Two points, many circles
Circles through two points
Two pins
Pin two points A and B on a sheet of paper. Now try to draw a circle that passes through both of them.
How many different circles pass through both A and B?
What this lesson covers
The idea
The perpendicular bisector of AB is the locus of points equidistant from A and B, so every point on it is the centre of a circle through A and B, and the centres of all such circles lie on it.
Two pins
Pin two points A and B on a sheet of paper. Now try to draw a circle that passes through both of them.
How many different circles pass through both A and B?
- Only one
- Exactly two
- Many, many circles
Find the centres
A and B are fixed. The circle has centre O and passes through A. Drag O to places where the circle passes through B as well, then let go to keep the circle.
The centres line up
The perpendicular bisector of AB is the locus of points equidistant from A and B. So every point on it is the centre of a circle through A and B, and the centres of all such circles lie on it.
A centre O must have OA = OB. The midpoint of AB is one such point: its circle has AB as a diameter. Every other point of the bisector gives one more circle.
Notes
The centres of all circles through A and B lie on the perpendicular bisector of AB, and every point of it is such a centre.
Check yourself
How many circles pass through two given points A and B?
Where do the centres of all the circles through A and B lie?
O is on the perpendicular bisector of AB, and OA = 7 cm. What is OB, in cm?
Answer: 7 cm
O is equidistant from A and B, so OB = OA = 7 cm.
A and B are 12 cm apart. The circle with centre at the midpoint of AB passes through A and B. What is its radius, in cm?
Answer: 6 cm
AB is a diameter, so the radius is half of it: 12 ÷ 2 = 6 cm.
- Only one. One circle is for three points that are not on a line. Two points have many circles.
- Exactly two. Every point on the perpendicular bisector gives a circle, so there are infinitely many.
- Infinitely many — correct. Yes. Every point of the perpendicular bisector of AB is the centre of one.
- On the perpendicular bisector of AB — correct. Yes. A centre has to be equidistant from A and B.
- Only at the midpoint of AB. The midpoint is one such centre, but every other point of the perpendicular bisector works too.
- On the line AB. Only the midpoint of AB is on the line AB and equidistant from A and B. The other centres are off that line.