Use circle symmetry and equal tangents
Use equal tangent lengths and chord symmetry to find lengths and locate a circle's centre.
Section 5 · Geometry
Use this pathway for Cambridge IGCSE International Mathematics 0607, 2028–2030. Core and Extended objectives are labelled in each lesson; Extended adds requirements beyond Core. Paper 1 and Paper 2 are non-calculator. Papers 3–6 require a graphic display calculator (GDC). These original lessons use self-checked practice; they are not a Cambridge grade, full timed paper or teacher review.
Before you begin
- Check whether your school has entered you for Core or Extended. Core and Extended share most Number ideas, but some notation and methods are Extended only.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Where do the equal tangent segments begin?
What you will learn
- Use equal tangents from one external point
- Relate equal chords to equal distances from the centre
- Locate the centre with a chord's perpendicular bisector
Tangents from one external point match
Two tangents drawn from the same point outside a circle have equal lengths. The contact points and external point form an isosceles triangle.
Core focuses on equal tangent lengths; Extended includes this with two chord-symmetry properties.
Equal chords share central geometry
Equal chords are equidistant from the centre. The perpendicular bisector of a chord passes through the centre; two such bisectors can locate it.
A radius perpendicular to a chord bisects it. Use a right triangle to calculate chord length or distance from the centre.
Pause and explain
Where does a chord's perpendicular bisector pass?
Worked example
Tangents from external point P touch a circle at A and B. If PA = 7.5 cm, find PB.
Show the worked solution
- PA and PB start at the same external point.
- Tangents from the same external point are equal.
- Therefore PB = 7.5 cm.
Answer PB = 7.5 cm.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A chord is 16 cm long and 6 cm from the centre. Find the radius.
Give me a hint
The perpendicular from the centre bisects the chord.
Compare my reasoning
- Half the chord is 8 cm.
- Radius² = 6² + 8² = 100.
- The radius is 10 cm.
10 cm.
Look for these in your work
- Chord halved
- Pythagoras used
- Positive radius selected
Two equal chords lie in the same circle. The first is 5 cm from the centre. Find the second chord's distance and name the theorem.
Give me a hint
Use the equal-chords symmetry property.
Compare my reasoning
- The chords have equal length.
- Equal chords in a circle are equidistant from the centre.
- The second chord is 5 cm from the centre.
5 cm; equal chords are equidistant from the centre.
Look for these in your work
- Given equality recognised
- Circle property named
- Distance transferred
A surveyor wants to locate a circular fountain's centre from two straight chord edges. Describe a construction using a right-angle tool.
Give me a hint
The perpendicular bisector of each chord passes through the centre.
Compare my reasoning
- Mark the midpoint of each chord.
- Draw a perpendicular bisector through each midpoint.
- Their intersection locates the centre, subject to accurate construction.
Draw two chord perpendicular bisectors; their intersection is the centre.
Look for these in your work
- Midpoints constructed
- Right angles used
- Intersection interpreted
Common mistakes
- Comparing tangents from different external points
- Assuming unrelated chords have equal lengths
- Forgetting the perpendicular bisector passes through the centre
- Confusing chord distance with chord length
What can you explain now?
Two tangents from P have lengths 9 cm and x cm. Find x.
Compare with the explanation
9 cm
Tangents from one external point are equal.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
How do equal tangent lengths and perpendicular chord bisectors reveal circle symmetry?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.