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) Yes, there are circles of all possible radii passing through A and B.
(2) Half the length of AB
(3) There is no largest circle passing through A and B.
Explanation
Yes, there are circles of all possible radii passing through A and B. Any point on the perpendicular bisector of AB can be the centre of a circle passing through A and B, and as we move along this bisector, we get circles of varying radii.
The radius of the smallest circle passing through A and B is half the length of AB. This occurs when the midpoint of AB is the centre, making AB the diameter of the circle.
There is no largest circle passing through A and B. As we move away from segment AB along its perpendicular bisector, the radii of the circles increase continuously without any upper limit.
Solution Steps
Step 1: Identify that the centres of all circles passing through A and B lie on the perpendicular bisector of AB.
Step 2: Determine the smallest radius by finding the circle with the midpoint of AB as its centre, where AB is the diameter and the radius is half the length of AB.
Step 3: Conclude there is no largest circle because moving infinitely along the perpendicular bisector increases the radius without limit.