5.1 Properties of Circles
A circle is a set of points that are equidistant from a fixed point called the centre.
Parts of a Circle
- Centre: The fixed point from which all points on the circumference are at an equal distance.
- Radius ($r$): The straight line distance from the centre to any point on the circumference.
- Diameter ($d$): A straight line passing through the centre, connecting two points on the circumference. $d = 2r$.
- Circumference: The perimeter or outer boundary of the circle.
- Chord: A straight line segment connecting any two points on the circumference. The diameter is the longest chord.
- Arc: A part of the circumference. A minor arc is smaller than a semicircle; a major arc is larger than a semicircle.
- Sector: The region bounded by two radii and an arc. (Minor sector vs Major sector).
- Segment: The region bounded by a chord and an arc. (Minor segment vs Major segment).
Symmetry Properties of Chords
- A radius that is perpendicular to a chord bisects the chord into two equal parts.
- The perpendicular bisector of any chord passes through the centre of the circle.
- Chords of equal length are equidistant from the centre of the circle and subtend equal arcs.
5.2 Properties of Angles in Circles
- The angle subtended by an arc at the centre is twice the angle subtended by the same arc at the circumference:
$$\theta_{\text{centre}} = 2 \times \theta_{\text{circumference}}$$
- Angles subtended at the circumference by the same arc or equal arcs are equal.
- The angle subtended at the circumference by a semicircle (or a diameter) is always $90^\circ$ (a right angle).
- Cyclic Quadrilaterals: A 4-sided polygon whose four vertices lie on the circumference of a circle:
- Opposite interior angles add up to $180^\circ$: $A + C = 180^\circ$, $B + D = 180^\circ$.
- An exterior angle of a cyclic quadrilateral is equal to its corresponding interior opposite angle.
5.3 Circumference and Area of a Circle
Circumference and Arc Length
- Circumference ($C$):
$$C = \pi d = 2\pi r$$
- Arc Length ($s$): Proportional to the angle subtended at the centre ($\theta$):
$$\frac{\text{Arc Length}}{2\pi r} = \frac{\theta}{360^\circ} \implies \text{Arc Length} = \frac{\theta}{360^\circ} \times 2\pi r$$
Area of a Circle and Area of a Sector
- Area of a Circle ($A$):
$$A = \pi r^2$$
- Area of a Sector: Proportional to the angle subtended at the centre ($\theta$):
$$\frac{\text{Area of Sector}}{\pi r^2} = \frac{\theta}{360^\circ} \implies \text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2$$
Note: Standard values used for $\pi$ are $\frac{22}{7}$ or $3.142$.