Need Help with Locus problems

Question

Parallel lines l and m are 8 centimeters apart and A is point on line l. How many points are equidistant from l and m and also 4 centimeters from A?

Answer 1

All points 4cm from A can be represented by a circle with its centre at A. So the equidistant point must lie on the circumference of the circle and it must be 4cm away from A on line l. The only point which is 4cm from A and 4cm away from parallel line m is the closest approach of the circle centre A to m. This point is 4cm along the perpendicular from A on l to a point B (directly opposite A) on m. AB is length 8cm, the distance between l and m. So the point is the midpoint of AB. All other points on the A circle are further away from m.