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CBSE Class 9 · Mathematics

I'm Up and Down, and Round and Round

Official NCERT chapter from Ganita Manjari Part I (book code iemh1). ExamMaster notes are original teaching at CBSE Class 9 depth.

This lesson follows the official chapter “I'm Up and Down, and Round and Round” in Ganita Manjari Part I. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 9
  • Medium level
  • 8 concepts

1Definitions

A circle is all points at a fixed distance (radius) from a centre. A chord joins two points on the circle; a diameter is a chord through the centre. Definitions are tests you can mark, not a round doodle.

A freehand oval is not a circle until equal radii are argued.

Figure. O is the centre. A radius joins O to the rim; a diameter is two radii in one line; a chord joins two rim points and need not pass through O.

How it works

  1. Mark the centre and a radiusThe definition.
  2. Name chord versus diameterThrough the centre or not.
  3. Refuse a look-only “round”Run a radius-test.

2Symmetries of a Circle

A circle’s symmetries: every rotation about the centre, and every reflection through a diameter. That is why a radius you rotate still lands on the circle. A chord is not a symmetry-line unless it is a diameter.

A pretty petal drawing is not a diameter.

Figure. Every diameter is a fold line of the circle. Two perpendicular diameters give four matching quarters. The rim is not drawn; the folds still cross at O.

How it works

  1. Rotate a radius about the centreIt stays a radius.
  2. Reflect through a diameterThe circle looks the same.
  3. Keep those as the taught symmetriesUp, down, and round.

3How Many Circles?

How many circles through given points: one circle through three non-collinear points (as taught); infinitely many through two points (many centres on the perpendicular bisector). Collinear three points do not sit on one finite circle.

Two points do not lock a unique circle.

Figure. Three non-collinear points A, B, C fix one centre: the perpendicular bisectors meet at a single O. A fourth point may or may not sit at the same distance from O.

How it works

  1. Count the given points2 or 3.
  2. Ask collinear or not if threeThe lock.
  3. Name unique / many / none as the lesson didHow many.

4Chords and the Angles They Subtend

A chord subtends an angle at the centre and an angle at the circumference. The centre-angle is twice the circumference-angle on the same arc, as taught. Subtend means “stand opposite that chord/arc”.

Any two angles that look related are not automatically this double.

Figure. Equal radii make triangle OAB isosceles. The base angles at A and B are equal, so a chord’s end-angles match without drawing the rim.

How it works

  1. Name the same arc or chordThe object.
  2. Mark the centre-angle and a circumference-angle on that arcThe pair.
  3. Use centre = 2 × circumference if that is the taught factThe relation.

5Midpoints and Perpendicular Bisectors of Chords

The perpendicular from the centre to a chord bisects the chord. Conversely, the perpendicular bisector of a chord runs through the centre. Midpoint plus right angle is the pair; a “looks middle” tick is not enough.

A freehand drop that misses 90° is not this bisector.

Figure. The line from the centre to a chord, drawn perpendicular, meets the chord at its midpoint. AM = MB is the test, not a tidy-looking sketch.

How it works

  1. From O drop a perpendicular to chord ABThe construction.
  2. The foot is the midpoint of ABThe fact.
  3. Or: start from the perp-bisector of AB and find O on itThe converse as taught.

6Distance of Chords from the Centre

Equal chords sit at equal distances from the centre; a longer chord sits closer to the centre (in the same circle). Distance here is the perpendicular distance. A far thin chord is short; a near fat chord is long.

A chord that looks longer because the pencil was thick is not the test.

Figure. Equal visual radius. The longer chord sits nearer O; the shorter chord sits farther. Distance is the perpendicular from O, not a sideways glance.

How it works

  1. Compare two chords’ lengthsThe given.
  2. Compare their perp-distances from OThe heading.
  3. Equal length ↔ equal distance, as taughtThe pair.

7Angles Subtended by an Arc

An arc subtends an angle at the centre (the pie-slice) and angles at the remaining circumference. Same arc, same circumference-angles, as taught. A different arc is a different angle.

Using a major-arc angle where the lesson used the minor is a silent swap.

Figure. The centre angle AOB is twice the inscribed angle on the same arc. Here AOB = 80°, so an angle at the rim on arc AB is 40°. The rim is not drawn.

How it works

  1. Name the arc (minor or major as given)The object.
  2. Mark the angle at O or on the remaining circumferenceThe subtend.
  3. Keep same-arc facts on that arc onlyThe job.

8Concyclicity of Points

Points are concyclic if one circle holds them all. Four points of a quadrilateral are concyclic when a taught opposite-angle sum is 180°, or when they see a segment at the same angle, as the lesson framed it. Concyclic is a test, not “they look round”.

Any four points you like are not concyclic.

Figure. A quadrilateral is cyclic when one pair of opposite angles adds to 180°. Here A + C = 80° + 100°. The rim is not drawn; the angle test is the exam mark.

How it works

  1. Name the four pointsThe candidates.
  2. Run the taught test (opposite angles, or equal segment-angles)The test.
  3. Pass or failOne circle or not.
The perpendicular from the centre to a chord
  1. Bisects the chord
  2. Is always a diameter
  3. Makes the chord longer

Midpoint plus 90°.

Notes

  • The official chapter title is “I'm Up and Down, and Round and Round”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • perp from centre to a chord bisects the chord
  • centre-angle = 2 × circumference-angle on the same arc (as taught)

Recap

Hold these pegs from the official chapter “I'm Up and Down, and Round and Round”. The wording is ExamMaster’s teaching, not a textbook recap.

Definitions
A circle is all points at a fixed distance (radius) from a centre.
Symmetries of a Circle
A circle’s symmetries: every rotation about the centre, and every reflection through a diameter.
How Many Circles?
How many circles through given points: one circle through three non-collinear points (as taught); infinitely many through two points (many centres on the perpendicular bisector).
Chords and the Angles They Subtend
A chord subtends an angle at the centre and an angle at the circumference.
Midpoints and Perpendicular Bisectors of Chords
The perpendicular from the centre to a chord bisects the chord.
Distance of Chords from the Centre
Equal chords sit at equal distances from the centre; a longer chord sits closer to the centre (in the same circle).

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