What is a reflection?
A reflection is an isometry that flips an object across a fixed straight line called the axis of reflection (a mirror line). Each point of the image is the same distance from the axis as the matching point of the object, but on the opposite side. The object and image are congruent, but the image is laterally inverted (like a mirror).
Key idea / 要点
Reflection in the x-axis: (a, b) → (a, −b). Reflection in the y-axis: (a, b) → (−a, b). Points ON the axis do not move (they are invariant).
Perpendicular distance
The axis of reflection is the perpendicular bisector of the line segment joining a point to its image. This means the mirror line cuts that segment in half at a right angle.
Worked example / 例题
Reflect point P(3, 5) in the x-axis, then reflect Q(3, 5) in the y-axis.
In the x-axis: keep x, change sign of y → P′ = (3, −5).
In the y-axis: change sign of x, keep y → Q′ = (−3, 5).
A point lying on the axis of reflection maps onto itself. For example, reflecting (0, 4) in the y-axis gives (0, 4) — unchanged, because it is on the y-axis. Reflecting twice in the same axis brings a shape back to its starting position.
Reflections also appear in everyday life: the mirror line of a butterfly, the letter M or the still surface of a lake all show reflective symmetry, where one half is the mirror image of the other.
Remember / 记住
- Reflection needs an axis (mirror line).
- x-axis: (a, b) → (a, −b).
- y-axis: (a, b) → (−a, b).
- Points on the axis stay fixed (invariant).