Chapter 5: I'm Up and Down, and Round and RoundClass 9 Mathematics — summary, notes, extra questions & NCERT solutions

Summary

This chapter studies the circle and its many elegant properties. A circle is the set (locus) of all points in a plane at a fixed distance, the radius, from a centre. A chord joins two points on the circle; a chord through the centre is a diameter, the longest chord. Circles have rotational symmetry about the centre and reflection symmetry across every diameter. Infinitely many circles pass through two given points, with centres on the perpendicular bisector of the segment; exactly one circle (the circumcircle) passes through three non-collinear points. Key theorems: equal chords subtend equal angles at the centre and conversely; the line from the centre to a chord's midpoint is perpendicular to the chord and conversely; equal chords are equidistant from the centre and conversely; the longer of two chords lies nearer the centre. The angle an arc subtends at the centre is double the angle it subtends at any point on the remaining circle, so the angle in a semicircle is \(90^{\circ}\). Four points are concyclic when a segment subtends equal angles at two points on the same side, and opposite angles of a cyclic quadrilateral sum to \(180^{\circ}\).

Key terms

Extra questions & answers

Think and Reflect

Think and Reflect

Think and Reflect

Exercise Set 5.1

Think, Draw and Infer

Exercise Set 5.2

Exercise Set 5.3

Exercise Set 5.4

Exercise Set 5.5

Exercise

Exercise Set 5.6

Exercise

End-of-Chapter Exercises

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