No, all the points in a given plane that are 1 inch from a given point lie on a circle, radius 1 inch. And the complete circle only measures inches all the way round.2. Let be a closed non-intersecting curve all of whose points are 1 inch from a given point. Is it possible that is longer than 1 mile? Justify your answer.
Yes it can. You can make the circumference of the circle as big as you like!3. Let be a triangle all of whose sides are shorter than 1 inch. Can the circle which passes through , , and have circumference which is larger than 1 mile? Justify your answer.
How? Well, you can find the centre of a circle by drawing the perpendicular bisectors of two of its chords, and finding where they meet. So if we make, say and , both a half-inch long (so that is still less than 1 inch) but very nearly lying in a straight line (in other words is very nearly ), we can make these perpendicular bisectors very nearly parallel. Hence we can make their point of intersection as far away as we like.