MathematicsChapter 11: Practical Geometry of Circles
Finding the Centre of A Given Circle
The centre is equidistant from every point on the circumference.
Learning objectives
- Explain finding the centre of a given circle using the correct definitions, conditions, and notation.
- Apply perp bisectors of two chords meet at O correctly in routine and exam-style questions.
- Show each important step, simplify the result, and check that it satisfies the original conditions.
Definition
The centre is equidistant from every point on the circumference. Construction: draw two (non-parallel) chords and erect their perpendicular bisectors; they meet at the centre. This is a named compass-and-straightedge outline, not a workshop for other tools.
Where / symbols
- FINDING THE CENTRE OF A GIVEN CIRCLE
- The centre is equidistant from every point on the circumference. Construction: draw two (non-parallel) chords and erect their perpendicular bisectors; they meet at the centre. This is a named compass-and-straightedge outline, not a workshop for other tools.
- WHY IT WORKS
- Perp bisector of a chord is the locus of points equidistant from the endpoints, and it passes through O (Chapter 9). Two such lines pin O.
- WHAT TO AVOID
- Angle bisectors of chords are the wrong lines. One chord is not enough (a whole diameter-line of possible centres). Parallel chords give parallel bisectors that never meet.
Overview
Two chords, two bisectors:
Why It Works
Perp bisector of a chord is the locus of points equidistant from the endpoints, and it passes through O (Chapter 9). Two such lines pin O.
Examples Two non-parallel chords · Their perp bisectors · Intersection = centre
What To Avoid
Angle bisectors of chords are the wrong lines. One chord is not enough (a whole diameter-line of possible centres). Parallel chords give parallel bisectors that never meet.
Examples Not angle bisectors · Not one chord only · Not a guessed radius
WHAT — FINDING THE CENTRE OF A GIVEN CIRCLE
The centre is equidistant from every point on the circumference. Construction: draw two (non-parallel) chords and erect their perpendicular bisectors; they meet at the centre. This is a named compass-and-straightedge outline, not a workshop for other tools. The symbols and conditions are part of the mathematics, so include them when stating the rule.
WHY — Why this topic matters
Finding The Centre Of A Given Circle matters because accurate compass-and-straightedge constructions apply circle theorems in a visible way. Each construction works because a locus or perpendicular condition fixes the required centre, circle, or tangent.
HOW — How to solve and explain it
Start by identifying what is given and what must be found. Work through WHY IT WORKS, then WHAT TO AVOID. Worked: steps to locate the centre (named). draw chord AB construct the perpendicular bisector of AB draw chord CD, not parallel to AB construct the perpendicular bisector of CD their intersection is the centre O Finish by checking signs, restrictions, units, and whether the answer matches the question.
Worked example
- steps to locate the centre (named).
- draw chord AB
- construct the perpendicular bisector of AB
- draw chord CD, not parallel to AB
- construct the perpendicular bisector of CD
- their intersection is the centre O
In simple words
The same idea finds the circumcentre of a triangle: perp bisectors of the sides.
Key points
- perp bisectors of two chords meet at O
- locus + Chapter 9
- need two, not parallel
Exam preparation
Exam Focus
Quick revision
Revision Zone
Review the key facts, formulas and checks before you move on.Finding the Centre of A Given Circle
Sticky Notes · Quick Revision
Core Idea
- The centre is equidistant from every point on the circumference. Construction: draw two (non-parallel) chords and erect their perpendicular bisectors; they meet at the centre. This is a named compass-and-straightedge outline, not a workshop for other tools.
WHY IT WORKS
- Perp bisector of a chord is the locus of points equidistant from the endpoints, and it passes through O (Chapter 9). Two such lines pin O.
WHAT TO AVOID
- Angle bisectors of chords are the wrong lines. One chord is not enough (a whole diameter-line of possible centres). Parallel chords give parallel bisectors that never meet.
Worked Method
- Worked: steps to locate the centre (named).
- draw chord AB
- construct the perpendicular bisector of AB
- draw chord CD, not parallel to AB
- construct the perpendicular bisector of CD
- their intersection is the centre O
Common Misconception
- The same idea finds the circumcentre of a triangle: perp bisectors of the sides.
FBISE Exam Check
- The same idea finds the circumcentre of a triangle: perp bisectors of the sides.
60-second revision
- The centre is equidistant from every point on the circumference.
- WHY IT WORKS: Perp bisector of a chord is the locus of points equidistant from the endpoints, and it passes through O (Chapter 9).
- WHAT TO AVOID: Angle bisectors of chords are the wrong lines.
- Worked-method reminder: their intersection is the centre O
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