Reflection (In x-axis, y-axis, x = a, y = a and the origin; Invariant Points)
298. Mirror Through the Origin · Flip both signs, keep the distance
O is always the midpoint of PP', and P' is always (-x, -y) relative to O.
Reflecting in the origin sends (x, y) → (−x, −y) — both signs change. The origin O is the midpoint of the segment joining each point to its image.
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For subscribers · Selina ICSE: Reflection (In x-axis, y-axis, x = a, y = a and the origin; Invariant Points)
What this lesson covers
Try to break it
Drag P anywhere. P' always lands at (−x, −y) — both coordinates flip. The origin O is the midpoint of PP', and OP = OP' at every position. Drag P to the origin and P' lands on top of P; the only fixed point of reflection through O is O itself.
How you build it
Reflect a point in the origin.
- Place point P. Its reflection P' will appear automatically.
The proof, step by step
Prove that reflection in the origin maps P(x, y) to P prime(−x, −y).
- Let P be (x, y) and O be (0, 0).
- By definition of reflection in origin, O is the midpoint of PP'.
- Midpoint formula: ((x + x')/2, (y + y')/2) = (0, 0).
- Solving gives x' = -x and y' = -y.
- Hence, P' is (-x, -y).
Worked example
If point A(3, -4) is reflected in the origin, what are the coordinates of its image A'?
Reflection in the origin negates both coordinates. A(3, -4) becomes A'(-3, 4).
- A'(3, 4)
- A'(-3, 4) — correct
- A'(-3, -4)
- A'(4, -3)