Transformation Geometry Notes | Junior Cycle Mathematics

Transformation geometry is the study of how shapes move across a plane by sliding, reflecting, or turning while keeping their size and shape intact. In the Junior Cycle Higher Level exam, you need to recognise and draw images under translations, axial symmetry, central symmetry, and rotations, and find all axes of symmetry in 2D shapes.

The Four Transformations

A transformation maps an original shape, called the object, onto a new shape, called the image. The four transformations on the Junior Cycle course are isometries: lengths, angles, and area stay the same, so the image is congruent to the object.

Transformation	What happens to the shape	Everyday picture
Translation	Slides every point the same distance and direction	Sliding a book
Axial symmetry	Flips over a mirror line (axis of symmetry)	Looking in a mirror
Central symmetry	Half-turn ( $180^\circ$ ) through a fixed centre point	Spinning a card halfway
Rotation	Turns around a fixed centre by a set angle	Turning clock hands

Identifying Transformations

To identify an unknown transformation, link corresponding points on the object and image to test for a translation vector, an axis of reflection, a centre of symmetry, or an angle of rotation.

Axial and Central Symmetry

In axial symmetry, a shape is reflected across a mirror line. For a point off the mirror line, the line crosses its joining segment halfway, at right angles. Points on the mirror line stay fixed.

A triangle and its reflection share a vertex on the mirror line. Corresponding vertices lie equally far from the line along perpendicular segments.

Reflection in a line: draw a perpendicular from each vertex to the mirror line; measure and continue equally far beyond it.
Axial symmetry in the $x$ -axis: keep $x$ , change the sign of $y$ : $(x, y) \to (x, -y)$ .
Axial symmetry in the $y$ -axis: change the sign of $x$ , keep $y$ : $(x, y) \to (-x, y)$ .

Central Symmetry

In central symmetry, the centre point is the midpoint of the line segment connecting each point to its image. Connect each vertex to the centre and extend the line the same distance beyond it.

Central symmetry in the origin $(0, 0)$ : change both signs: $(x, y) \to (-x, -y)$ .

A triangle and its half-turn image lie on opposite sides of the origin, which bisects every segment joining corresponding vertices.

Translations and Rotations

A translation slides every point of a shape by a fixed distance and direction. The translation $\overrightarrow{AB}$ moves every point by the vector from $A$ to $B$ .

If $A(1, 2)$ moves to $B(4, 1)$ , the shift is $+3$ horizontally and $-1$ vertically. Every point transforms as $(x, y) \to (x + 3, y - 1)$ .

Two congruent triangles on a square grid are connected by equal parallel arrows, each moving three units right and one unit down.

Rotations

A rotation requires three details: a centre of rotation, an angle of turn (such as $90^\circ$ or $180^\circ$ ), and a direction (clockwise or anticlockwise).

To draw a rotation:

Draw a straight line from the centre to a vertex.
Measure the required angle from this line using a protractor in the given direction.
Mark the image vertex at the exact same distance from the centre.
A rotation of $180^\circ$ about any point produces the same image as central symmetry in that point.

A scalene triangle rotates 90° anticlockwise about O. Equal radii connect O to a vertex and its image.

Axes of Symmetry in 2D Shapes

An axis of symmetry is a line dividing a shape into two mirror-image halves that match exactly when folded. Halving the area alone is not enough; the outer edges must coincide.

Shape	Number of Axes	Position of Axes
Equilateral triangle	$3$	From each vertex to the midpoint of the opposite side
Isosceles triangle (not equilateral)	$1$	Bisects the angle between the two equal sides
Scalene triangle	$0$	None
Square	$4$	Two midlines and two corner diagonals
Non-square rectangle	$2$	Two midlines through opposite sides (not diagonals)
Rhombus (not a square)	$2$	The two corner diagonals
Kite	$1$	The main diagonal between the unequal angles
Parallelogram (not rectangle/rhombus)	$0$	Has central symmetry, but no axis of symmetry
Regular $n$ -sided polygon	$n$	Regular pentagon has $5$ , regular hexagon has $6$
Circle	Infinite	Any straight line passing through the centre

A square has four dashed symmetry axes; a non-square rectangle has two midlines; a non-square rhombus has two diagonals.

Key terms

Object: The original geometric shape or point before any transformation is applied.
Image: The resulting position of a point or shape after undergoing a transformation, typically written with prime marks (such as A').
Translation: Sliding an object a specific distance in a given direction without turning or flipping it.
Axial Symmetry: Reflecting a shape across a mirror line: points on the line stay fixed, while for points off the line, the line perpendicularly bisects the segment to its image.
Central Symmetry: A transformation through a point where the centre is the exact midpoint between each point and its corresponding image point.
Rotation: Turning an object about a fixed centre point through a specified angle and direction.
Axis of Symmetry: A line dividing a 2D shape into two mirror-image halves that match perfectly when folded.
Centre of Rotation: The fixed point about which every point of an object turns during a rotation.

Check yourself

What is the image of $(-3, 5)$ under axial symmetry in the $x$ -axis?
$(-3, -5)$ , because reflection in the $x$ -axis changes only the sign of the $y$ -coordinate.
How many axes of symmetry does a regular hexagon have?
Six, because every regular polygon with $n$ sides has exactly $n$ axes of symmetry.
How many axes of symmetry does a parallelogram have if it is neither a rectangle nor a rhombus?
Zero, although it has central symmetry in the point where its diagonals intersect.
Which single transformation produces the exact same image as a rotation of $180^\circ$ about the origin?
Central symmetry in the origin $(0, 0)$ .

Transformation Geometry