Welcome to your ultimate and comprehensive study guide for Chords and Arcs. Geometry provides essential insights into spatial properties and circular measurements. This chapter explores foundational circle geometry theorems relating congruent arcs, matching chords, central angles subtended by chords, and their corresponding relationships in congruent or identical circles.
1. Congruent Arcs and Chords
- Congruent Arcs & Chords Theorem 1: If two arcs of a circle (or of congruent circles) are congruent, then their corresponding chords are equal in length.
- Theorem 2 (Converse): If two chords of a circle (or of congruent circles) are equal, then their corresponding arcs (minor, major, or semi-circular) are congruent.
2. Central Angles Subtended by Chords
- Theorem 3: Equal chords of a circle (or of congruent circles) subtend equal angles at the centre (or at the corresponding centres).
- Theorem 4 (Converse): If the angles subtended by two chords of a circle (or congruent circles) at the centre (corresponding centres) are equal, then the chords are equal.
- Corollaries: In congruent circles or in the same circle, if central angles are equal, then the corresponding sectors are equal, and unequal arcs subtend unequal central angles.
3. Step-by-Step Example: Working with Central Angles
Example: Understanding the relation between central angles and congruent chords.
Step-by-Step Solution:
- Step 1: Identify given properties: Consider two congruent circles with centers O and O’ having chords AC and A’C’.
- Step 2: Apply angle equivalence: If angle AOC equals angle A’O’C’, the chords are corresponding components.
- Step 3: Conclude side congruence: Using triangle congruence (SAS postulate on triangles AOC and A’O’C’), we prove that chord AC equals chord A’C’.
Essential Conceptual Review Questions
Q1: What happens to corresponding chords if two arcs of a circle are congruent?
Answer: The corresponding chords are strictly equal in length.
Q2: Do equal chords in congruent circles subtend equal angles at their respective centres?
Answer: Yes, equal chords of a circle or congruent circles always subtend equal angles at their corresponding centres.
⭕ Central Angles, Minor/Major Arcs & Chord-Arc Equivalence Theorems
Understanding the relationship between central angles, subtended arcs, and chords in a circle is critical for solving geometric riders and objective MCQs:
- Unique Circle Through Three Non-Collinear Points: One and only one circle can pass through three non-collinear points A, B, and C, found by intersecting the perpendicular bisectors of segments AB and BC.
- Equal Chords Subtend Equal Central Angles: In the same circle (or congruent circles), if two chords are congruent (
|AB| = |CD|), then their corresponding minor arcs are congruent (m(arc AB) = m(arc CD)) and the central angles they subtend at the center O are equal (m∠AOB = m∠COD). - Arc Length & Sector Ratio: The ratio of the lengths of two arcs of a circle is directly proportional to the ratio of the central angles they subtend at the center.