Reflection and Invariant Points

Reflection is a transformation that flips a point across a fixed line called the mirror line (or axis of reflection). The image P' is the same perpendicular distance from the mirror as the object P, but on the opposite side, and the mirror line is the perpendicular bisector of the segment PP'. The shape and size never change - only the position. In ICSE Class 10 the mirror lines are the x-axis, the y-axis, the origin (a point), and the special lines x = a and y = b.

Three reflections are used constantly. Reflection in the x-axis keeps x and flips the sign of y: (x, y) becomes (x, -y). Reflection in the y-axis keeps y and flips x: (x, y) becomes (-x, y). Reflection in the origin flips both: (x, y) becomes (-x, -y). A clean way to remember: the coordinate named after the mirror axis is the one that survives unchanged.

A point is invariant under a reflection if its image is exactly itself - it does not move. This happens only when the point lies on the mirror line. So under reflection in the x-axis, invariant points are those with y = 0 (every point of the x-axis). Under reflection in the y-axis, invariant points have x = 0. Under reflection in the origin, only the origin (0, 0) is invariant. The word invariant simply means unchanged.

Reflections can be done one after another. A key Selina result: reflecting in the x-axis and then in the y-axis gives the same final image as a single reflection in the origin - because flipping the sign of y and then of x flips both. Mapping notation is used to record this: M-x(P) means the image of P after reflection in the x-axis. Questions often ask you to find such an image, name the single transformation that replaces two, or state which points stayed invariant.