Reflections –

Reflections are one of the four transformations you need to know for GCSE Maths. A reflection flips a shape over a mirror line so that the image is the same distance from the line as the original, on the opposite side. This guide covers reflecting in the axes, the lines y = x and y = −x, and other lines, with step-by-step methods for both performing and describing reflections.

What Is a Reflection?

A reflection creates a mirror image of a shape across a given line, called the mirror line or line of reflection. Every point on the original shape is the same perpendicular distance from the mirror line as the corresponding point on the image, but on the opposite side.

Key Formulas

Step-by-Step Method

1. Draw the mirror line on the coordinate grid if it is not already shown.
2. For each vertex of the shape, count the perpendicular distance from the vertex to the mirror line.
3. Plot the reflected vertex the same distance on the other side of the line.
4. Connect the reflected vertices to form the image.
5. Label the image with dashed lines or primes (A', B', C') to distinguish it from the original.

Worked Example 1 — Foundation Level

Question: Reflect the triangle with vertices A(1, 3), B(4, 3), and C(4, 1) in the x-axis.

Working:

Reflection in the x-axis: (x, y) maps to (x, −y).

A(1, 3) maps to A'(1, −3)

B(4, 3) maps to B'(4, −3)

C(4, 1) maps to C'(4, −1)

Answer: The image has vertices A'(1, −3), B'(4, −3), C'(4, −1).

Worked Example 2 — Higher Level

Question: Reflect the point P(3, 5) in the line y = x. State the coordinates of the image.

Working:

Reflection in y = x: swap the x and y coordinates.

P(3, 5) maps to P'(5, 3).

Answer: P' = (5, 3).

Worked Example 3 — Exam Style

Question: Shape A has been reflected to give shape B. Shape A has a vertex at (2, 4) and the corresponding vertex on shape B is at (2, −2). Describe the single transformation fully.

Working:

The x-coordinate stays the same (2 to 2), so the mirror line is horizontal.

The midpoint of the y-coordinates is (4 + (−2)) ÷ 2 = 1.

The mirror line is y = 1.

Answer: A reflection in the line y = 1.

Common Mistakes

• Not describing the transformation fully. You must state the type of transformation (reflection) and the equation of the mirror line. Saying just "reflection" without the line loses marks.
• Counting diagonally instead of perpendicularly. Always measure the shortest (perpendicular) distance from each point to the mirror line.
• Confusing reflection with rotation. A reflected shape is a mirror image — it appears "flipped". A rotated shape keeps the same orientation.

Exam Tips

• When describing a reflection, always write: "Reflection in the line [equation]." Both parts are needed for full marks.
• To find the mirror line when given the object and image, find the midpoint of each pair of corresponding points — the mirror line passes through all midpoints.
• Use tracing paper in the exam to check your reflection is accurate.
• Reflections do not change the size or shape of the figure — the object and image are congruent.

Practice Questions

Q1 (Foundation): Reflect the point (5, 2) in the y-axis.

Q2 (Foundation): Reflect the point (3, −4) in the x-axis.

Q3 (Higher): A shape is reflected so that the point (1, 6) maps to (6, 1). What is the mirror line?

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Related Topics

• Rotations
• Translations and Column Vectors
• Enlargement and Centre of Enlargement
• Transformations — Reflection, Rotation, Translation

Summary

• A reflection flips a shape over a mirror line. Every point on the image is the same perpendicular distance from the mirror line as the corresponding point on the original, but on the opposite side. You must know the coordinate rules for reflecting in the x-axis, y-axis, y = x, and y = −x. When describing a reflection, always state the transformation type and the equation of the mirror line.