Symmetry
55. Symmetric Point · where every point finds its twin
OP always equals OQ, and AB is always perpendicular to PQ.
Two points P and Q are symmetric about a line AB when AB is the perpendicular bisector of segment PQ. That means OP = OQ (equidistant) AND AB ⊥ PQ (perpendicular).
What this lesson covers
Try to break it
What happens if P lands exactly on line AB? Try it! O and Q will merge with P. The symmetry line becomes a mirror that folds the point onto itself.
How you build it
Construct the mirror image of P across line AB.
- Place point A — one point on the mirror line.
- Place point B — the mirror line runs through A and B.
- Draw the mirror line through A and B.
- Place point P, clearly off the mirror line — this is the point to reflect.
- Draw an arc centred at P, opened wide enough to cross the mirror line at two points — click P, then click past the line.
- Mark point U where the arc crosses the mirror line on one side.
- Mark point V where the arc crosses the mirror line on the other side.
- With the same opening, draw an arc centred at U.
- With the same opening, draw an arc centred at V — it crosses the arc from U.
- Mark point Q where the arcs from U and V cross, on the opposite side of the mirror line from P. Q is the mirror image of P.
- Join P and Q — PQ crosses the mirror line at a right angle, and the line cuts PQ in half.
The proof, step by step
Prove that P and Q are symmetric about line AB.
- AB is perpendicular to PQ by construction.
- O is the midpoint of PQ because OP = OQ by construction.
- A line perpendicular to a segment at its midpoint is its perpendicular bisector.
- Therefore, P and Q are symmetric with respect to AB.
Worked example
In the diagram, line AB is the perpendicular bisector of segment PQ. If OP = 4.5 cm, what is the length of PQ?
Since AB is the perpendicular bisector, it cuts PQ exactly in half at O. Each half (OP and OQ) is 4.5 cm, so the full length is 4.5 + 4.5 = 9 cm.
- 4.5 cm
- 9 cm — correct
- 18 cm
- 2.25 cm