F3 Physics Reflection of light

Q: Why the size of his image remains unchanged and he can still see himself in the mirror when the person stands farther away from the mirror?

This is just the result of the Law of Reflection. The size of the image formed by reflection of light on a plan mirror is the same as the size of the object. Therefore, the image size should always be the same as the size of the person. Since the size of the person doesn't change, the image size would not be changed too.

Q: I tried to draw the diagram out and found out that the minimum size of mirror is the same. But can I know how to explain that situation?

This is again the result of the Law of Reflection. The distance of the image behind the mirror is the same as the object (i.e. the person) in front of the mirror.

When the person moves nearer to or farther away from the mirror, the image would be moved accordingly, so as to maintain equal distances between image-mirror and object-mirror. Hence, by geometrical symmetry, you would find that the mirror is always situated mid-way between the person and his image This gives the same length of mirror disregard the where the image and object are from the mirror.

Therefore, following the "mid-point thereom"in gemetry, the length of the mirror is half that of the image. As explained above, because the size of the image, which equals to the size of the person, is always unchanged. It follows that the minimum length of the mirror must be equal to half of the height of the person.