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

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – examination boards hold copyright in their own papers. Use these alongside the official past papers from your board or school.

These questions cover 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. Core content is also assessable on the Extension tier. Questions marked (Extension only) use combined transformations or fractional and negative scale factors. Read the study guide or the revision notes first if you need to. Course links: the hub, the printable checklist and the free diagnostics.

Questions

1. Triangle ABC has A(0, 1), B(2, 1) and C(2, 4). It is translated by the vector (−3, −2). Write down the coordinates of A′, B′ and C′. [2]

2. The triangle with vertices A(2, 3), B(5, 3) and C(5, 1) is reflected.

(a) Reflect it in the line y = −1 and write down the image coordinates. [2] (b) A different reflection maps A to (−4, 3). Find the equation of its mirror line. [2]

3. The triangle with vertices (3, 2), (6, 2) and (6, 4) is called T.

(a) T is rotated through 180° about (1, 1). Write down the image coordinates. [2] (b) A rotation maps P(4, 3) to P′(3, −2) and Q(4, 1) to Q′(1, −2). Describe the rotation fully. [4]

4. Triangle R has vertices (1, 1), (3, 1) and (3, 2).

(a) R is enlarged by scale factor 2, centre (0, −1). Find the image coordinates. [2] (b) Find the area of R and the area of the image. [2]

5. A triangle has area 9.5 cm² and perimeter 21 cm. It is enlarged by scale factor 4.

(a) Find the perimeter of the image. [1] (b) Find the area of the image. [2] (c) State whether the two triangles are congruent or similar. [1]

6. Triangle T has vertices (0, 2), (2, 2) and (2, 3). Triangle U has vertices (−2, 0), (−2, −2) and (−3, −2). Triangle V has vertices (4, −1), (6, −1) and (6, 0). Triangle W has vertices (2, 0), (0, 0) and (0, −1). The vertices are listed in matching order.

(a) Describe fully the single transformation that maps T to U. [2] (b) Describe fully the single transformation that maps T to V. [2] (c) Describe fully the single transformation that maps T to W. [3]

7. (Extension only) The triangle with vertices (3, 1), (4, 1) and (4, 3) is enlarged.

(a) Use scale factor −2 and centre (2, 0). Find the image coordinates. [3] (b) Triangle H has vertices (2, 5), (8, 5) and (8, −1). Enlarge H by scale factor ⅓, centre (−1, 2). Find the image coordinates. [2]

8. (Extension only) Triangle J has vertices (4, 1), (8, 1) and (8, 3). Triangle K has vertices (1, 1), (−1, 1) and (−1, 0). The vertices are listed in matching order. Describe fully the single transformation that maps J to K. [4]

9. (Extension only) Triangle M has vertices (1, 2), (3, 2) and (3, 5).

(a) M is rotated 90° anticlockwise about the origin and the image is then reflected in the line y = x. Find the final vertices and describe the single transformation that maps M to them. [4] (b) A point is reflected in x = 1 and then in x = 5. Describe the single transformation that has the same effect. [2]

10. Triangle A has vertices (3, 1), (5, 1) and (5, 4).

(a) A is rotated through 90° anticlockwise about (2, 0) to give B. Find the vertices of B. [3] (b) B is enlarged by scale factor 2, centre (1, 1), to give C. Find the vertices of C. [3] (c) Find the area of A and the area of C. [2] (d) State which two of the triangles A, B and C are congruent. [1]

11. (Extension only) Triangle A has vertices (2, 0), (3, 0) and (3, 2). It is enlarged by scale factor 3, centre (1, −1), to give B. Then B is enlarged by scale factor −⅓, centre (1, −1), to give C.

(a) Find the vertices of B. [2] (b) Find the vertices of C. [2] (c) Describe the single transformation that maps A to C. [2]

Answers

1. Add −3 to each x and −2 to each y: (x, y) → (x − 3, y − 2). [1] A′(−3, −1), B′(−1, −1), C′(−1, 2). [1] [2] Examiner insight: Label each image point with its own letter and give full coordinates. A list of numbers with no labels makes it hard to see which vertex went where.

2. (a) For y = −1 the rule is (x, y) → (x, −2 − y). [1] A′(2, −5), B′(5, −5), C′(5, −3). [1] (b) The mirror line is halfway between A and its image. Midpoint of x-values = (2 + (−4))/2. [1] The mirror line is x = −1. [1] [4] Examiner insight: Give a mirror line as an equation such as x = −1, not in words.

3. (a) Relative to (1, 1) the points are (2, 1), (5, 1), (5, 3). Turn 180°: (−2, −1), (−5, −1), (−5, −3). [1] Add (1, 1): (−1, 0), (−4, 0), (−4, −2). [1] (b) PP′ has midpoint (3.5, 0.5), and the perpendicular bisector is x + 5y = 6. [1] QQ′ has midpoint (2.5, −0.5), and the perpendicular bisector is x + y = 2. [1] Subtracting gives 4y = 4, so y = 1 and x = 1. The centre is (1, 1). [1] P is (3, 2) from the centre and P′ is (2, −3) from it, so the turn is 90° clockwise. [1] The rotation is 90° clockwise about (1, 1). [6] Examiner insight: A description of a rotation has three parts: angle, direction and centre. Check your final answer by rotating one point, so that you catch a clockwise and anticlockwise mix-up.

4. (a) Image = centre + 2 × (point − centre), so (1, 1) gives (0, −1) + 2(1, 2). [1] The image is (2, 3), (6, 3), (6, 5). [1] (b) Area of R = ½ × 2 × 1 = 1 square unit. [1] The image has base 4 and height 2, so its area is ½ × 4 × 2 = 4 square units. [1] [4] Examiner insight: Measure from the centre of enlargement and not from a corner of the shape. Using the area scale factor 2² = 4 as a check shows you the answer to part (b) is consistent.

5. (a) Lengths scale by 4: 21 × 4 = 84 cm. [1] (b) Areas scale by 4² = 16. [1] 9.5 × 16 = 152 cm². [1] (c) The angles are equal but the sides are different sizes, so the triangles are similar and not congruent. [1] [4] Examiner insight: A common slip is to multiply the area by 4. Write the area scale factor as 4² first, and include units.

6. (a) It is a reflection. [1] The mirror line is y = −x, since (0, 2) maps to (−2, 0), using (x, y) → (−y, −x). [1] (b) It is a translation. [1] The vector is (4, −3), since (0, 2) maps to (4, −1). [1] (c) It is a rotation. [1] The angle is 180°, which needs no direction. [1] The centre is the midpoint of (0, 2) and (2, 0), which is (1, 1). [1] [7] Examiner insight: Name the transformation first, then give each detail it needs. If you name two transformations in one description, the answer is not a single transformation.

7. (a) Relative to (2, 0): (1, 1), (2, 1), (2, 3). [1] Multiply by −2: (−2, −2), (−4, −2), (−4, −6). [1] Add (2, 0): (0, −2), (−2, −2), (−2, −6). [1] (b) Relative to (−1, 2): (3, 3), (9, 3), (9, −3). Multiply by ⅓: (1, 1), (3, 1), (3, −1). [1] Add (−1, 2): (0, 3), (2, 3), (2, 1). [1] [5] Examiner insight: With a negative scale factor the image lands across the centre from the object. A quick sketch shows at once if your answer is on the wrong side.

8. The base of J has length 4 and the base of K has length 2, so the size factor is ½. [1] K lies across the centre from J, so the scale factor is −½. [1] The line through (4, 1) and (1, 1) is y = 1. The line through (8, 3) and (−1, 0) is y = x/3 + 1/3. They meet where x/3 + 1/3 = 1, so x = 2. [1] The centre is (2, 1). It is an enlargement, scale factor −½, centre (2, 1). [1] [4] Examiner insight: A scale factor of ½ with the wrong sign describes a different image. Say which side of the centre the image is on before you decide on the sign.

9. (a) After the rotation, (x, y) → (−y, x): (−2, 1), (−2, 3), (−5, 3). [1] After the reflection in y = x, swap coordinates: (1, −2), (3, −2), (3, −5). [1] Comparing with M, only the y-values change sign. [1] The single transformation is a reflection in the x-axis (y = 0). [1] (b) x → 2 − x → 10 − (2 − x). [1] This equals x + 8, so it is a translation by (8, 0). [1] [6] Examiner insight: The order of transformations is part of the question. Follow each vertex through both steps in the order given, then compare the final positions with the starting positions.

10. (a) Relative to (2, 0): (1, 1), (3, 1), (3, 4). [1] Turn 90° anticlockwise, (x, y) → (−y, x): (−1, 1), (−1, 3), (−4, 3). [1] Add (2, 0): B(1, 1), (1, 3), (−2, 3). [1] (b) Relative to (1, 1): (0, 0), (0, 2), (−3, 2). [1] Multiply by 2: (0, 0), (0, 4), (−6, 4). [1] Add (1, 1): C(1, 1), (1, 5), (−5, 5). [1] (c) Area of A = ½ × 2 × 3 = 3 square units. [1] Area of C = 3 × 2² = 12 square units. [1] (d) A and B are congruent, because a rotation preserves size and shape. [1] [9] Examiner insight: In a multi-step question, show each image clearly so that your method can be followed and you can spot slips early.

11. (a) Relative to (1, −1): (1, 1), (2, 1), (2, 3). [1] Multiply by 3 and add the centre: B(4, 2), (7, 2), (7, 8). [1] (b) Relative to (1, −1) the points of B are (3, 3), (6, 3), (6, 9). Multiply by −⅓: (−1, −1), (−2, −1), (−2, −3). [1] Add the centre: C(0, −2), (−1, −2), (−1, −4). [1] (c) The combined scale factor is 3 × (−⅓) = −1. [1] A scale factor of −1 about (1, −1) is a rotation of 180° about (1, −1). [1] [6] Examiner insight: Multiplying the scale factors only works because both enlargements use the same centre. Check the result with one vertex, for example (2, 0) mapping to (0, −2).

Where marks are usually lost

• Describing a transformation in one word, with no mirror line, centre, angle, direction or vector.
• Leaving the direction out of a rotation that is not 180°.
• Measuring a reflection from the wrong line, or using the shape’s edge instead of the mirror line.
• Writing the translation vector with the numbers swapped or the signs reversed.
• Measuring an enlargement from a corner instead of from the centre.
• Using k instead of k² for the area scale factor.
• Putting a negative enlargement on the same side of the centre as the object.
• Combining transformations in the wrong order.

Next steps

• Condense the topic with the revision notes.
• Revisit the explanations in the study guide.
• Return to the course hub and the printable checklist.
• Try all the free 10-minute diagnostics.
• Book a free trial class.

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.