IGCSE Maths / Transformation and similarity

Reflecting in y = x and y = -x

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

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A reflection in a diagonal mirror places each point the same perpendicular distance on the other side of the line. The coordinate rules for y = x and y = −x let you check your image and describe the transformation precisely.

Reflecting in the line y = x

The line y = x passes through the origin and points where the coordinates are equal, such as (1, 1) and (2, 2). When you reflect a point in this line, its coordinates swap over. Each point and its image lie on a line at right angles to the mirror, at equal distances from it. Coordinate rule: (x, y) → (y, x). A(1, 3) maps to A′(3, 1), and B(1, 5) maps to B′(5, 1).

Reflecting in the line y = −x

The line y = −x slopes down through the origin, passing through (1, −1) and (2, −2). Here you swap the coordinates first, then change the sign of each one. The image is still the same perpendicular distance from the mirror as the original point. Coordinate rule: (x, y) → (−y, −x). For A(2, 1), the image is A′(−1, −2).

Count towards the diagonal mirror

Do not count straight across the page. That measures from a different mirror line, so it gives the wrong image. For example, reflecting P(3, 1) in the y-axis would give (−3, 1), but reflecting it in y = x gives (1, 3). The original point and its image must be on opposite sides of the mirror. The mirror must cut the line between them in half at a right angle. A quick check for y = x is to swap the coordinates and compare with your drawing.

Describe the reflection precisely

A reflection is described by naming the transformation and giving the equation of its mirror line. The mirror line is the key information that fixes the reflection. Use: reflection in the line y = x. Or: reflection in the line y = −x.

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