Class 9 Mathematics – Chapter 5
I'm Up and Down, and Round and Round
Ganita Manjari – Part 1
This chapter develops the geometry of circles through construction, symmetry, congruence, Baudhāyana–Pythagoras reasoning, chord properties, arcs and cyclic quadrilaterals. The aim is to understand why the results are true, not simply memorise them.
📚 Complete Exercise Solutions
All six exercise sets and the 26 End-of-Chapter Exercises are now available in step-by-step form.
🔑 Important Results
| Result | What it tells us |
|---|---|
| Circumcircle | Three non-collinear points determine one unique circle. |
| Equal chords | Equal chords subtend equal angles at the centre, and the converse also holds. |
| Perpendicular to a chord | The perpendicular from the centre bisects the chord. |
| Distance of chords | Equal chords are equidistant from the centre; the longer chord is nearer the centre. |
| Chord formula | $L=2\sqrt{r^2-d^2}$, where $d$ is the perpendicular distance from the centre. |
| Arc angle | The angle at the centre is twice the angle subtended by the same arc at a point on the circle. |
| Semicircle | An angle subtended by a diameter at the circle is $90^\circ$. |
| Cyclic quadrilateral | Opposite angles are supplementary: their sum is $180^\circ$. |
📐 The 12 Theorems at a Glance
- A unique circle passes through three non-collinear points.
- Equal chords subtend equal angles at the centre.
- Chords subtending equal angles at the centre are equal.
- The line joining the centre to the midpoint of a chord is perpendicular to the chord.
- The perpendicular from the centre to a chord bisects the chord.
- Equal chords are equidistant from the centre.
- Chords equidistant from the centre are equal.
- The longer of two unequal chords is nearer the centre.
- The angle subtended by an arc at the centre is twice the angle subtended at a point on the circle.
- If a segment subtends equal angles at two suitable points on the same side, the four points are concyclic.
- Opposite angles of a cyclic quadrilateral sum to $180^\circ$.
- If opposite angles of a quadrilateral sum to $180^\circ$, the quadrilateral is cyclic.
🔗 Chapter Navigation
← Chapter 4: Exploring Algebraic Identities