IGCSE Maths / Transformation and similarity

Reflecting a shape in a line revision notes

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

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A reflection flips a shape across a mirror line while keeping its size and shape unchanged. Each corner and its image must be the same perpendicular distance from the given line.

How to reflect a shape in a line

The mirror line is given in the question, and it may not be an axis. Draw it on the grid before moving the shape. For each corner, measure to the mirror at right angles, then mark the image the same distance on the other side. Joining the image corners gives a shape with the same size and shape, but facing the other way. Name the reflection using the given line, for example, reflection in the line x = 1. The corner-to-mirror gap and mirror-to-image gap are equal. The full distance between a corner and its image is twice the distance from the corner to the mirror.

Reflecting in an upright line

For an upright mirror such as x = 1, count horizontally. The x-coordinate changes because the point moves sideways, while the y-coordinate stays the same. For example, the vertices of triangle ABC are (3, 1), (6, 1) and (3, 4). A at (3, 1) is 2 squares to the right of x = 1, so A′ is 2 squares to its left, at (−1, 1). B at (6, 1) is 5 squares to the right of x = 1, so B′ is at (−4, 1). C at (3, 4) is 2 squares to the right of x = 1, so C′ is at (−1, 4).

Reflecting in a flat line

For a flat mirror such as y = 1, count vertically. The y-coordinate changes because the point moves up or down, while the x-coordinate stays the same. Triangle ABC has vertices (2, 3), (5, 3) and (2, 6). A at (2, 3) is 2 squares above the line, so A′ is at (2, −1). B at (5, 3) is 2 squares above the line, so B′ is at (5, −1). C at (2, 6) is 5 squares above the line, so C′ is at (2, −4).

Checking the mirror line and fixed corners

Use the line stated in the question, not a familiar axis by habit. A point at (3, 1) reflected in x = 1 is 2 squares from the mirror, so its image is at (−1, 1). Reflecting it in the y-axis instead would give (−3, 1), which is a different reflection. Any corner on the mirror line is already the same distance from both sides, so it does not move. An upright mirror changes x only. A flat mirror changes y only. If a corner lies on the mirror line, leave it in place.

Quick check: reflect a point in y = 2

The point (4, 5) is 3 units above the flat mirror y = 2. Keep its x-coordinate and place it 3 units below the mirror. The image is (4, −1).