EQN-04-11

Transformations & Symmetry

Translate, reflect, and rotate figures, and recognise lines of symmetry — what changes and what stays the same.

~5 min read•Difficulty 2/5

The three rigid motions

These move a shape without changing its size — distances and angles are preserved.

Translation: slide; every point moves the same amount.
Reflection: flip across a line (the mirror line).
Rotation: turn about a fixed point by an angle.

Coordinate rules (common cases)

Reflect across the x -axis : (x, y) \to (x, - y)

Reflect across the y -axis : (x, y) \to (- x, y)

Rotate 180 ° about origin : (x, y) \to (- x, - y)

Translate by (a, b) : (x, y) \to (x + a, y + b)

What stays the same

Rigid motions preserve lengths, angles, and area. Only position (and, for reflection, orientation) changes.

Lines of symmetry

A figure has a line of symmetry if folding along it matches the two halves. A square has 4, a (non-square) rectangle has 2, an equilateral triangle has 3, a circle has infinitely many.

Diagram

Reflecting across the x-axis flips the sign of the y-coordinate.

Worked examples

Example 1 (difficulty 2/5)

The point

(3, - 5)

is reflected across the x-axis. What is the image?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

How many lines of symmetry does a (non-square) rectangle have?

◎ Predict the answer first:

Tips

Rigid motions preserve length, angle, and area — only position/orientation change.
Memorize the axis-reflection rules: x-axis negates y, y-axis negates x.
Count symmetry lines by imagining the folds that match the halves.

Common mistakes

Negating the wrong coordinate when reflecting

Why: The two axis rules look symmetric.

Fix: x-axis reflection changes y; y-axis reflection changes x.

Counting a rectangle’s diagonals as lines of symmetry

Why: They look like they split it evenly.

Fix: Folding along a diagonal does not match the halves of a non-square rectangle.

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