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Shape vocabulary, symmetry and quadrilaterals

8 exam-style questions, grades 1 to 4. Worked solutions and the marks are on the last page.

  1. Question 1Non-calculator · 1 mark

    (a) A rhombus is not a square. What is its order of rotational symmetry? (1)

    1. 1
    2. 2
    3. 4
    4. 0
  2. Question 2Non-calculator · 3 marks

    This question is about special quadrilaterals.

    (a) Write down the name of the quadrilateral that has exactly one pair of parallel sides. (1)

    1. Parallelogram
    2. Kite
    3. Rhombus
    4. Trapezium

    (b) Write down the number of lines of symmetry of a rhombus that is not a square. (1)

    (c) Rosa says, "A rectangle is a special kind of rhombus." Explain why Rosa is wrong. (1)

  3. Question 3Non-calculator · 2 marks

    This question is about symmetry.

    (a) Write down the order of rotational symmetry of a regular hexagon. (1)

    (b) Sam says, "Any shape with rotational symmetry of order 2 has 2 lines of symmetry." Use a parallelogram to explain why Sam is wrong. (1)

  4. Question 4Non-calculator · 3 marks

    A rhombus has one interior angle of 60∘60^\circ. It is not a square.

    (a) Find the sizes of the other three angles. (2)

    (b) A pupil says each diagonal is a line of symmetry. Is this correct? Explain. (1)

  5. Question 5Non-calculator · 4 marks

    A quadrilateral has vertices (−3,−2)(-3,-2), (3,−2)(3,-2), (3,2)(3,2) and (−3,2)(-3,2) in order, with k=3k=3.

    (a) Name the quadrilateral as precisely as possible and justify your answer. (2)

    (b) Find its perimeter. (2)

  6. Question 6Non-calculator · 3 marks

    A kite has adjacent equal sides of 7 cm and 7 cm, and its other two sides are each 10 cm.

    (a) Find its perimeter. (2)

    (b) Must this kite have two pairs of parallel sides? Explain. (1)

  7. Question 7Non-calculator · 3 marks

    A quadrilateral has exactly one pair of parallel sides, and its two non-parallel sides are equal in length.

    (a) Which quadrilateral is this? Select one answer. (1)

    1. An isosceles trapezium
    2. A parallelogram
    3. A kite
    4. A rhombus

    (b) How many lines of symmetry does it have? (1)

    (c) What is its order of rotational symmetry? (1)

  8. Question 8Non-calculator · 4 marks

    Quadrilateral PQRS has vertices P(0, 0), Q(4, 3), R(9, 3) and S(5, 0).

    (a) Show that PQRS is a rhombus. (2)

    (b) Work out the area of PQRS. (2)

Worked solutions and marks

Question 1

(a) 2

  1. A non-square rhombus matches itself after 180∘180^\circ and 360∘.360^\circ.
  2. There are two matching positions in a full turn, so its order is 2.
  • B1 Correct answer: 2

Question 2

(a) Trapezium

  1. A trapezium is defined by having exactly one pair of parallel sides.
  • B1 Naming the trapezium.

(b) 2

  1. The two diagonals of a rhombus are its mirror lines. The lines joining midpoints of opposite sides are not, unless all its angles are right angles.
  • B1 Correct answer: Answer 2.

(c) A rhombus has four equal sides, but a rectangle's sides need not all be equal.

  1. A rhombus must have four equal sides.
  2. A rectangle only needs four right angles; its length and width can differ, so it is not always a rhombus.
  • C1 A correct statement that a rectangle need not have four equal sides (or that only a square is both).

Question 3

(a) 6

  1. A regular hexagon fits onto itself after every turn of 60∘60^\circ, and 360∘÷60∘=6360^\circ \div 60^\circ = 6.
  • B1 Correct answer: Order 6.

(b) A parallelogram (not a rectangle or rhombus) has rotational symmetry of order 2 but no lines of symmetry.

  1. A half turn about the point where the diagonals cross maps a parallelogram onto itself, so it has order 2.
  2. Folding along either diagonal or through the midpoints of opposite sides does not fit, so it has no lines of symmetry.
  • C1 Stating that a parallelogram has rotational symmetry of order 2 but no lines of symmetry.

Question 4

(a) 120∘,60∘,120∘120^\circ,60^\circ,120^\circ

  1. Adjacent angles in a parallelogram sum to 180 degrees; opposite angles are equal.
    180−60180-60
  2. Therefore 120∘,60∘,120∘120^\circ,60^\circ,120^\circ.
  • M1 Adjacent angles in a parallelogram sum to 180 degrees; opposite angles are equal.
  • A1 Correct answer: 120∘,60∘,120∘120^\circ,60^\circ,120^\circ

(b) Yes. A rhombus has equal sides, and each diagonal bisects the angles at its endpoints and reflects one half onto the other.

  1. Yes. A rhombus has equal sides, and each diagonal bisects the angles at its endpoints and reflects one half onto the other.
  • C1 Correct conclusion with supporting reasoning: Yes. A rhombus has equal sides, and each diagonal bisects the angles at its endpoints and reflects one half onto the other.

Question 5

(a) It is a rectangle: adjacent sides are horizontal and vertical, all angles are right angles, and the side lengths are unequal.

  1. The horizontal side is twice k and the vertical side is four units.
    2×32\times 3
  2. It is a rectangle: adjacent sides are horizontal and vertical, all angles are right angles, and the side lengths are unequal.
  • M1 The horizontal side is twice k and the vertical side is four units.
  • C1 Correct conclusion with supporting reasoning: It is a rectangle: adjacent sides are horizontal and vertical, all angles are right angles, and the side lengths are unequal.

(b) 2020

  1. Add two horizontal and two vertical sides.
    2(2×3+4)2(2\times 3+4)
  2. Therefore 2020.
  • M1 Add two horizontal and two vertical sides.
  • A1 Correct answer: 2020

Question 6

(a) 3434 cm

  1. Add both copies of each distinct side length.
    2×7+2×102\times 7+2\times 10
  2. Therefore 3434 cm.
  • M1 Add both copies of each distinct side length.
  • A1 Correct answer: 3434 cm

(b) No. A kite has two pairs of adjacent equal sides. This does not imply that opposite sides are parallel.

  1. No. A kite has two pairs of adjacent equal sides. This does not imply that opposite sides are parallel.
  • C1 Correct conclusion with supporting reasoning: No. A kite has two pairs of adjacent equal sides. This does not imply that opposite sides are parallel.

Question 7

(a) An isosceles trapezium

  1. An isosceles trapezium.
  • B1 Correct answer: An isosceles trapezium

(b) 11

  1. Therefore 11.
  • B1 Correct answer: 11

(c) 11

  1. Therefore 11.
  • B1 Correct answer: 11

Question 8

(a) PQ = √(4² + 3²) = 5, QR = 5, RS = √(4² + 3²) = 5 and SP = 5, so all four sides are equal.

  1. Use Pythagoras for each sloping side.
    42+32=5\sqrt{4^{2}+3^{2}}=5
  2. Therefore PQ = √(4² + 3²) = 5, QR = 5, RS = √(4² + 3²) = 5 and SP = 5, so all four sides are equal.
  • M1 Use Pythagoras for each sloping side.
  • C1 Correct conclusion with supporting reasoning: PQ = √(4² + 3²) = 5, QR = 5, RS = √(4² + 3²) = 5 and SP = 5, so all four sides are equal.

(b) 1515 square units

  1. Multiply the base PS by the perpendicular height.
    5×35\times 3
  2. Therefore 1515 square units.
  • M1 Multiply the base PS by the perpendicular height.
  • A1 Correct answer: 1515 square units

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Shape vocabulary, symmetry and quadrilaterals

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