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

These notes condense reference G21 in section 3.3.3 of the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1, for exams from May/June 2018 onwards. For full explanations and worked examples, read the study guide. Test yourself with the practice questions, tick topics off on the printable checklist, find the course hub here, and try a free diagnostic to find weak spots. Content labelled Extension only is combined transformations and fractional or negative enlargements. Everything else is Core and is also assessable on the Extension tier.

Describing each transformation

Transformation	You must state
Translation	the column vector (a, b)
Reflection	the equation of the mirror line
Rotation	angle, direction (clockwise or anticlockwise), centre
Enlargement	scale factor and centre

Use one transformation per description. If a question asks for a single transformation, do not name two.

Properties preserved

	Lengths	Angles	Area	Congruent?
Translation	same	same	same	yes
Rotation	same	same	same	yes
Reflection	same	same	same	yes
Enlargement, factor k	× k	same	× k²	no, similar

• Reflection reverses the orientation, so the image is a mirror image. Translation, rotation and positive enlargement do not.
• Lengths scale by k and areas by k². The perimeter scales by k.
• Invariant points do not move. On a reflection they are the points on the mirror line. On a rotation it is the centre only. On an enlargement it is the centre only. A translation has none.

Translation

• Vector (a, b): a is the horizontal move (right is positive) and b is the vertical move (up is positive).
• Image = (x + a, y + b).
• Find the vector by taking the image point minus the object point.
• The inverse translation uses (−a, −b).

Reflection

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

To find the mirror line: take any point and the point it maps to, find their midpoint, then draw the line through it at right angles to the join. For (2, 7) and (2, 1) the mirror line is horizontal through the midpoint (2, 4), so it is y = 4.

Rotation

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

Method for another centre C in steps:

1. Subtract C from the point to get its relative position.
2. Turn the relative position using the table.
3. Add C back.

To find the centre: draw the perpendicular bisector of two different point and image pairs. They cross at the centre. To find the angle, compare the direction of a point from the centre with the direction of its image.

Quick check: a point and its image must be equally far from the centre.

Enlargement (positive scale factor)

• Image = centre + k × (point − centre).
• If k > 1 the image is larger. The image is on the same side of the centre as the object.
• To find the centre, join each point to its image with a straight line. The lines meet at the centre.
• To find k, divide an image length by the matching object length.
• Area and volume: multiply areas by k², not k.

Worked reminder. Enlarge (3, 2) by scale factor 4, centre (1, 1). The relative position is (2, 1). Multiply by 4 to get (8, 4). Add the centre to get (9, 5).

Extension only: fractional and negative scale factors

Scale factor	Effect on the image
k > 1	larger, same side of centre
0 < k < 1	smaller, same side of centre
−1 < k < 0	smaller, opposite side of centre
k < −1	larger, opposite side of centre
k = −1	same size, opposite side: a rotation of 180° about the centre

• The same formula applies for every k: image = centre + k × (point − centre).
• The image size is |k| times the object, whatever the sign.
• To find the sign of k, look at which side of the centre the image is on.

Worked reminder. Centre (0, 0), k = −½, point (6, −4). The image is −½ × (6, −4) = (−3, 2).

Extension only: combined transformations

• Work through the transformations in the order given. The first one is applied first.
• Swapping the order often changes the answer. Reflect in y = 0 then rotate 90° anticlockwise about the origin gives reflection in y = x. Rotating first and then reflecting in y = 0 gives reflection in y = −x.
• Two reflections in parallel lines give a translation. Its length is twice the gap between the mirror lines.
• Two reflections in lines that cross give a rotation about the crossing point. The angle is twice the angle between the lines.
• Two translations add their vectors. Two enlargements about the same centre multiply their scale factors.

To describe the single transformation: follow two or three points through every step. Then compare their final positions with their starting positions. Decide which single transformation fits, and check it with a third point.

Identifying a single transformation from a diagram

1. Compare the size. If lengths have changed, it is an enlargement. If not, it is a translation, rotation or reflection.
2. If the lengths are the same, check whether the image is a mirror image. Walk round the vertices of the object in order, then round the matching vertices of the image. If the direction of travel has reversed, it is a reflection.
3. If the shape is the same way round and every side stays parallel to the original, it is a translation, or a rotation of 180°.
4. If the sides have turned to a new direction, it is a rotation. Find the centre with two perpendicular bisectors.
5. Write the full description, then test one vertex against it.

Worked reminder. Point (5, 2) is reflected in x = −1. The distance from the line is 5 − (−1) = 6, so the image is 6 units on the other side, at x = −1 − 6 = −7. The image is (−7, 2). The formula agrees: 2(−1) − 5 = −7.

Must-know distinctions

• Object and image: the object is the original. The image is the result.
• Congruent and similar: congruent means identical in size and shape. Similar means the same shape with a constant ratio of lengths.
• Scale factor and area factor: scale factor k, area factor k².
• Describe and carry out: describing needs every defining detail. Carrying out needs the image drawn or its coordinates.

Quick self-test

1. Translate (1, 1), (3, 1), (3, 2) by the vector (−2, −4). State the image coordinates.
2. Reflect (1, 0), (3, 0), (3, 2) in the line y = 2. State the image coordinates.
3. Rotate the point (2, 5) through 90° clockwise about the origin.
4. Enlarge the point (1, 2) by scale factor 3 from the origin.
5. (Extension only) Enlarge the point (2, 3) by scale factor −3, centre (1, 1).
6. (Extension only) Enlarge the point (8, 0) by scale factor ¼, centre (0, 4).
7. Rotate (5, 3) through 180° about (2, 2).
8. Reflect (3, −4) in the line y = −x.
9. (Extension only) The point (1, 1) is reflected in x = 3 and then in x = 7. State its final position and the single transformation that has the same effect.
10. A rectangle has area 20 cm² and perimeter 18 cm. It is enlarged by scale factor 1.5. State the new area and perimeter.

Answers

1. (−1, −3), (1, −3), (1, −2)
2. (1, 4), (3, 4), (3, 2)
3. (5, −2), using (x, y) → (y, −x).
4. (3, 6)
5. (−2, −5). The relative position is (1, 2), and −3 times that is (−3, −6). Adding (1, 1) gives (−2, −5).
6. (2, 3). The relative position is (8, −4), and ¼ of that is (2, −1). Adding (0, 4) gives (2, 3).
7. (−1, 1). The relative position is (3, 1), the turn gives (−3, −1), and adding (2, 2) gives (−1, 1).
8. (4, −3), using (x, y) → (−y, −x).
9. (9, 1). In x = 3 the point goes to (5, 1) and in x = 7 it goes to (9, 1). The single transformation is a translation by (8, 0), which is twice the gap of 4 between the lines.
10. Area 20 × 1.5² = 45 cm². Perimeter 18 × 1.5 = 27 cm.

Where marks are usually lost

• Giving a one-word description, such as “rotation”, when the centre, angle and direction are also needed.
• Leaving out the direction of a rotation that is not 180°.
• Swapping the two numbers in a translation vector.
• Reflecting in x = k or y = k by moving a fixed distance rather than doubling the distance to the line.
• Multiplying area by k instead of k².
• Starting an enlargement from the object corner instead of from the centre.
• Calling a reduction “not an enlargement”. A fractional scale factor is still called an enlargement.
• Getting the sign of a negative scale factor wrong, or putting the image on the same side of the centre.
• Applying combined transformations in the wrong order.
• Giving an unlabelled image. Mark every vertex clearly so that your working can be followed.

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.