Section A — Chapters 1–22

Transformation Geometry

Junior Cycle — 1st Year

✓Identify and draw axes of symmetry for 2D shapes.
✓Understand the meaning of a transformation in geometry.
✓Perform axial symmetry (reflection) of points and 2D shapes about a given line.
✓Perform central symmetry (rotation by 180°) of points and 2D shapes about a given centre point.
✓Perform translation of points and 2D shapes along a given vector.

Key concepts

Transformation

A transformation is a movement of a shape or point from one position to another. The new position is called the image, and the original position is called the object. The size and shape of the object usually remain the same after a transformation.

Axis of Symmetry

An axis of symmetry is a line that divides a shape into two identical halves, such that if you fold the shape along this line, the two halves match exactly.

Axial Symmetry (Reflection)

Axial symmetry, also known as reflection, is a transformation where every point of an object is moved to an image point on the opposite side of a line (the axis of symmetry), and is the same perpendicular distance from the line. The object and its image are congruent (same size and shape).

Central Symmetry (Rotation by 180°)

Central symmetry is a transformation where every point of an object is rotated 180 degrees about a fixed point called the centre of symmetry. For any point P, its image P' is such that P, the centre, and P' are collinear, and the distance from P to the centre is equal to the distance from the centre to P'. The object and its image are congruent.

Translation

Translation is a transformation where every point of an object is moved the same distance in the same direction. This movement is often described by a vector. The object and its image are congruent.

Key facts to remember

1Transformations move shapes or points without changing their size or shape (they are congruent).
2An axis of symmetry divides a shape into two identical mirror images.
3In axial symmetry, the object and its image are equidistant from the axis of symmetry.
4In central symmetry, the object, the centre of symmetry, and the image are collinear.
5In translation, every point moves by the same vector.
6The image of a point (x, y) reflected about the x-axis is (x, -y).
7The image of a point (x, y) reflected about the y-axis is (-x, y).
8The image of a point (x, y) under central symmetry about the origin (0, 0) is (-x, -y).

Worked examples

Example 1

Reflect the point A(2, 3) about the x-axis to find its image A'.

IIdentify the object point A(2, 3) and the axis of symmetry (the x-axis).

IIThe x-axis is the line y = 0.

IIIFor axial symmetry about the x-axis, the x-coordinate remains the same, and the y-coordinate changes sign.

IVSo, A(2, 3) becomes A'(2, -3).

Answer

A'(2, -3)

Remember that the object and image are equidistant from the axis of symmetry.

Example 2

Find the image of the triangle with vertices P(1, 1), Q(3, 1), R(2, 4) under central symmetry about the origin (0, 0).

IIdentify the centre of symmetry as the origin (0, 0).

IIFor central symmetry about the origin, each point (x, y) maps to (-x, -y).

IIIApply this rule to each vertex:

IVP(1, 1) maps to P'(-1, -1).

VQ(3, 1) maps to Q'(-3, -1).

VIR(2, 4) maps to R'(-2, -4).

VIIConnect the image points P', Q', R' to form the image triangle.

Answer

The image triangle has vertices P'(-1, -1), Q'(-3, -1), R'(-2, -4).

Each object point, the centre, and its image point are collinear.

Example 3

Translate the point B(1, 2) by the vector (3, -1) to find its image B'.

IIdentify the object point B(1, 2) and the translation vector (3, -1).

IITo translate a point (x, y) by a vector (a, b), the new point is (x+a, y+b).

IIIAdd the x-component of the vector to the x-coordinate of B: 1 + 3 = 4.

IVAdd the y-component of the vector to the y-coordinate of B: 2 + (-1) = 1.

Answer

B'(4, 1)

A translation moves every point the same distance in the same direction.

Common mistakes

✗Confusing axial symmetry with central symmetry.
✗Incorrectly identifying the axis or centre of symmetry.
✗Not measuring distances accurately when drawing images.
✗Applying the wrong sign change for coordinates in reflections (e.g., reflecting about x-axis and changing x-coordinate).
✗Translating in the wrong direction or by the wrong magnitude.

Exam tips

★Always use a ruler and pencil for drawing transformations to ensure accuracy.
★Label the object points (e.g., A, B, C) and their corresponding image points (e.g., A', B', C') clearly.
★For axial symmetry, remember that the line connecting an object point to its image point is perpendicular to the axis of symmetry.
★For central symmetry, ensure the object point, centre, and image point form a straight line.

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