OxfordAQA International GCSE Mathematics: Transformations of 2D shapes (9260)

This study guide covers reference G21 in section 3.3.3 (Transformations, Matrices and Vectors) of the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1, for exams from May/June 2018 onwards. Core content is also assessable on the Extension tier. Content marked Extension only is combined transformations and enlargements with fractional or negative scale factors. The specification notes that translations are specified by a vector.

Related pages: revision notes, practice questions, the course hub, the printable checklist and the free diagnostics. Vectors and matrices have their own pages, so this guide does not teach them.

What G21 asks you to do

Skill	Tier
Describe and carry out a single translation, given as a column vector	Core
Describe and carry out a single reflection in a line	Core
Describe and carry out a single rotation (centre, angle, direction)	Core
Describe and carry out an enlargement with a positive scale factor	Core
Say which properties a transformation preserves	Core
Describe and carry out a combination of transformations	Extension only
Enlarge by a fractional or negative scale factor	Extension only

The object and the image

The original shape is the object. After a transformation it becomes the image. A point P maps to P′ (“P dash”). Always label points this way so that you can see which point went where.

Properties preserved

Transformation	Lengths	Angles	Shape turned over?	Image congruent to object?
Translation	same	same	no	yes
Rotation	same	same	no	yes
Reflection	same	same	yes (mirror image)	yes
Enlargement, scale factor k	multiplied by k	same	no	no, but similar

Congruent shapes have identical size and shape. Similar shapes have equal angles, and each side of one is the same multiple of the matching side of the other. Under an enlargement with scale factor k, areas are multiplied by k², not k. A description must be complete, so a single word such as “reflection” is never enough.

Translation

In a translation, every point of the shape moves by the same vector. The vector (a, b) means a units right (left if negative) and b units up (down if negative). Orientation and size do not change.

To describe a translation, subtract the coordinates of a point from those of its image.

Example 1. Triangle ABC has A(−3, 1), B(−1, 1) and C(−1, 4). It is translated by the vector (5, −2).

A′ = (−3 + 5, 1 − 2) = (2, −1)
B′ = (−1 + 5, 1 − 2) = (4, −1)
C′ = (−1 + 5, 4 − 2) = (4, 2)

To return A′B′C′ to ABC you use the opposite vector (−5, 2).

Reflection

A reflection flips the shape over a mirror line. Every point and its image lie on opposite sides of the line, each at the same perpendicular distance from it. Points on the line do not move.

Useful rules for the lines you meet most often:

Mirror line	Point (x, y) maps to
x = k	(2k − x, y)
y = k	(x, 2k − y)
y = x	(y, x)
y = −x	(−y, −x)

Example 2. Triangle L(1, 1), M(4, 2), N(2, 5) is reflected in x = −1. Separately, the same triangle is reflected in y = x. Find both images.

In x = −1:  (x, y) → (−2 − x, y)
L′(−3, 1), M′(−6, 2), N′(−4, 5)

In y = x:   (x, y) → (y, x)
L″(1, 1), M″(2, 4), N″(5, 2)

L lies on the line y = x, so it does not move.

Finding the mirror line. Find the midpoint of any point and its image. The mirror line passes through it at right angles to the join. If (6, 1) maps to (2, 1), the points differ only in x and the midpoint has x = (6 + 2)/2 = 4. The mirror line is x = 4.

Rotation

A rotation turns every point through the same angle about a fixed point, the centre. To describe one you must give three things: the angle, the direction (clockwise or anticlockwise) and the centre. For 180° the direction does not matter.

About the origin you can use these rules:

Rotation	Image of the point (x, y)
a 90° turn, anticlockwise	(−y, x)
a 90° turn, clockwise	(y, −x)
a 180° turn	(−x, −y)

For another centre C, work from the centre: write each point as a position relative to C, turn it, then add C back.

Example 3. Rotate triangle (2, 1), (5, 1), (5, 3) through 90° clockwise about (1, 0).

Relative to (1, 0):  (1, 1), (4, 1), (4, 3)
Turn 90° clockwise (x, y) → (y, −x):  (1, −1), (1, −4), (3, −4)
Add (1, 0) back:  (2, −1), (2, −4), (4, −4)

Finding the centre. A point and its image are equally far from the centre, so the centre must lie on the perpendicular bisector of the segment joining them. Use two pairs of points: the two bisectors cross at the centre.

Example 4. A rotation maps P(4, 3) to P′(0, 5) and Q(4, 1) to Q′(2, 5). Find the centre.

PP′: midpoint (2, 4), direction (−4, 2).
     Perpendicular bisector: y = 2x
QQ′: midpoint (3, 3), direction (−2, 4).
     Perpendicular bisector: y = 0.5x + 1.5
Solve 2x = 0.5x + 1.5:  x = 1, y = 2

The centre is (1, 2). Check the angle: P is (3, 1) from the centre and P′ is (−1, 3) from it. That is a turn of 90° anticlockwise.

Enlargement with a positive scale factor

To enlarge a shape you must know its centre and its scale factor k. Draw a ray from the centre through an object point. The image point sits on that ray, k times as far from the centre as the object point. If k is greater than 1 the image is bigger. A positive k keeps the image on the same side of the centre as the object.

Example 5. Enlarge triangle (2, 1), (3, 1), (3, 3) by scale factor 3, centre (1, 0).

Image = centre + 3 × (point − centre)
(2, 1): (1, 0) + 3(1, 1) = (4, 3)
(3, 1): (1, 0) + 3(2, 1) = (7, 3)
(3, 3): (1, 0) + 3(2, 3) = (7, 9)

The object has area ½ × 1 × 2 = 1 and the image has area 9, which is 3² times as big.

Finding the centre. Draw a line through each pair of corresponding points. The lines meet at the centre. Here (2, 1) to (4, 3) is the line y = x − 1, and (3, 3) to (7, 9) is the line y − 3 = 1.5(x − 3). They meet at (1, 0). The scale factor is the ratio of corresponding lengths: image ÷ object.

Extension only: fractional scale factors

If 0 < k < 1 the enlargement makes the shape smaller, and the image is still on the same side of the centre. Say “enlargement, scale factor ½” even though the shape shrinks.

Example 6 (Extension only). Enlarge the triangle (0, 0), (4, 0), (0, 6) by scale factor ½, centre (6, 2).

(0, 0): (6, 2) + ½(−6, −2) = (3, 1)
(4, 0): (6, 2) + ½(−2, −2) = (5, 1)
(0, 6): (6, 2) + ½(−6, 4)  = (3, 4)

Extension only: negative scale factors

With a negative k the image lies on the opposite side of the centre from the object, and is turned through 180°. Its size is |k| times the object. Every image point is found by the same formula, centre + k × (point − centre).

Example 7 (Extension only). Enlarge (2, 2), (4, 2), (4, 3) by scale factor −2, centre (1, 1).

(2, 2): (1, 1) − 2(1, 1) = (−1, −1)
(4, 2): (1, 1) − 2(3, 1) = (−5, −1)
(4, 3): (1, 1) − 2(3, 2) = (−5, −3)

To find k from a diagram, compare lengths to get |k|, then decide the sign. If object and image are on opposite sides of the centre, k is negative.

Extension only: combined transformations

A combination applies one transformation, then another to the image. The order matters. Take S with corners (1, 1), (3, 1), (3, 2).

Example 8 (Extension only). Compare “reflect in the x-axis, then rotate 90° anticlockwise about the origin” with the same two steps in the other order.

Reflect, then rotate:
(x, y) → (x, −y) → (y, x)
S → (1, 1), (1, 3), (2, 3)    a reflection in y = x

Rotate, then reflect:
(x, y) → (−y, x) → (−y, −x)
S → (−1, −1), (−1, −3), (−2, −3)    a reflection in y = −x

The two results differ, so the order cannot be swapped. A combination of two transformations can often be replaced by one. Three results are worth knowing:

• Reflections in two parallel lines give a translation. Reflecting in x = 1 and then x = 5 gives x → 2 − x → 10 − (2 − x) = x + 8, a translation by (8, 0). The distance is twice the gap between the lines.
• Reflections in the y-axis and then the x-axis give (x, y) → (−x, y) → (−x, −y), a rotation of 180° about the origin.
• Reflections in two lines that cross give a rotation about the crossing point, through twice the angle between the lines.

To describe the single equivalent transformation, track two or three points and compare the final positions with the start.

Common errors

• Writing only “rotation” or “enlargement”. A full description needs the angle, direction and centre, or the scale factor and centre.
• Reading a vector as (up, right). The top number is the horizontal move.
• Reflecting in x = 3 by moving the point 3 units instead of measuring its distance from the line.
• Multiplying areas by k instead of k².
• Measuring enlargement distances from a corner instead of from the centre.
• Rotating “by eye” without checking one point’s distance from the centre before and after.
• Forgetting that the first transformation in a combination is applied first.

Official syllabus

OxfordAQA International GCSE Mathematics (9260), Version 5.1, for exams May/June 2018 onwards, published by OxfordAQA. Reference G21 in section 3.3.3.