Leave lesson

Geometry & measures · Transformations

1 / 10

Reflections & rotations

Reflecting in named mirror lines including y = x and y = −x, rotating about a given centre with a sense, and the exact wording needed to DESCRIBE a transformation for full marks.

Geometry & measures · Transformations

Reflections & rotations

Reflecting in named mirror lines including y = x and y = −x, rotating about a given centre with a sense, and the exact wording needed to DESCRIBE a transformation for full marks.

Why it works

A transformation moves every point by the same rule. Reflections and rotations are congruent transformations: the image is the same shape and the same size as the object — only its position or orientation changes.

Reflection: every point crosses the mirror at right angles, keeping its distance. If AA is 3 squares above the mirror line, A′A' is 3 squares below it, measured perpendicular to the line. So reflect one vertex at a time, then join up. In the two diagonal cases the coordinates swap:

in y=x:(a,b)↦(b,a),in y=−x:(a,b)↦(−b,−a).\text{in } y = x: (a, b) \mapsto (b, a), \qquad \text{in } y = -x: (a, b) \mapsto (-b, -a).

Before reflecting in y=xy = x or y=−xy = -x, draw the mirror line — most lost marks here come from reflecting in an imagined line that was never drawn.

Rotation: every point turns through the same angle about the centre. The centre is the one point that doesn't move. Distance from the centre is preserved, so tracing paper is a legitimate exam tool: pin it at the centre, trace the shape, turn it by the angle, and read off where it lands. Positive turns are anticlockwise by convention, so the sense must be stated: 90°90° clockwise and 90°90° anticlockwise land in different places (though 180°180° needs no sense — both ways agree).

Finding a centre of rotation you weren't given: join each point to its image and construct the perpendicular bisector of two of those segments — the centre is where the bisectors cross. In practice, most GCSE centres can be found by trial with tracing paper.

Describing a transformation is a marked skill with a fixed checklist. Name the transformation, then give exactly the information it needs — no more, no less:
  • Reflection → the equation of the mirror line ("reflection in the line
x=2x = 2", not "reflection in the vertical line").
  • Rotation → the angle, the direction, and the centre ("rotation 90°90°
clockwise about (0,1)(0, 1)").

One transformation only. If the question says "describe fully the single transformation", writing "reflect then translate" scores zero even if it gets the shape there. And naming two transformations at once ("rotation and reflection") is a common way to lose an otherwise-earned mark.