Chapter 5: I’m Up and Down, and Round and Round | Grade 9 Mathematics

5.1 Foundations

The Locus of Perfection

Shapes like the full moon, raindrops on water, and the sun are likely inspirations for the geometric circle. Every circle, big or small, shares one property: every point on it is at an equal distance from a fixed center.

$ \text{Circle: } \{ P \in \text{Plane} \mid \text{dist}(P, \text{Center}) = \text{Radius} \} $

Chord

A line segment joining any two points on a circle.

Diameter

The longest chord; it passes through the center.

5.2 Mirror & Motion

Symmetries of a Circle

Rotational

A circle has **Complete Rotational Symmetry**. Rotate it by any angle, and it looks exactly the same.

Reflectional

Every **diameter** is a line of symmetry. Fold a circular paper along any diameter, and the halves overlap perfectly.

5.3 Construction

How Many Circles?

The number of circles depends on the points you provide:

Two Points: Infinitely many circles. The centers of all these circles lie on the perpendicular bisector of the segment joining the two points.
Three Points:
- If Collinear: **Zero** (no circle exists).
- If Non-Collinear: **Exactly One** unique circle (Circumcircle).

Theorem 1: The Circumcircle

The unique circle passing through three non-collinear points is the circumcircle of the triangle they form.

Since OA = OB, the center O lies on the perpendicular bisector of AB. Since OA = OC, O also lies on the perpendicular bisector of AC. These two lines meet at exactly one unique point O.

5.4 Geometry of Chords

Chords & Angles

Theorem 2

Equal chords subtend equal angles at the centre.

If AB = DE, then ∠ACB = ∠DCE.

In ∆ACB and ∆DCE: AC=DC (radius), BC=EC (radius), AB=DE (given). By SSS congruence, ∆ACB ≅ ∆DCE. Hence ∠ACB = ∠DCE.

Theorem 3

The converse: Equal angles $\rightarrow$ Equal chords.

If ∠ACB = ∠DCE, then AB = DE.

5.6 Distance Mastery

Distance from the Centre

$ \text{Equal Chords} \iff \text{Equidistant from Center} $

Theorem 8: The Length-Distance Paradox

The **longer** the chord, the **closer** it is to the center. The diameter (the longest chord) has a distance of zero from the center.

5.7 Arc Dynamics

Angles Subtended by an Arc

One of the most beautiful properties of a circle is how it preserves angles across its boundary.

$ \angle \text{Centre} = 2 \times \angle \text{Circumference} $

Theorem 9

The angle subtended by an arc at the centre is double the angle subtended at any point on the remaining part of the circle.

Semi-circle Rule

The angle subtended by a diameter at any point on the circle is **90°** (Right Angle).

5.8 Concyclicity

Cyclic Quadrilaterals

A quadrilateral is called **cyclic** if all its vertices lie on a single circle.

$ \angle \text{Opposite}_1 + \angle \text{Opposite}_2 = 180^\circ $

Summary of Cyclic Properties

Opposite angles are supplementary (Sum = 180°).
If the sum of opposite angles is 180°, the quadrilateral is cyclic.
An exterior angle at any vertex equals the interior opposite angle.