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GRADE 7 · MATHEMATICS

Transformations and Symmetry

A transformation changes the position or orientation of a shape. Explore translations, reflections and rotations, then identify lines of symmetry and rotational symmetry while keeping track of lengths and angles.

Learning time: approximately 30–40 minutes · Curriculum-neutral learning support

What You Will Learn

Describe translations, reflections and rotations accurately.
Use direction, distance, centre and angle to specify a transformation.
Recognise line symmetry and rotational symmetry.
Understand that a rigid transformation preserves lengths and angles.

1. 1. What Is a Transformation?

A transformation maps every point of a shape to a new position. The original shape is the object; its image is the shape after the transformation. In a rigid transformation, corresponding side lengths and angles stay equal, so the image is congruent to the object.

Key idea: A transformation can move or turn a shape without stretching or changing its size.

2. 2. Translation: Slide a Shape

A translation slides every point the same distance in the same direction. It does not turn or flip the shape.

On a coordinate grid, a translation by vector (a, b) maps (x, y) to (x + a, y + b).

Example: Translate point A(2, 3) by vector (4, −1). New coordinates are (2 + 4, 3 − 1) = (6, 2).

Positive horizontal movement is to the right; negative is to the left. Positive vertical movement is up; negative is down.

3. 3. Reflection: Flip Across a Line

A reflection flips a shape across a mirror line. Each point and its image lie on opposite sides of the mirror line at equal perpendicular distances.

Reflection line	Coordinate rule
x-axis	(x, y) → (x, −y)
y-axis	(x, y) → (−x, y)
line y = x	(x, y) → (y, x)

Example: Reflect (−3, 5) in the y-axis. Change the sign of x: image = (3, 5).

4. 4. Rotation: Turn Around a Centre

A rotation turns a shape around a fixed centre through a specified angle and direction. A complete turn is 360°. Common turns are 90°, 180° and 270°.

For rotations about the origin (0, 0), anticlockwise rules include:

Rotation	Coordinate rule
90° anticlockwise	(x, y) → (−y, x)
180°	(x, y) → (−x, −y)
270° anticlockwise	(x, y) → (y, −x)

Example: Rotate (2, 1) by 90° anticlockwise about the origin: (2, 1) → (−1, 2).

Always state the centre, angle and direction. A clockwise 90° turn is equivalent to an anticlockwise 270° turn.

5. 5. Line Symmetry

A shape has line symmetry if it can be folded along a line so that the two halves match exactly. The fold line is called a line of symmetry.

A square has 4 lines of symmetry.
A non-square rectangle has 2 lines of symmetry.
An equilateral triangle has 3 lines of symmetry.
A general scalene triangle has no line of symmetry.

Some shapes have no line symmetry; others have several. Check whether each half is a mirror image, not merely similar in appearance.

6. 6. Rotational Symmetry

A shape has rotational symmetry if it matches its original position after a turn of less than 360°. The order is the number of times it matches during one complete turn.

Example: A square matches at 90°, 180°, 270° and 360°. Its rotational symmetry has order 4. Its smallest positive angle of rotation is 90°.

For a shape with rotational symmetry, smallest angle = 360° ÷ order.

7. 7. Describe a Transformation Clearly

Use precise mathematical language so another person can reproduce the transformation.

Translation: give the vector or direction and distance.
Reflection: name the mirror line.
Rotation: give the centre, angle and direction.

Common mistake: Saying only “turn the shape” is incomplete. State how far it turns, which way, and around which centre.

8. Practice Exercises

Try these questions before revealing the worked solutions.

Translate (−1, 4) by vector (3, −2).
Reflect (5, −2) in the x-axis.
Reflect (−4, 1) in the y-axis.
Rotate (3, 2) by 90° anticlockwise about the origin.
Rotate (−2, 5) by 180° about the origin.
How many lines of symmetry does a non-square rectangle have?
A shape has rotational symmetry of order 6. Find its smallest positive angle of rotation.
Explain why a translation is a rigid transformation.

Show practice answers and explanations

(−1 + 3, 4 − 2) = (2, 2).
Change the sign of y: (5, 2).
Change the sign of x: (4, 1).
(x, y) → (−y, x), so (3, 2) → (−2, 3).
(x, y) → (−x, −y), so (−2, 5) → (2, −5).
2 lines of symmetry.
360° ÷ 6 = 60°.
Every point moves by the same vector, so distances and angles between points are unchanged.

10. Printable Worksheet

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