Strand 2 — Geometry & Trigonometry

Transformations: Translation, Axial & Central Symmetry, Rotation

1st Year · 2nd Year · 3rd Year (Junior Cert)

✓By the end of this lesson students will be able to define and identify translation, axial symmetry, central symmetry, and rotation.
✓By the end of this lesson students will be able to perform translations of points and 2D shapes using translation vectors.
✓By the end of this lesson students will be able to construct images of points and 2D shapes under axial symmetry.
✓By the end of this lesson students will be able to construct images of points and 2D shapes under central symmetry.
✓By the end of this lesson students will be able to construct images of points and 2D shapes under rotation about a given centre, angle, and direction.

Key concepts

Translation

A translation is a transformation that moves every point of a figure or object by the same distance in a given direction. It is defined by a translation vector, which specifies the horizontal and vertical shift. The image of a point P(x, y) translated by a vector (a, b) is P'(x+a, y+b).

P(x, y) → P'(x+a, y+b) for vector (a, b)

Axial Symmetry

Axial symmetry (or reflection) is a transformation where each point of a figure is mapped to an image point such that the line segment joining the point and its image is perpendicular to, and bisected by, the axis of symmetry. The axis of symmetry acts like a mirror.

Central Symmetry

Central symmetry (or point reflection) is a transformation where each point of a figure is mapped to an image point such that the centre of symmetry is the midpoint of the line segment joining the point and its image. It is equivalent to a rotation of 180° about the centre of symmetry.

P(x, y) about centre C(a, b) → P'(2a-x, 2b-y)

Rotation

A rotation is a transformation that turns a figure about a fixed point called the centre of rotation, through a given angle and in a given direction (clockwise or anti-clockwise). The distance from the centre of rotation to any point on the figure is the same as the distance from the centre of rotation to its image.

Key facts to remember

1Transformations are movements of geometric figures. They are isometries, meaning they preserve size and shape.
2A translation is defined by a vector, indicating direction and distance.
3Axial symmetry requires an axis of symmetry (a line). The image is a mirror reflection.
4Central symmetry requires a centre of symmetry (a point). It is equivalent to a 180° rotation.
5Rotation requires a centre of rotation, an angle of rotation, and a direction (clockwise or anti-clockwise).
6The image of a point P under a transformation is denoted P' (P prime).
7For axial symmetry, the axis is the perpendicular bisector of the line segment connecting a point and its image.
8For central symmetry, the centre is the midpoint of the line segment connecting a point and its image.

Worked examples

Example 1

Triangle ABC has vertices A(1, 2), B(4, 1), and C(2, 5). Find the coordinates of the image of △ABC, denoted △A'B'C', under a translation by the vector (2, -3).

ITo find the image of each vertex, add the components of the translation vector (2, -3) to the coordinates of the original vertex.

IIFor A(1, 2): A' = (1+2, 2-3) = (3, -1)

IIIFor B(4, 1): B' = (4+2, 1-3) = (6, -2)

IVFor C(2, 5): C' = (2+2, 5-3) = (4, 2)

Answer

The vertices of the translated triangle are A'(3, -1), B'(6, -2), and C'(4, 2).

A translation shifts the entire shape without changing its orientation or size.

Example 2

Find the image of the point P(3, -2) under axial symmetry in the line y = x.

IFor axial symmetry in the line y = x, the x and y coordinates of the point are swapped.

IIOriginal point P(x, y) = P(3, -2).

IIIImage point P'(y, x) = P'(-2, 3).

Answer

The image of P(3, -2) under axial symmetry in the line y = x is P'(-2, 3).

For axial symmetry in the x-axis, P(x, y) → P'(x, -y). For axial symmetry in the y-axis, P(x, y) → P'(-x, y).

Example 3

Point D(5, 1) is rotated 90° anti-clockwise about the origin (0, 0). Find the coordinates of its image, D'.

IFor a rotation of 90° anti-clockwise about the origin, the rule is P(x, y) → P'(-y, x).

IIApply this rule to point D(5, 1).

IIID'(x', y') = D'(-1, 5).

Answer

The image of D(5, 1) after a 90° anti-clockwise rotation about the origin is D'(-1, 5).

For a 90° clockwise rotation about the origin, the rule is P(x, y) → P'(y, -x). For a 180° rotation (clockwise or anti-clockwise) about the origin, the rule is P(x, y) → P'(-x, -y).

Common mistakes

✗Confusing clockwise and anti-clockwise directions for rotations, especially when rotating about the origin.
✗Incorrectly applying the rules for axial symmetry in specific lines (e.g., swapping x and y for y=x, but negating y for x-axis).
✗Not using a ruler, protractor, and compass accurately when performing geometric constructions for transformations.
✗Forgetting to label the image points with prime notation (e.g., A' instead of A).
✗Miscalculating coordinates when translating, especially with negative numbers in the vector.

Exam tips

★Always use a sharp pencil, ruler, protractor, and compass for constructions. Accuracy is key.
★Read the question carefully to identify the type of transformation, the specific parameters (vector, axis, centre, angle, direction).
★Label all original points and their image points clearly (e.g., A and A').
★If working on a coordinate plane, draw and label the axes, and use a consistent scale.
★Check your final image to ensure it is congruent (same size and shape) to the original figure, and that its orientation matches the transformation.

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