GCSE MATHS • HIGHER TIER
Tangent Theorems Made Simple
Learn how to use tangent properties to find missing angles and lengths, recognise tangent diagrams, and solve algebraic GCSE circle-theorem questions.
Clear method
Exam practice
Worked proofs
Tangent Theorems in Four Moves
1
Identify
Identify the tangent and the point where it touches the circle.
2
Check
Look for a radius or a second tangent from the same external point.
3
Apply
Use the correct tangent theorem.
4
Solve
Find the missing angle, length or algebraic value.
What You’ll Learn
Understand Tangents
Understand that a tangent touches a circle at exactly one point.
Use the Radius-Tangent Rule
Recognise that a radius is perpendicular to a tangent at the point of contact.
Use Equal Tangents
Know that two tangents from the same external point are equal in length.
Solve Algebraic Questions
Form and solve equations using equal tangent lengths.
Connect Circle Theorems
Use tangent rules alongside other GCSE circle theorems.
Key Rules
Tangent Theorem 1: Radius and Tangent
The radius of a circle is always perpendicular to the tangent at the point of contact.
Angle between a radius and a tangent = 90°
Tangent Theorem 2: Two Tangents from the Same Point
If two tangents are drawn from the same external point, they are equal in length..
Tangent length 1 = Tangent length 2
Worked Proof: Arc Length
Step-by-Step Method
- 1. Identify the tangent.
- 2. Look for a radius or a second tangent.
- 3. Recall the correct theorem.
- 4. Write the angle or equality rule.
- 5. Solve any calculations or algebra.
- 6. Check the answer is sensible.
More Worked Examples
1
Question: A radius meets a tangent. Find the angle between them.
A radius is perpendicular to a tangent at the point of contact.
2
Question: An angle between a radius and a tangent is labelled x.
x = 90°
3
Question: One tangent from an external point measures 8 cm.
The second tangent must also measure 8 cm.
4
Question: One tangent from an external point measures 12.5 cm.
The second tangent has the same length.
- Common GCSE Mistakes
- 90° rule: Remember the radius-tangent angle is always 90°.
- Equal tangents: Two tangents from the same external point have equal lengths.
- Tangent vs chord: A tangent touches the circle only once.
- Point of contact: Identify where the tangent touches the circle.
- Algebra errors: Set equal tangent lengths correctly.
- Exam Tips
- Spot the tangent first: It is the main clue.
- Check for a radius: Use the 90° rule at the contact point.
- Check for two tangents: Their lengths are equal.
- State the theorem: Name the rule in reasoning questions.
- Show algebra clearly: Write the equality before solving.
Skills Used in Tangent Theorems
Alternate Segment Theorem
Angles at the Centre
Angles in the Same Segment
Cyclic Quadrilaterals
Radius and Tangent Theorem
Circle Geometry
Ready to Practise?
Printable Worksheet
Download and print practice questions with answers.
Interactive Quiz
Test your knowledge with instant feedback and hints.
Exam-Style Questions
Timed questions to build confidence for the real exam.
Watch the Tangent Theorems Tutorial
Step-by-step walkthroughs of Tangent Theorems with clear methods and exam tips.
Frequently Asked Questions
A tangent is a straight line that touches a circle at exactly one point.
A radius is perpendicular to the tangent at the point of contact, so the angle is 90°.
The two tangent lengths are equal.
Look for a line touching a circle once, a radius to the contact point, or two tangents from one external point.
Yes. Equal tangent lengths can be used to form and solve equations.
They commonly link with the Alternate Segment Theorem, Angles at the Centre, Angles in the Same Segment and Cyclic Quadrilaterals.
