Two points: infinitely many circles, centres on the perpendicular bisector

What to assume they know

• Turning an observation into a definition: the circle as a locus — a circle as the locus of points at a fixed distance from a fixed point; centre, radius, chord, diameter
• Total rotational symmetry, and why every diameter is an axis of reflection — the two-point locus question, and why one direction of it is not enough
• Constructing the perpendicular bisector of a segment, and that it passes through the segment's midpoint at right angles
• That a point equidistant from two given points lies on their perpendicular bisector
• The idea that a set can be infinite, and that a line contains infinitely many points

What they should be able to do

• Restate a question about circles through given points as a question about the location of centres, and say why the restatement loses nothing
• Produce one circle through two given points, by taking the midpoint of the segment as centre, and say why AB is then a diameter
• State the smallest radius available to a circle through two given points, in terms of the distance between them
• State the perpendicular bisector as the locus of points equidistant from two points, and explain what each of the two directions of that claim contributes
• Explain why the circles through two given points are infinitely many, by matching them one-to-one with the points of a line
• Describe how the radius and the visible curvature change as the centre travels out along the bisector, and say whether either has a largest value
• Read Fig. 5.4 and say which family of circles it shows on the left and what the labelled centres K, J, L on the right are doing
• Contrast the circle count with the count of squares having two given points on the boundary, or as two corners

Where it usually goes wrong

• "Two points determine a circle." Two points determine a line; they leave a whole line's worth of circles. Students transfer the two-points-one-line fact and get the wrong count. The chapter is deliberately asking about two before asking about three.
• "The circle through two points is the one with AB as diameter." That is one circle, and the smallest. It is the natural first find, which is exactly why the chapter goes looking for more immediately afterwards.
• "There must be a biggest circle through A and B." There is not. Push the centre far enough out and the radius exceeds any number you name, with the arc through A and B looking almost straight.
• "Almost straight means straight." However flat the arc looks, the centre is a definite point at a definite finite distance and the figure is a circle. The chapter's Q4 invites the "less curved" observation and it must not slide into "eventually a line".
• "Every point on the bisector works, so the bisector is the locus." Not yet — that is one implication. The set could in principle be larger. §5.2's hint says so in as many words.
• "Infinitely many means any circle at all will do." Radii below half of AB are impossible, and every one of the infinitely many circles has its centre on one particular line. An infinite family can still be tightly constrained.
• "The two clusters in Fig. 5.4 show the same thing twice." They do not: the left shows circles with no bisector, the right shows the bisector with lettered centres. Redrawing them as one panel destroys the figure's argument.

Questions to check understanding

• How many circles pass through two given points? — with a reason, not just a count
• The smallest radius available to a circle through two given points (the form Exercise Set 5.1 Q4, p. 98 takes)
• Where do the centres of all circles through two given points lie?
• Given two points and a stated radius, decide whether such a circle exists and how many there are
• Prove that a point equidistant from two given points lies on their perpendicular bisector, and prove the converse
• Construct, with ruler and compasses, three different circles through two marked points, and mark the line their centres lie on
• Given two points, count the squares having them as adjacent corners, as opposite corners, or merely on the boundary

Examples worth working on the board

The chapter prints no answers, so every value below marked verified is worked out here on the chapter's own inputs.

• Fig. 5.4 (p. 95, caption names circles through two points). Two separate clusters share the one figure and they are doing different jobs, which a teacher must not merge. On the left: points A and B with several circles drawn through both, and no bisector drawn — the picture of the phenomenon. On the right: points C and D with the perpendicular bisector of CD drawn as a vertical dashed line, and three of its points marked and lettered as centres — K and J above the segment CD and L below it — each carrying its own circle through C and D. So the right-hand cluster is the explanation: the centres are on the line, and different heights on the line give different circles.
• The midpoint circle, worked. Input: A and B, 6 units apart. Verified: taking the midpoint as centre gives radius 3, and AB is then a diameter of that circle. This is the smallest circle through the pair, so the least possible radius is half the distance between the points — which is exactly what Exercise Set 5.1 Q4, p. 98 asks for.
• Walking out along the bisector. Input: A and B, 6 units apart, so the half-chord is 3. Take the centre at heights 0, 4 and 12 above AB on the bisector. Verified: radii 3, 5 and about 12.37; so the radius grows without limit and there is no largest circle. Growth is not proportional to the height — from height 0 to 4 the radius rises by 2, from 4 to 12 by about 7.37 — which is worth showing, because students expect a straight-line relationship.
• The Think and Reflect questions (§5.3, p. 95). Q1 asks how many circles pass through two points; Q2 asks whether every radius is available and what the smallest and largest are; Q3 asks whether the radii increase or decrease as the centre moves away from AB along the bisector; Q4 asks whether the circle looks more or less curved as you go; Q5 turns to squares. Verified: radii available are every value from half of AB upwards, the smallest is half of AB, and there is no largest; moving out, radii increase and the arc through A and B looks flatter.
• The squares contrast (Think and Reflect Q5, §5.3, p. 95). Verified: with A and B required to be two corners of the square, there are exactly three squares — AB can be a side, with a square on either side of it, or AB can be a diagonal, giving one more. With A and B merely required to be on the boundary, there are infinitely many. State both counts as arithmetic added here; the chapter poses the question and stops. The point of the contrast is that "infinitely many" is not the automatic answer to a two-point question — it depends on how tightly the shape is pinned.
• The both-directions statement, spelled out for section 6. Direction one: take any point on the perpendicular bisector of AB and its distances to A and B come out equal. Direction two: take any point whose distances to A and B are equal and it turns out to lie on that bisector. §5.2's hint on p. 94 gives the first and explicitly flags that the second is still owed; §5.3 on p. 95 asserts both.

Figures to have open

• Fig. 5.4 redrawn, and redrawn as two panels, not one. Left: the family through two points with no bisector. Right: the bisector drawn dashed with three lettered centres and their circles. This is the chapter's own figure (p. 95).
• A step-by-step centre sliding along the bisector with a live radius readout, and the arc through the two fixed points flattening. Standard schematic; the chapter poses this in words at Think and Reflect Q3 and Q4 and draws nothing.
• A two-distance test figure: one candidate centre on the bisector, one off it, each with its distances to A and B measured. Standard schematic.
• The squares panel for section 10: the three squares having A and B as corners, drawn on one pair of points. Standard schematic; the chapter asks and does not draw.
• No photograph is needed anywhere in this topic.

Where this sits in the book

• NCERT Ganita Manjari, Class 9 Mathematics, printed Chapter 5, "I’m Up and Down, and Round and Round", §5.3, whose printed heading is "How Many Circles?", from the foot of p. 94 through p. 95 and continuing to the three-point question at the top of p. 96.
• Fig. 5.4 with its caption, p. 95.
• Think and Reflect, §5.3, p. 95 — five questions, the last about squares.
• Exercise Set 5.1, Q4, p. 98 — the least possible radius through two points; the rest of that exercise set belongs to Three points not in a line: exactly one circle (Theorem 1).
• Back-reference inside the chapter: the perpendicular-bisector hint at Think and Reflect Q3, §5.2, p. 94, which supplies one direction and asks for the other.
• Chapter Summary, p. 117 — the bullet on infinitely many circles through two points and where their centres lie.