Symmetry

107. Mirror on the X-Axis · flipping coordinates with symmetry

The image P' always shares P's x-coordinate and has the exact opposite y-coordinate.

OPP'MxyPM = 200PM = 200MP' = 200MP' = 200x = 300x = 300y = 200y = 200x = 300x = 300y = -200y = -200
Reflection across the x-axis: point P(x, y) maps to P′(x, −y). The x-coordinate stays; the y-coordinate flips sign. The x-axis itself is the line of symmetry.

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Selina ICSE: Symmetry

What this lesson covers

Try to break it

Try dragging P far to the left or right. Does P' stay perfectly aligned vertically? What happens to the distance from the x-axis?

How you build it

Reflect a point across the x-axis.

  • Pick the Point tool. Click on the x-axis to drop point A — anywhere on it. Use a grid intersection if you want a clean coordinate.
  • Click again on the x-axis, at a different spot, to drop point B. A and B together identify the mirror line.
  • Click anywhere above the x-axis 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 past the x-axis. 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 the x-axis, the two arcs cross at exactly one point.
  • Switch to the Point tool. Below the x-axis, the two arcs cross at exactly one point. Click that intersection — this is P', the mirror image of P. Because AP' = AP and BP' = BP, both A and B are equidistant from P and P'. If P = (x, y), then P' = (x, -y).
  • Pick the Line tool. Click P, then P'. The line PP' crosses the x-axis at a right angle, and it is split exactly in half by the x-axis. Both A and B being equidistant from the endpoints means the x-axis IS the perpendicular bisector of PP'.

The proof, step by step

Prove that reflection in the x-axis keeps the x-coordinate and negates the y-coordinate.

  • The x-axis acts as a horizontal mirror line for the reflection.
  • The line segment PP' is drawn perpendicular to the x-axis.
  • The x-axis bisects PP' at point M, meaning PM = MP'.
  • Since horizontal position is unchanged, P and P' share the exact same x-coordinate.
  • Since vertical distance is equal but directions are opposite, P' has coordinate -y.

Worked example

If point A(4, 7) is reflected in the x-axis, what are the coordinates of its image A'?

Reflection in the x-axis keeps the x-coordinate unchanged and negates the y-coordinate. Thus, (4, 7) becomes (4, -7).

  • (-4, 7)
  • (4, -7) — correct
  • (-4, -7)
  • (7, 4)
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