MathematicsFoundation20 min read

Transformations and Symmetry

Moving a shape, and describing exactly what you did

This topic appears in:

O Level Mathematics (4024) · ch.4O Level Mathematics (4024) · ch.7
01

Four transformations, and what each preserves

A transformation maps every point of a shape to a new position. The original is the object and the result is the image, and the whole of this topic is knowing which properties survive.

Three of the four are congruent transformations: translation, reflection and rotation all produce an image identical in size and shape. Only enlargement changes the size, and even then the angles are unchanged, which is why the image is similar to the object.

TransformationPreserves size?Preserves orientation?Must state
Translationyesyesthe column vector
Reflectionyesno — reversedthe equation of the mirror line
Rotationyesyes, but turnedangle, direction and centre
Enlargementnoyes (unless k < 0)scale factor and centre

Try Enlargement with a negative amount. The image lands on the opposite side of the centre and is turned upside down — a negative scale factor is a genuine enlargement, not an error.

02

Describing a transformation fully

Exam questions almost always say "describe fully", and that phrase is doing real work: an incomplete description scores no marks even when the transformation named is correct. Each type has a fixed list of things that must be given.

A translation needs its column vector. A reflection needs the equation of the mirror line — "the y-axis" is acceptable, "a vertical line" is not. A rotation needs three things: the angle, the direction, and the centre. An enlargement needs the scale factor and the centre.

Worked example

Triangle A has vertices (1,1), (3,1), (1,4). Triangle B has vertices (−1,1), (−3,1), (−1,4). Describe fully the transformation mapping A to B.

  1. The two triangles are the same size, so it is not an enlargement.Check the size first — it eliminates one of the four immediately.
  2. The y-coordinates are unchanged and the x-coordinates have changed sign.A pattern in the coordinates usually identifies the transformation faster than a sketch.
  3. That is the effect of reflecting in the y-axis: (x, y) → (−x, y).Reflection in the x-axis would have changed the y values instead.
  4. Full description: a reflection in the line x = 0, that is the y-axis.Giving the equation of the mirror line is what makes the description complete.

A reflection in the y-axis (the line x = 0)

Rotation needs all three

Writing "a rotation of 90°" scores nothing without the direction and the centre — a 90° rotation clockwise about (0, 0) and one anticlockwise about (2, 1) land the shape in completely different places. Whenever a question says "describe fully", count the items in your answer against the list before moving on.

03

Enlargement, and what it does to area

An enlargement needs a centre and a scale factor k. Every point moves so that its distance from the centre is multiplied by k, along the line joining it to the centre.

Three cases are examinable. When k > 1 the image is larger. When 0 < k < 1 it is smaller — still called an enlargement, however odd that sounds. When k < 0 the image appears on the opposite side of the centre and is turned through 180°.

Lengths multiply by k, so areas multiply by k² and volumes by k³ — the same rule as for any similar figures.

lengths × kareas × k²volumes × k³to find the centre: join each point to its image and extend;the lines all meet at the centre of enlargementthe scale factor is the ratio of any image length to the matching object length

Finding the centre from the picture

Join each vertex of the object to the matching vertex of the image and extend the lines. They all pass through the centre of enlargement, so two lines are enough to find it and a third confirms it. This is quicker and more reliable than trying to deduce the centre from coordinates, and it works for negative scale factors too — the lines then cross between the two shapes.

04

Symmetry

A shape has a line of symmetry if reflecting it in that line leaves it looking unchanged. It has rotational symmetry of order n if it fits onto itself n times during one complete turn.

Every shape has rotational symmetry of order at least 1, since a full turn always returns it — so "no rotational symmetry" is properly written as order 1. A regular n-sided polygon has n lines of symmetry and rotational symmetry of order n, which is the neatest connection in the topic.

ShapeLines of symmetryRotational order
Square44
Rectangle22
Rhombus22
Parallelogram02
Equilateral triangle33
Isosceles triangle11
Regular hexagon66
Circleinfiniteinfinite

The parallelogram is the one to remember

A parallelogram has no lines of symmetry but rotational symmetry of order 2 — turn it 180° and it fits onto itself. Students assume the two kinds of symmetry go together, and this is the standard counterexample used to show they do not.

Before you leave this chapter

  1. Translation, reflection and rotation preserve size; only enlargement changes it.
  2. "Describe fully" means the complete list: rotation needs angle, direction AND centre.
  3. A reflection needs the equation of the mirror line, not a description of it.
  4. Enlargement: lengths × k, areas × k², volumes × k³. A negative k flips through the centre.
  5. A parallelogram has no lines of symmetry but rotational symmetry of order 2.

Practice questions

6 questions · 20 marks · full working on every one

Try each one on paper first, then open the working. The marks are shown where they are actually awarded, because that is where they are actually lost.

Short questions

3 · 6 marks

Two marks each, in the style of the short-question section of the paper. Answer in two or three lines.

SQ1[2 marks]
State the three pieces of information needed to describe a rotation fully.
Model answer

The angle of rotation, the direction (clockwise or anticlockwise), and the centre of rotation.

Examiner tip. All three or the description is incomplete. A rotation of 90° about two different centres puts the shape in two different places.

SQ2[2 marks]
A shape is enlarged by scale factor 3. What happens to its area?
Model answer

The area is multiplied by 3² = 9, since area involves two lengths and each has been tripled.

Examiner tip. Areas scale by k², volumes by k³. Answering "×3" is the standard error and the reason this is asked so often.

SQ3[2 marks]
How many lines of symmetry does a parallelogram have, and what is its rotational symmetry order?
Model answer

No lines of symmetry, but rotational symmetry of order 2 — a half turn maps it onto itself.

Examiner tip. This is the standard example showing the two kinds of symmetry are independent of each other.

Solved numericals

2 · 8 marks

Full working, one step per line, with the marks shown where they are awarded.

N1[4 marks]
Triangle P has vertices (2,1), (5,1), (2,3). It is enlarged by scale factor 2, centre (0,0), to give triangle Q.
  1. Write down the coordinates of Q.
  2. State the ratio of the area of Q to the area of P.
Full working
  1. With centre at the origin, each coordinate is multiplied by the scale factorthis shortcut only works when the centre is (0, 0)[1]
  2. Q has vertices (4,2), (10,2) and (4,6)[1]
  3. Areas scale by k² = 2² = 4[1]
  4. Ratio = 4 : 1; check by calculating both areas — P is 3 and Q is 12 ✓[1]

(a) (4,2), (10,2), (4,6) (b) 4 : 1

Examiner tip. Multiplying coordinates directly only works from the origin. From any other centre you must work along the line from the centre to each point.

N2[4 marks]
Shape A maps to shape B under a transformation. Every point of B is 3 units right and 2 units down from the matching point of A, and B is the same size as A. Describe the transformation, and state what would change if B were also upside down.
Full working
  1. Same size and same orientation with every point moving equally means a translation[1]
  2. Column vector (3, −2)right is positive, down is negative[1]
  3. If B were upside down as well, it could not be a translation, since a translation never changes orientation[1]
  4. It would be a rotation of 180° about some centre, which produces the same displacement with the orientation reversedaccept an enlargement with k = −1[1]

A translation by (3, −2); if inverted, a 180° rotation instead.

Examiner tip. Orientation is the quickest test. If the image is turned or flipped, it is not a translation whatever the displacement looks like.

Long questions

1 · 6 marks

Theory and numerical together, as they appear in the long-question section.

LQ1[6 marks]
A logo is designed from a regular hexagon.
  1. State its lines of symmetry and its rotational symmetry order.
  2. The hexagon is enlarged by scale factor 1.5 about one of its vertices. Describe what happens to its perimeter and area.
  3. Explain why an enlargement with scale factor −1 gives the same result as a rotation of 180° about the same centre.
Mark scheme
  1. 6 lines of symmetrythree through opposite vertices, three through opposite edge midpoints[1]
  2. Rotational symmetry of order 6 — a regular n-gon always has n of each[1]
  3. Perimeter is a length, so it is multiplied by 1.5[1]
  4. Area is multiplied by 1.5² = 2.25[1]
  5. A scale factor of −1 keeps every distance from the centre the same but reverses the directionmagnitude 1 means no change of size[1]
  6. Reversing the direction of every point about a centre is exactly a half turn, so the two produce identical images[1]

(a) 6 and order 6 (b) perimeter × 1.5, area × 2.25 (c) k = −1 reverses direction without changing distance, which is a 180° rotation

Examiner tip. Part (c) is worth understanding rather than memorising: a negative scale factor sends each point through the centre to the far side, and doing that to every point is a half turn.