Question 4

The center of any circle passing through two points lies on the perpendicular bisector of the segment connecting them.

Step 1 — Understand the geometry

Let's consider two fixed points, A and B. Any circle passing through A and B has a center. Let this center be C. The distance from C to A is the radius. The distance from C to B is also the radius. So, CA must be equal to CB. This means C is equidistant from A and B. All points equidistant from A and B lie on the perpendicular bisector of AB. Therefore, the center C must lie on the perpendicular bisector of AB.

Step 2 — Find the minimum radius

We want to find the least possible radius. The radius is the distance from the center C to point A. The center C lies on the perpendicular bisector of AB. We need to make the distance CA as small as possible. This happens when C is the midpoint of the segment AB. When C is the midpoint of AB, the segment AB itself becomes the diameter of the circle. The radius is half of the diameter. So, the least possible radius is half the length of AB.

$\text{Least Radius} = \frac{\text{Length of AB}}{2}$

$\boxed{\frac{\text{AB}}{2}}$

Answer

The least possible radius of a circle through A and B is half of AB.