum of angles in a triangle is 180°.

Exam Tips & Common Pitfalls

• Pitfall 1: Forgetting the 90° Angle. The first thing you should look for is a radius meeting a tangent. Not marking the right angle is the most common missed opportunity.

• Pitfall 2: Misidentifying the Alternate Segment. Be careful to match the angle between the tangent and the chord with the angle in the correct opposite segment.

• Exam Tip 1: Two Tangents = Isosceles Triangle. When two tangents meet at an external point, connecting the two points of contact on the circle forms an isosceles triangle. This is a very common pattern in exams.

• Exam Tip 2: Chain of Reasoning. Start with the most obvious fact (like a 90° angle), use it to find a new angle or side, and then use that new piece of information to apply another theorem. Write down every step and your reason clearly (e.g., "Radius ⊥ Tangent").