Symmetry
106. The Mirror's Promise · Reflection and the perpendicular bisector
Line AB always perpendicularly bisects the segment PP'.
A reflection about a line AB maps each point P to a point P′ such that AB is the perpendicular bisector of segment PP′. Distance from AB is preserved; the figure flips to its mirror image.
What this lesson covers
Try to break it
Try dragging P right across line AB. Watch P' jump to the other side instantly! The perpendicular bisector rule never breaks, even when P crosses the mirror. Cool, right?
How you build it
Reflect a point across line AB.
- Pick the Point tool. Click on the canvas to drop point A — one end of the mirror line.
- Click again, away from A, to drop point B.
- Pick the Segment tool. Click A, then click B.
- Switch to the Point tool. Click above (or below) line AB to drop point P — the point we will reflect.
- Pick the Arc tool. Click A for the centre, then click P to set the radius. The arc passes through P and curves down across AB. Every point on this arc is exactly the distance AP from A.
- Stay on the Arc tool. Click B for the centre, then click P for the radius. This second arc also passes through P. Below line AB, the two arcs cross at exactly one point — that crossing is the reflection of P.
- Switch to the Point tool. On the opposite side of AB from P, the two arcs cross at exactly one point. Click that intersection and call it P'. Because AP' = AP and BP' = BP by construction, both A and B sit at equal distances from P and P'.
- Pick the Segment tool. Click P, then P'. Now look: A is equidistant from P and P', and so is B. Two distinct points equidistant from the endpoints of a segment must lie on its perpendicular bisector — therefore line AB IS the perpendicular bisector of PP'. AB crosses PP' at a right angle and splits it in half. P' is the mirror image of P across AB.
The proof, step by step
Prove that the mirror line is the perpendicular bisector of P and its image P prime.
- By definition, reflection across a line is an isometry that fixes every point on the mirror line.
- The mirror line is exactly the locus of all points equidistant from P and P'.
- Therefore, the mirror line AB must perpendicularly bisect the segment PP'.
Worked example
In the given figure, line l is the mirror line. If point P is at a distance of 5 cm from l, what is the distance between P and its reflection P'?
The mirror line bisects the segment joining P and P'. Thus, PP' = 2 × distance(P, l) = 2 × 5 cm = 10 cm.
- 2.5 cm
- 5 cm
- 10 cm — correct
- 15 cm