Math101 · Independent Ontario learning library
Math101Reflections
A reflection maps every point across a mirror line so that the line is the perpendicular bisector of the segment joining point and image.
Reflections model symmetry, optics, design, and congruence. Their perpendicular-bisector definition supports constructions and coordinate rules.
Intuition and core definition
A reflection maps every point across a mirror line so that the line is the perpendicular bisector of the segment joining point and image. Reflections preserve lengths and angles but reverse orientation. Points on the mirror line remain fixed.
Notation, language, and conditions
Common coordinate rules are across the $x$-axis $(x,y)\mapsto(x,-y)$, $y$-axis $(x,y)\mapsto(-x,y)$, line $y=x$: $(x,y)\mapsto(y,x)$, and vertical line $x=a$: $(x,y)\mapsto(2a-x,y)$.
Why this idea matters
A reflection reverses orientation while preserving length and angle, with the mirror line perpendicularly bisecting every point-image segment.
A dependable method
- Identify or construct the line of reflection.
- From each point, draw a perpendicular to the mirror line.
- Place the image the same distance on the opposite side.
- Apply an exact coordinate rule when the mirror is a standard line.
- Check midpoint and perpendicularity for point-image segments.
