Reflections and Rotations

Learn how to reflect shapes in a mirror line and rotate shapes around a centre point on a coordinate grid.

What Are Transformations?

A transformation moves or changes a shape. The original shape is called the object and the new shape is the image. In a transformation, the image is congruent to (the same size and shape as) the object.

Two key transformations are reflection and rotation.

Note

Transformations appear in art, architecture, and nature. Patterns on wallpaper, tiles, and fabric all use reflections and rotations to create repeating designs.

Reflections

A reflection flips a shape across a mirror line (line of reflection). Every point in the image is the same distance from the mirror line as the corresponding point in the object, but on the opposite side.

Common mirror lines on a coordinate grid:

The $y$ -axis (the vertical axis): point $(x, y)$ maps to $(-x, y)$
The $x$ -axis (the horizontal axis): point $(x, y)$ maps to $(x, -y)$
The line $y = x$ : point $(x, y)$ maps to $(y, x)$

Example: Reflect the point $A(3, 2)$ in the $y$ -axis.

The $y$ -axis has equation $x = 0$ . The image is $A'(-3, 2)$ .

Tip

To reflect a shape, reflect each vertex (corner) individually, then join the image vertices to complete the image shape.

Rotations

A rotation turns a shape around a fixed point called the centre of rotation. You need to know three things to describe a rotation:

The centre of rotation (a coordinate)
The angle (e.g. $90°$ , $180°$ , $270°$ )
The direction (clockwise or anticlockwise)

Common rotations about the origin $(0, 0)$ :

Rotation	$(x, y)$ maps to
$90°$ clockwise	$(y, -x)$
$90°$ anticlockwise	$(-y, x)$
$180°$	$(-x, -y)$

Example: Rotate point $B(4, 1)$ by $90°$ anticlockwise about the origin.

Apply the rule $(-y, x)$ : $B' = (-1, 4)$

Describing Transformations

When describing a reflection, state: "Reflection in the line $y = \ldots$ " (or $x = \ldots$ ).

When describing a rotation, state the centre, angle, and direction.

Warning

A rotation of $90°$ clockwise is NOT the same as $90°$ anticlockwise. Always state the direction. A $180°$ rotation is the same in both directions.

Real world: architecture and facade design

The front elevation of a building is often designed with bilateral symmetry — a reflection across a vertical centre line. Architects apply a reflection transformation to ensure that windows, pillars, and ornaments on the left perfectly mirror those on the right.

Real world: textile pattern design

A fabric designer creates a motif at $(1, 1)$ and reflects it in the line $y = x$ to produce a complementary motif at $(1, 1)$ . Rotating the motif by $90°$ four times tiles the fabric with a consistent repeating pattern used in scarves and wallpaper.

Tip Always state three details when describing a rotation: the centre, the angle, and the direction (clockwise or anticlockwise). Missing any one makes the description incomplete.

Practice Problems

Reflect the point $C(5, -3)$ in the $x$-axis.

Show answer

Reflecting in the $x$-axis maps $(x, y) \to (x, -y)$. Image: $C'(5, 3)$.

Rotate point $D(2, 3)$ by $180°$ about the origin.

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$180°$ rotation maps $(x, y) \to (-x, -y)$. Image: $D'(-2, -3)$.