6.1 Angle at the Centre and Subtended Angles
Subtended Angles by Arcs
- Angles subtended at the circumference by the same arc (or equal arc lengths) are equal:
$$\angle APB = \angle AQB$$
- The angle subtended by an arc at the centre of a circle is twice the angle subtended by the same arc at the circumference:
$$\text{Angle at centre} = 2 \times \text{Angle at circumference}$$
- An angle subtended at the circumference by a semicircle (diameter) is always $90^\circ$:
$$\angle APB = 90^\circ$$
6.2 Cyclic Quadrilaterals
Properties of Cyclic Quadrilaterals
A cyclic quadrilateral is a four-sided polygon whose vertices all lie on the circumference of a circle.
- Opposite Angles: The sum of opposite interior angles of a cyclic quadrilateral is $180^\circ$:
$$A + C = 180^\circ \quad \text{and} \quad B + D = 180^\circ$$
- Exterior Angle: The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle:
$$\theta_{\text{exterior}} = \theta_{\text{interior opposite}}$$
6.3 Tangents to Circles
Properties of Tangents
- A tangent to a circle is a straight line that touches the circle at only one point (point of tangency).
- The angle between a tangent and the radius drawn to the point of contact is always $90^\circ$:
$$\text{Radius} \perp \text{Tangent}$$
- Tangents from an External Point: If two tangents $TP$ and $TQ$ are drawn to a circle from an external point $T$:
- $TP = TQ$ (The tangent lengths are equal).
- $\angle TOP = \angle TOQ$ (Line $TO$ bisects the angle at the centre).
- $\angle OTP = \angle OTQ$ (Line $TO$ bisects the angle at the external point $T$).
6.4 Angles in Alternate Segments
Alternate Segment Theorem
The angle between a tangent and a chord through the point of contact is equal to the angle subtended by the chord in the alternate segment:
$$\angle \text{between tangent and chord} = \angle \text{in alternate segment}$$