5. I’m Up and Down, and Round and Round

13 topics2 h

Chapter 5 of NCERT Mathematics for Class 9: I’m Up and Down, and Round and Round. Five sections — What a circle is, How many circles pass through given points, Chords, Angles standing on an arc and When four points share a circle. 13 videos, 2 h in all.

Worked answers to the 44 exercise questions in this chapter →

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Part 1

What a circle is

10 minTurning an observation into a definition: the circle as a locusRound is something you recognise. It is not something you can argue from — which is why the chapter turns the observation into an entrance requirement.10 min · 12 partsहिंदी10 minTotal rotational symmetry, and why every diameter is an axis of reflectionWatch a wheel go past twice and say which observation came first. You cannot, and that is the circle's symmetry doing its work.10 min · 12 partsहिंदी

Part 2

How many circles pass through given points

9 minTwo points: infinitely many circles, centres on the perpendicular bisectorHow many circles pass through two given points? Most people say one, because two points fix a straight line. It is a whole line's worth.9 min · 12 partsहिंदी10 minThree points not in a line: exactly one circle (Theorem 1)Two points leave a whole line of possible centres. Add a third and the answer collapses to one — unless all three sit in a row.10 min · 12 partsहिंदी

Part 3

Chords

10 minChords of equal length cut off equal central angles, and the converse (Theorems 2–3)Draw a chord and you have drawn more than a chord. Two radii are already there, waiting, and they leave the triangle only one degree of freedom.10 min · 12 partsहिंदी10 minThe centre-to-midpoint line is perpendicular, and the converse (Theorems 4–5)Join the centre to the middle of a chord and the join comes out square. Two equal angles on a straight line is where that actually happens.10 min · 12 partsहिंदी10 minLength and distance from the centre are the same fact twice (Theorems 6–8)A chord carries two numbers: how long it is, and how far it stands from the centre. Fix the circle and choosing one fixes the other.10 min · 12 partsहिंदी

Part 4

Angles standing on an arc

9 minMajor and minor arcs, and why an arc's central angle is double what it subtends on the circle (Theorem 9)Two points cut a circle into two arcs, so “the angle of arc AB” has not said which. Answering that properly means letting an angle exceed 180°.9 min · 12 partsहिंदी9 minEqual angles in the same segment: the arc looks the same from every point beyond itAn arc's angle at the centre is fixed by the arc alone. So the angle it makes at the circle cannot depend on where you stand.9 min · 12 partsहिंदी10 minThe Corollary: a diameter stands on a right angle wherever you take the pointThe right angle in a semicircle is not a discovery. It is the doubling relation evaluated at the one arc whose central angle is a straight angle.10 min · 12 partsहिंदी9 minOff the circle the angle changes: points inside and outside compared (Fig. 5.25)Every point of one arc sees a chord at the same angle. The half that matters is that no other point does.9 min · 12 partsहिंदी

Part 5

When four points share a circle

10 minEqual angles on the same side force concyclicity (Theorem 10)You cannot show a fourth point lies on a circle by drawing it there — the circle is the thing in dispute. So the argument goes the other way.10 min · 12 partsहिंदी10 minCyclic quadrilaterals: opposite angles sum to 180°, and the converse (Theorems 11–12)Opposite corners of a cyclic quadrilateral always add to a straight angle. The reason is smaller than the result: 180 is 360 halved.10 min · 12 partsहिंदी

What you will be able to do

  • State the defining condition of a circle in terms of a fixed point and a fixed distance, and say which of the two is the centre and which the radius
  • Explain why a rotating wheel gives no way to tell one ground-contact point from another, and state the property this demonstrates
  • Restate a question about circles through given points as a question about the location of centres, and say why the restatement loses nothing
  • Explain why three points lying on one straight line cannot all be on one circle
  • Explain why the triangle formed by any chord and the centre is isosceles, and name its equal sides
  • Explain why the segment from the centre to a chord's midpoint is the axis of symmetry of the triangle the chord makes with the centre
  • State what a chord's distance from the centre means, and say why the perpendicular is the only distance worth measuring
  • Say what an arc is, and identify the two arcs a given pair of points determines
  • State that all points of a circle lying off a given arc see that arc at one and the same angle, and derive it from the doubling relation
  • Say what a corollary is, in the sense the chapter defines
  • State that the equal-angle property holds only for points of the circle, and say what would follow if it held everywhere
  • State what concyclic means, and say why four points is the first interesting case

What this chapter assumes you already know

  • Measuring the distance between two points, and the idea that two distances can be compared without either being computed
  • Angle, and how an angle is named by three letters with the vertex in the middle
  • That a triangle with two equal sides has two equal angles
  • The word set, and reading a description of a set as a condition its members must satisfy
  • Rotational symmetry and line symmetry from earlier classes, including the order of a rotational symmetry
  • That a point equidistant from two given points lies on the perpendicular bisector of the segment joining them
  • Angles on a straight line add to 180°
  • Reading a fold in paper as a line of reflection

It builds directly on Where coordinates came from: grid cities, meridians, and the road to the Cartesian plane.

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "A circle is the round region."
  • "The centre is on the circle."
  • "Radius means the segment from the centre to the rim."
  • "A definition is a description, so any true statement about circles could have been the definition."
  • "Diameter is a number, chord is a segment."
  • "The plane clause is legal boilerplate."
  • "Circles in nature are circles."
  • "Rotational symmetry means a few special angles."