📖 Theory Reference
Theorem 9.3
Statement: The perpendicular from the centre of a circle to a chord bisects the chord.
Given: A circle with centre O and chord AB. OM is perpendicular to AB.
To Prove: AM = MB
Result: The perpendicular from centre bisects the chord into two equal parts.
Theorem 9.4 (Converse)
Statement: The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord.
Given: A circle with centre O and chord AB. M is the midpoint of AB, so AM = MB.
To Prove: OM ⊥ AB
Result: The line from centre to the midpoint of a chord is perpendicular to the chord.
Theorem 9.5
Statement: Equal chords of a circle (or of congruent circles) are equidistant from the centre (or centres).
Given: A circle with centre O. Chords AB and CD are equal in length (AB = CD).
To Prove: OM = ON (where OM ⊥ AB and ON ⊥ CD)
Result: Equal chords are at equal distances from the centre.
Theorem 9.6 (Converse)
Statement: Chords equidistant from the centre of a circle are equal in length.
Given: A circle with centre O. Chords AB and CD are equidistant from O (OM = ON).
To Prove: AB = CD
Result: Chords at equal distances from centre are equal in length.
Distance from Point to Line
Definition: The length of the perpendicular from a point to a line is the distance of the line from the point.
Key Properties:
• The perpendicular distance is always the shortest distance from a point to a line
• Any other line segment from the point to the line will be longer than the perpendicular distance
• This concept is fundamental in geometry and is used in many theorems and proofs
Proof Method: RHS Congruence
Method: Right-Hand-Side (RHS) Congruence Rule for proving Theorem 9.3
Proof Steps:
1. Join OA and OB to form triangles OMA and OMB
2. OA = OB (both are radii of the circle)
3. OM = OM (common side)
4. ∠OMA = ∠OMB = 90° (OM is perpendicular to AB)
5. Therefore, △OMA ≅ △OMB (by RHS congruence rule)
6. Hence, AM = BM (by CPCT - Corresponding Parts of Congruent Triangles)
Key Properties and Relationships
Important Properties:
Bisection Property: The perpendicular from the centre of a circle to a chord always bisects the chord into two equal parts.
Equidistance Property: If two chords are equal in length, then they are equidistant from the centre of the circle.
Converse Property: If two chords are equidistant from the centre, then they are equal in length.
Distance Property: The perpendicular distance from the centre to a chord is always the shortest distance.
Symmetry Property: The perpendicular creates two congruent right-angled triangles, showing the symmetrical nature of the relationship.
Applications and Uses
Real-world Applications:
Construction: Finding the centre of a circle using perpendicular bisectors of chords.
Problem Solving: Proving that chords are equal or finding unknown lengths in circle problems.
Geometry: Foundation for many other circle theorems and geometric proofs.
Trigonometry: Calculating distances and angles in circular problems.
Engineering: Designing circular structures, wheels, and mechanical components.
Architecture: Creating symmetrical designs and calculating structural elements in circular buildings.
Important Notes and Tips
Study Tips:
Remember: The perpendicular from centre to chord always bisects the chord, regardless of the chord's position or length.
Key Insight: Equal chords create equal distances from centre, and vice versa.
Proof Strategy: Use RHS congruence when you have right angles and equal sides.
Common Mistakes: Don't confuse the perpendicular distance with other line segments from centre to chord.
Practice: Draw multiple examples with different chord positions to understand the concept better.