Form 2 Mathematics Chapter 5: Circles

5.1 Properties of Circles

Definition: A circle is a curved pathway locus of a point equidistant from a fixed point.
Parts of a Circle:
- Radius: Straight line from the center to any point on the circumference.
- Center: The fixed point where all circumference points are equidistant.
- Segment: Region enclosed by a chord and an arc.
- Sector: Region enclosed by two radii and an arc.
- Arc: Part of the circumference.
- Chord: Straight line joining two circumference points.
- Diameter: Straight line through the circle's center touching circumference at both ends.
Examples & Constructions:
- Construct a circle given radius or diameter.
- Construct a diameter or chord through a given point.
- Construct a sector given angle and radius.

Symmetry Features:
- Diameter as an axis of symmetry.
- Infinite axes of symmetry in a circle.
- Radius perpendicular to chord bisects it.
- Perpendicular bisectors of two chords meet at the center.
Examples and Solutions:
- Calculate segment lengths using Pythagorean theorem in chord-related problems.
- Use symmetry to determine equal arcs or chord distances from the center.

Relationships and Formulas:
- Circumference: ( C = \pi d ) or ( C = 2\pi r )
- Area: ( A = \pi r^2 )
Examples and Problems:
- Calculate circumference given diameter or radius.
- Calculate diameter or radius from circumference.
- Calculate area of a circle given diameter or radius.

Length of Arc: Proportional to angle at the center.
- Formula: ( \text{Length of arc} = \frac{\theta}{360^\circ} \times 2\pi r )
Area of Sector: Proportional to area of the circle.
- Formula: ( \text{Area of sector} = \frac{\theta}{360^\circ} \times \pi r^2 )
Examples:
- Calculate arc length given angle and radius.
- Calculate area of sector given angle and radius or given area to find angle.

Example Problem: Calculate the area covered with grass in a park with quadrant flower beds and a central pond.
- Use area formulas for circle, sector, and rectangle.

Concept Map: Understanding the relationship and geometry of circles involves recognizing properties, symmetry, area, and circumference.

This summary covers the key concepts of Chapter 5 on Circles, useful for revising and understanding the essential geometry principles.

Practice constructing circles and solving geometry problems with given formulas and examples.
Understand the symmetry concept to simplify problem-solving in geometry.
Apply the concepts of circumference and area in real-world contexts to strengthen understanding.