Concept Notes: Circle Anatomy, Chords, Radii & Perpendicular Bisection
1. Definitions & Basic Terminology
A Circle is the set of all points in a plane that are at a constant distance (radius $r$) from a fixed point (center $O$).
Figure 5.1: Anatomy of a Circle — Chords, Radii, Tangents, Sectors, Segments, and the Perpendicular Bisector Theorem
- Chord: Line segment joining two points on the circle.
- Diameter: Longest chord passing through the center ($d = 2r$).
- Arc: A piece of a circle between two points. Minor arc ($< 180^\circ$) vs Major arc ($> 180^\circ$).
- Segment: Region enclosed by a chord and its corresponding arc.
- Sector: Region enclosed by two radii and the arc between them.
2. The Perpendicular from Center to Chord Theorems
- Theorem 5.1: The perpendicular drawn from the center of a circle to a chord bisects the chord.
- If $OM \perp AB \implies AM = MB = \frac{1}{2}AB$.
- Theorem 5.2 (Converse): The line joining the center to the midpoint of a chord is perpendicular to the chord.
3. Worked Computational Example
Problem: A chord of length $16\text{ cm}$ is drawn in a circle of radius $10\text{ cm}$. Find its distance from the center.
- Draw $OM \perp AB$. By Theorem 5.1, $AM = \frac{1}{2} \times 16 = 8\text{ cm}$.
- In right-angled $\Delta OMA$: $OA = r = 10\text{ cm}$, $AM = 8\text{ cm}$.
- By Pythagoras:
$$OM^2 = OA^2 - AM^2 = 10^2 - 8^2 = 100 - 64 = 36 \implies OM = \sqrt{36} = \mathbf{6\text{ cm}}$$