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Circles through given points · 5 marks

1. How many circles pass through two points on a plane?
2. Are there circles of all possible radii passing through A and B? What is the radius of the smallest circle passing through A and B? What is the radius of the largest circle passing through A and B?
3. As you move away from segment AB along its perpendicular bisector, do the radii of the circles containing A and B increase or decrease?
4. As you go along the perpendicular bisector, will the circle drawn from that point through A and B appear more curved or less curved?
5. You are given two points A and B on a plane. How many squares can you draw on the same plane with A and B on the boundary? How many squares can you draw on the plane with A and B as the corners of the square?

  1. 1. How many circles pass through two points on a plane?
  2. 2. Are there circles of all possible radii passing through A and B? What is the radius of the smallest circle passing through A and B? What is the radius of the largest circle passing through A and B?
  3. 3. As you move away from segment AB along its perpendicular bisector, do the radii of the circles containing A and B increase or decrease?
  4. 4. As you go along the perpendicular bisector, will the circle drawn from that point through A and B appear more curved or less curved?
  5. 5. You are given two points A and B on a plane. How many squares can you draw on the same plane with A and B on the boundary? How many squares can you draw on the plane with A and B as the corners of the square?
Answer: 1. Infinitely many. 2. No: the radius must be at least ½AB. Smallest radius = ½AB (centre at the midpoint); there is no largest. 3. The radii increase. 4. The circles look less curved (flatter near AB). 5. Infinitely many squares have A and B on the boundary; exactly 3 squares have A and B as corners.

Step-by-step solution

Idea: A circle passes through A and B exactly when its centre O is equidistant from A and B, that is, when O lies on the perpendicular bisector of AB. If O is at distance d from the midpoint M of AB, then by the Baudhāyana–Pythagoras theorem the radius is r = √((½AB)2 + d2).

ABperpendicular bisectorSquares with A and B as cornersABABAB

1. How many circles pass through two points on a plane?

  1. The centre of a circle through A and B must be equidistant from A and B, so it lies on the perpendicular bisector of AB. Every point of that line works as a centre, and the line has infinitely many points. So infinitely many circles pass through A and B.1 mark
Infinitely many.

2. Are there circles of all possible radii passing through A and B? What is the radius of the smallest circle passing through A and B? What is the radius of the largest circle passing through A and B?

  1. Let M be the midpoint of AB and O a centre at distance d from M. In right triangle OMA: r2 = AM2 + d2, so r ≥ AM = ½AB. A radius smaller than ½AB is impossible, so not all radii occur.½ mark
  2. r is smallest when d = 0, that is, O = M: smallest radius = ½AB, and AB is then a diameter. As d grows without limit, so does r: there is no largest circle.½ mark
No; every radius from ½AB upwards occurs. Smallest radius = ½AB; there is no largest.

3. As you move away from segment AB along its perpendicular bisector, do the radii of the circles containing A and B increase or decrease?

  1. From r2 = (½AB)2 + d2: as the distance d of the centre from AB increases, r2 and so r increase. (This is true on either side of AB.)1 mark
The radii increase.

4. As you go along the perpendicular bisector, will the circle drawn from that point through A and B appear more curved or less curved?

  1. A bigger circle bends more gently: the larger the radius, the flatter each small piece of it looks (a very large circle looks almost straight near A and B). Since the radius increases as we move away, the circles appear less curved.1 mark
Less curved (flatter).

5. You are given two points A and B on a plane. How many squares can you draw on the same plane with A and B on the boundary? How many squares can you draw on the plane with A and B as the corners of the square?

  1. A and B on the boundary: a square of any size and position whose sides pass through A and B will do (for example, any large square with A and B on one of its sides, or on two different sides). So infinitely many squares.½ mark
  2. A and B as corners: either AB is a side or AB is a diagonal. With AB as a side, the square can be on either side of AB: 2 squares. With AB as a diagonal, the square is fixed (its other diagonal is the perpendicular bisector of AB, of the same length): 1 square. Total 3 squares.½ mark
Infinitely many squares with A and B on the boundary; exactly 3 squares with A and B as corners.
1. Infinitely many circles. 2. No; the smallest radius is ½AB, and there is no largest radius. 3. The radii increase. 4. The circles look less curved. 5. Infinitely many squares with A and B on the boundary; exactly 3 squares with A and B as corners.

Check: Take AB = 6 units. Centres at distances 0, 4 and 8 from the midpoint give radii √(9 + 0) = 3, √(9 + 16) = 5 and √(9 + 64) ≈ 8.5: the radius never goes below 3 = ½AB and keeps growing.

Answer to write in the exam

1.

Centre O of a circle through A and B: OA = OB ⇒ O lies on the perpendicular bisector of AB

Every point of this line can be a centre.

∴ Infinitely many circles pass through A and B.

2.

r2 = AM2 + d2 (M = midpoint of AB, d = OM)

⇒ r ≥ AM = ½AB

∴ Not all radii; smallest radius = ½AB (centre at M, AB a diameter); no largest radius.

3.

r2 = (½AB)2 + d2; d increases ⇒ r increases

∴ The radii increase.

4.

Larger radius ⇒ flatter arc

∴ The circles appear less curved.

5.

A, B on the boundary: infinitely many squares

A, B as corners: AB as a side → 2 squares; AB as a diagonal → 1 square

∴ 3 squares

Common mistakes that cost marks

  • Answering “one circle” or “two circles” for part 1. Any point on the perpendicular bisector can be the centre, so there are infinitely many.
  • Giving the smallest radius as AB instead of ½AB. The smallest circle has AB as a diameter.
  • In part 5, counting only the 2 squares with AB as a side and missing the square with AB as a diagonal.

How this can come in the exam

MCQ (1 mark)

Two points P and Q are given. How many squares can be drawn with both P and Q as vertices?

  1. 1
  2. 2
  3. 3
  4. 4
Show answer

(C) 3
PQ can be a side with the square on either side of PQ (2 squares), or PQ can be a diagonal (1 square): 3 squares.

Assertion–Reason (1 mark)

Assertion (A): Infinitely many circles pass through two given points A and B.
Reason (R): Every point on the perpendicular bisector of AB is equidistant from A and B.

  1. Both A and R are true, and R is the correct explanation of A.
  2. Both A and R are true, but R is not the correct explanation of A.
  3. A is true, but R is false.
  4. A is false, but R is true.
Show answer

(A) Both A and R are true, and R is the correct explanation of A.
Both are true, and each point of the perpendicular bisector (R) can be the centre of a circle through A and B, which gives infinitely many circles (A).

Try one yourself

A and B are 10 cm apart. A circle through A and B has its centre 12 cm from the midpoint of AB. Find its radius.

Show answer

r = √(52 + 122) = √169 = 13 cm.

More questions like this

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