Use chord symmetry and equal external tangents
Locate a chord midpoint using a perpendicular from the center · Relate equal chord lengths to equal center distances · Use equal tangents from one external point
Exam scope and assessment notes
Extended content; not a Core requirement. Historical study of Cambridge IGCSE Mathematics (US) 0444, 2025 Version 2. The final assessment was June 2025; the syllabus was available only to schools in Arizona, USA. This is not a current exam-entry option. Original self-check teaching and explained practice are not Cambridge papers, official marks or predicted grades. Linked teaching foundations do not establish full syllabus mastery. Calibrated cumulative assessment, reliable drawn-work review and qualified mathematics-teacher review remain unfinished.
Before you begin
- Perpendicular bisectors
- Right-triangle relation a²+b²=c²
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
What should you establish before applying the method in this lesson?
What you will learn
- Locate a chord midpoint using a perpendicular from the center
- Relate equal chord lengths to equal center distances
- Use equal tangents from one external point
Chord and center
A chord's perpendicular bisector passes through the circle center. A perpendicular from the center to the chord meets its midpoint. If the radius is R, half-chord a and perpendicular center distance d, the right triangle gives a²+d² = R².
Within the same circle, or congruent circles, equal chords have equal perpendicular center distances. That statement cannot be transferred to circles with different radii. For example, an 8-unit chord has half-length 4, so its center distance is 3 in a radius-5 circle but √84 in a radius-10 circle.
One point, two tangents
Tangents from one external point P to a circle have equal lengths. Both contact triangles share OP, have the same radius and a right angle at contact, giving equal tangent lengths. The points must refer to the same external point and circle.
If a chord passes through the center, it is a diameter with midpoint O; draw the perpendicular through O rather than attempting to define a line from O to itself. Keep a named reason alongside each calculation.
Sources for this lesson
Pause and explain
Do equal chords in circles of different radii necessarily have equal center distances?
Worked example
A radius-10 circle has a chord of length 12. Find its perpendicular center distance.
Show the worked solution
- The perpendicular bisects the chord into two lengths 6.
- d² = 10²−6² = 64, so d = 8.
Answer 8 units.
Try a topic test
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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 radius-13 circle has a chord of length 10. Find its perpendicular center distance.
Give me a hint
Use a half-chord of 5.
Compare my reasoning
- d² = 13²−5² = 144.
- d = 12.
12 units.
Look for these in your work
- Bisect the chord first
- Use the positive distance
A radius-10 circle has a chord 6 units from the center. Find the full chord length.
Give me a hint
Solve for a half-chord, then double.
Compare my reasoning
- a² = 100−36 = 64, so a = 8.
- Full length = 16.
16 units.
Look for these in your work
- Distinguish half and full length
- Check the radius triangle
Two tangents from one external point to one circle have lengths x+3 and 2x−1. Find x and the common length.
Give me a hint
Set the tangent lengths equal.
Compare my reasoning
- x+3 = 2x−1 gives x = 4.
- Both lengths are 7, positive as required.
x = 4; both tangents have length 7.
Look for these in your work
- Use the shared-point/shared-circle condition
- Check positive lengths
Common mistakes
- Use half the chord in the right triangle.
- Equal-chord distance comparisons need the same radius.
What can you explain now?
Do equal chords in circles of different radii necessarily have equal center distances?
Compare with the explanation
No; the radius also determines the right-triangle distance.
The equal-chord theorem is used within one circle or congruent circles.
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Make it stick
Do equal chords in circles of different radii necessarily have equal center distances?
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.