OxfordAQA International GCSE Mathematics: Symmetry, congruence, similarity and 3D solids (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 references G6, G7, G10 and G11 in section 3.3.1 of the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1 (for exams May/June 2018 onwards). Conditions for congruent triangles and constructing plans and elevations are Extension only, so questions on them carry the label “(Extension only)”. Every Core statement can also be assessed on the Extension papers.

If a topic is unfamiliar, start with the study guide. The revision notes give quick recall. The course page is OxfordAQA IGCSE Mathematics and the printable list is the checklist. The free 10-minute diagnostics show which topics need work first.

Questions

1. State how many mirror lines each shape has, and give its order of rotation symmetry.

(a) A kite that is not a rhombus [2] (b) A regular octagon [2]

2. Triangle PQR has vertices P(1, 2), Q(7, 2) and R(4, 6).

(a) Give the equation of the mirror line of triangle PQR. [2] (b) The point S is added below PQ so that PRQS is a rhombus. Find the coordinates of S. [2] (c) Give the number of mirror lines of PRQS and its order of rotation symmetry. [2]

3. Triangles ABC and DEF are congruent, and the letters correspond in alphabetical order. AB = 9.5 cm, BC = 7 cm, angle A = 64° and angle B = 53°. Find DE, EF and angle F. [3]

4. (Extension only)

(a) Triangle 1 has sides of 7 cm and 9 cm with an angle of 40° between them. Triangle 2 has sides of 9 cm and 7 cm with an angle of 40° between them. Are the triangles congruent? Give the condition. [2] (b) Triangle 3 and triangle 4 each have angles of 50°, 60° and 70°. Does this prove they are congruent? Give a reason. [1] (c) In triangle ABC, angle B = 90°, AC = 13 cm and AB = 5 cm. In triangle DEF, angle E = 90°, DF = 13 cm and DE = 5 cm. Write a congruence statement for the triangles and give the condition. [2]

5. (Extension only) In triangle ABC, AB = AC. M is the midpoint of BC. Prove that angle AMB = 90°. [4]

6. Triangle ABC has AB = 8 cm, BC = 12 cm and CA = 10 cm. Triangle PQR is similar to triangle ABC, with PQ = 6 cm matching AB.

(a) Find QR. [2] (b) Find PR and the perimeter of triangle PQR. [2]

7. Rectangle A measures 15 cm by 9 cm. Rectangle B measures 25 cm by 15 cm. Rectangle C measures 24 cm by 16 cm.

(a) Show that rectangles A and B are similar. [2] (b) Explain why rectangles A and C are not similar. [2]

8. Points D and E lie on sides AB and AC of triangle ABC, and the line DE is parallel to BC. AD = 4 cm, DB = 6 cm and DE = 5 cm.

(a) Find BC. [2] (b) AC = 15 cm. Find EC. [2]

9. The lines AE and BD cross at C. AB is parallel to DE. AC = 6 cm, CE = 9 cm, DE = 12 cm and CD = 7.5 cm.

(a) Show that triangles ABC and EDC are similar. [2] (b) Find AB. [2] (c) Find BC. [2]

10.

(a) A prism has a regular hexagon as its cross-section. Give its numbers of faces, edges and vertices. [3] (b) Another solid is a pyramid with 16 edges. Find the number of sides of its base and the total number of faces. [3] (c) A solid has a plan that is a circle with a dot at its centre, and a front elevation that is an isosceles triangle. Name the solid, and explain why it is not a pyramid. [2]

11. A model is built from cubes. Each number is the count of cubes stacked at that position of the plan, and the back row is at the top.

Back:    2  0  3
Middle:  1  1  0
Front:   0  2  2

(a) Give the heights, in cubes, of the front elevation from left to right. [2] (b) Give the heights of the side elevation seen from the right, from left to right. [2] (c) How many cubes are in the model? [1]

12. (Extension only) A solid is made from two cuboids. The lower cuboid is 6 cm wide, 4 cm deep and 2 cm high. The upper cuboid is 3 cm wide, 4 cm deep and 2 cm high, sits on top at the left-hand end, and is flush with the front and back. On squared paper, draw to a scale of 1 cm to 1 cm:

(a) the plan [2] (b) the front elevation [2] (c) the side elevation seen from the right [2]

Answers

1. (a) One line of symmetry [1] and rotation symmetry of order 1 [1] [2] (b) 8 lines of symmetry [1] and order 8 [1] [2] Examiner insight: A shape with no rotation symmetry has order 1, not 0, so writing order 0 is a likely way to lose the second mark.

2. (a) PQ is horizontal, so the midpoint has x-coordinate (1 + 7)/2 = 4 [1]. R is directly above it, so the line is x = 4 [1] [2] (b) The midpoint of PQ is (4, 2), and it is also the midpoint of RS [1]. S = (2 × 4 − 4, 2 × 2 − 6) = (4, −2) [1] [2] (c) 2 lines of symmetry [1] and order 2 [1] [2] Examiner insight: Giving the line as an equation, x = 4, answers the question as asked; saying “the vertical line through R” is less precise.

3. DE matches AB, so DE = 9.5 cm [1]. EF matches BC, so EF = 7 cm [1]. Angle C = 180° − 64° − 53° = 63°, and F matches C, so angle F = 63° [1] [3] Examiner insight: Matching the letters in order (A with D, B with E) rather than by position on the page prevents answers being swapped.

4. (a) Congruent [1], by SAS (two sides and the included angle) [1] [2] (b) Equal angles fix only the shape, so one triangle could be an enlargement of the other [1] [1] (c) Triangle ABC is congruent to triangle DEF [1], by RHS [1] [2] Examiner insight: Naming the condition in words or letters is part of the answer; saying “they look the same” does not show a reason.

5. AB = AC (given) [1]. BM = CM, since M is the midpoint [1]. AM is common, so the triangles ABM and ACM are congruent by SSS [1]. Then angle AMB = angle AMC. They lie on a straight line and sum to 180°, so each is 90° [1] [4] Examiner insight: A proof is easiest to credit when each equal pair has its own line with a reason, then the condition, then the conclusion.

6. (a) k = 6/8 = 0.75 [1]. QR = 0.75 × 12 = 9 cm [1] [2] (b) PR = 0.75 × 10 = 7.5 cm [1]. Perimeter = 6 + 9 + 7.5 = 22.5 cm [1] [2] Examiner insight: Writing the scale factor first lets a slip in the multiplication be seen separately from a wrong choice of matching sides.

7. (a) 25/15 = 5/3 [1] and 15/9 = 5/3, so the ratios are equal and the rectangles are similar [1] [2] (b) Long side to short side: 24/16 = 1.5, but 15/9 = 1.667 (to 3 decimal places) [1]. The ratios differ, so C is not similar to A [1] [2] Examiner insight: Quoting both ratios with their values is stronger than stating that the sides “do not match”.

8. (a) AB = 4 + 6 = 10 cm [1]. BC = (10/4) × 5 = 12.5 cm [1] [2] (b) AE = (4/10) × 15 = 6 cm [1]. EC = 15 − 6 = 9 cm [1] [2] Examiner insight: A scale factor built from the part DB instead of the whole side AB is a common wrong route, so writing the matching sides down first helps.

9. (a) Angle ACB = angle ECD, vertically opposite [1]. Angle BAC = angle DEC, alternate angles as AB is parallel to DE [1]. The triangles have equal angles, so they are similar [2] (b) A matches E, B matches D and C matches C, so AB matches ED. k = AC/EC = 6/9 = 2/3 [1]. AB = 2/3 × 12 = 8 cm [1] [2] (c) BC matches DC, so BC = 2/3 × 7.5 = 5 cm [1] [1] [2] Examiner insight: In crossing-line diagrams the pairing of letters decides the answer, so naming A matches E and B matches D before calculating avoids using the wrong side.

10. (a) 8 faces [1], 18 edges [1], 12 vertices [1] [3] (b) A pyramid with an n-sided base has 2n edges, so 2n = 16 [1] and n = 8 [1]. Faces = 8 + 1 = 9 [1] [3] (c) Cone [1]. The plan of a pyramid is a polygon, not a circle [1] [2] Examiner insight: An explanation that names the actual view, such as “a pyramid’s plan is a polygon”, gives a clear reason; “it looks like a cone” does not.

11. (a) Column maxima: first column 2, 1, 0 gives 2; second column 0, 1, 2 gives 2; third column 3, 0, 2 gives 3 [1]. 2, 2, 3 [1] [2] (b) Row maxima: back 3, middle 1, front 2 [1]. From the right the front is on the left, so 2, 1, 3 [1] [2] (c) 2 + 0 + 3 + 1 + 1 + 0 + 0 + 2 + 2 = 11 cubes [1] [1] Examiner insight: A side elevation reads in the opposite order from the plan rows, and stating which side you are looking from makes the order clear.

12. (a) A rectangle 6 cm by 4 cm [1], with a vertical line 3 cm from the left-hand edge [1] [2] (b) An L-shaped outline: the left 3 cm is 4 cm high and the right 3 cm is 2 cm high [1], with a total width of 6 cm and the base along one line [1] [2] (c) A rectangle 4 cm wide and 4 cm high [1], with a horizontal line across it at a height of 2 cm [1] [2] Examiner insight: Aligning the views so that the widths line up, and writing the scale and dimensions on the drawing, makes a construction easier to check.

Plan (6 by 4)       Front elevation      Side elevation (from right)
+-----+-----+       +-----+              +-----------+
|     |     |       |     |              |           |
|     |     |       |     |              +-----------+
|     |     |       |     +-----+        |           |
+-----+-----+       +-----+-----+        +-----------+

Where marks are usually lost

• Writing order 0 for a shape with no rotation symmetry.
• Giving a rhombus or parallelogram four lines of symmetry.
• Matching triangle vertices by position on the page rather than by the letters in the statement.
• Giving AAA or SSA as the reason for congruence.
• Stating that triangles are similar or congruent without naming the equal angles, sides or condition.
• Using a part of a side, such as DB, in a scale factor.
• Dividing by the scale factor when the answer should be larger.
• Counting a curved surface as a face, or forgetting the edge of a cone.
• Reading a side elevation in the wrong order.
• Leaving a drawn view without a scale or labelled dimensions.

Next steps

• Review the revision notes and the study guide.
• Return to the OxfordAQA IGCSE Mathematics course page and tick off topics on the checklist.
• Take one of 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. Section 3.3.1, references G6, G7, G10 and G11.