IGCSE Maths / Transformation and similarity

Equations of common mirror lines revision notes

Edexcel International GCSE Maths (4MA1), chapter 14: Transformation and similarity, specification point 5.2d. Free lesson.

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A reflection is described by the equation of its mirror line. Recognising the common mirror lines helps you name the transformation accurately and reflect points correctly.

What a mirror line tells you

A mirror line is the line a shape flips over. Each point and its image lie on opposite sides of the line, the same distance away. This distance is measured at right angles to the mirror, and points already on the line stay still. To describe a reflection, give the mirror line’s equation, not just a drawing. For example, write reflection in the line x = 3.

How to reflect a shape in x = 3

The line x = 3 is upright, so every point on it has x-coordinate 3. Each vertex moves straight across to the mirror and the same distance beyond it. The x-coordinate changes, while the y-coordinate stays the same. Triangle ABC has vertices (4, 1), (7, 1) and (4, 3). A is 1 square from the mirror, so its image is 1 square on the other side. B is 4 squares from the mirror, so its image is 4 squares on the other side.

How to recognise the common mirror lines

An upright line fixes the x-coordinate, while a flat line fixes the y-coordinate, or height. The diagonal y = x has matching coordinates. The other diagonal, y = −x, has opposite coordinates. x = 3 is upright. Points on it include (3, 0) and (3, 5). A reflection maps (x, y) to (6 − x, y). y = −1 is flat. Points on it include (0, −1) and (4, −1). A reflection maps (x, y) to (x, −2 − y). y = x rises to the right. Points on it include (2, 2) and (−3, −3). A reflection swaps the coordinates: (x, y) maps to (y, x). The axis versions are x = 0, the y-axis, and y = 0, the…

How to name a line from its direction

Choose the coordinate that stays the same along the line. An upright line has a constant x-coordinate, so its equation begins x =. A flat line has a constant y-coordinate, so its equation begins y =. The upright line through 3 on the x-axis is x = 3, not y = 3. Its height changes as you move along it, but x stays 3. The flat line through 3 on the y-axis is y = 3.

How to find a mirror line from two points

Compare the coordinates of the two points. If their y-coordinates match, they are at the same height and lie on a flat line. If their x-coordinates match, they lie on an upright line. The points (0, 4) and (5, 4) have the same y-coordinate, so their line is flat and has equation y = 4.