Chapter 5 of Class 9 Maths Ganita Manjari Part 1 is I’m Up and Down, and Round and Round — the chapter on circles. It explores the geometric properties of circles, chords, arcs, and cyclic quadrilaterals. These Class 9 Mathematics Chapter 5 solutions are also useful as quick revision notes before exams.
Basic Definitions
A circle is the set of all points equidistant from a fixed point (the centre). The radius is the distance from the centre to any point on the circle. A chord is a line segment joining two points on the circle; the diameter is the longest chord, passing through the centre.
Chord Properties
- Equal chords of a circle subtend equal angles at the centre.
- The perpendicular from the centre to a chord bisects the chord.
- Conversely, the line joining the centre to the midpoint of a chord is perpendicular to the chord.
- The perpendicular bisector of a chord always passes through the centre.
Angles Subtended by Arcs
The angle subtended by an arc at the centre is exactly twice the angle it subtends at any point on the remaining part of the circle (the circumference).
Angle in a Semicircle
The angle subtended by a diameter at any point on the circle is always 90° (a direct consequence of the angle-at-centre rule, since a diameter subtends 180° at the centre).
Circumcircle and Circumcentre
A circumcircle is a circle passing through all three vertices of a triangle. Its centre, the circumcentre, is found by drawing the perpendicular bisectors of any two sides of the triangle — the point where they intersect is equidistant from all three vertices.
Cyclic Quadrilaterals
A cyclic quadrilateral has all four vertices lying on a single circle. Its key property: opposite angles are always supplementary (sum to 180°). Also, the exterior angle at any vertex of a cyclic quadrilateral equals the interior opposite angle.
Class 9 Mathematics Chapter 5 – Notes and Extra Questions
Along with these NCERT Solutions, students can also use the Class 9 Mathematics Chapter 5 Extra Questions and Class 9 Mathematics Chapter 5 Revision Notes for quick revision and extra practice.
