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About Symmetry — Class 7 Olympiad

Identify rotational and reflective symmetry; solve pattern completion and symmetry-based reasoning problems. This topic is part of the Olympiad Class 7 mathematics syllabus (chapter: Module 12). On this page you can practice 60 questions across three difficulty levels — 20 easy, 20 medium, and 20 hard — each with a visual step-by-step solution, plus a timed 35-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Symmetry — solved examples for Class 7 Olympiad

Example 1easy

Consider the following figure formed by joining the midpoints of the sides of a square and then connecting its vertices to the center. How many lines of symmetry does this figure have?
  1. A)4
  2. B)2
  3. C)6
  4. D)8

Step-by-step solution

  1. A square itself has 4 lines of symmetry: two passing through opposite vertices and two passing through midpoints of opposite sides.
  2. Joining the midpoints of the sides creates another smaller square, which also has 4 lines of symmetry, coinciding with the parent square's lines.
  3. Connecting the vertices of the outer square to the center also maintains these 4 lines of symmetry. The entire figure will be symmetrical about these same 4 lines.

Answer: 4

Example 2medium

A figure is formed by arranging 5 identical squares. Four squares form a larger 2×2 square, and the fifth square is attached to the middle of the top side of this 2×2 arrangement. How many lines of reflectional symmetry does this composite figure have?
  1. A)A) 0
  2. B)B) 1
  3. C)C) 2
  4. D)D) 4

Step-by-step solution

  1. Imagine the 2×2 square formed by four unit squares. Let its vertices be (0,2), (2,2), (2,0), (0,0). The fifth square is attached to the middle of the top side, meaning its vertices could be (0.5, 2), (1.5, 2), (1.5, 3), (0.5, 3) (assuming unit squares and (1,2) is the midpoint of the top side).
  2. A horizontal line of symmetry would require the top attached square to have a mirror image below the 2×2 square, which it doesn't. So, no horizontal line of symmetry.
  3. A vertical line of symmetry passing through the center of the 2×2 square and the center of the top attached square (x=1) would perfectly divide the figure into two identical halves.
  4. Therefore, the figure has exactly one line of reflectional symmetry.

Answer: B) 1

Example 3hard

A complex figure is formed by overlapping an equilateral triangle, a square, and a regular hexagon, all centered at the same point. The vertices of the triangle lie on the midpoints of the square's sides, and the vertices of the square lie on the midpoints of the hexagon's sides. How many lines of symmetry does this combined figure have?
  1. A)1
  2. B)2
  3. C)4
  4. D)3

Step-by-step solution

  1. A regular hexagon has 6 lines of symmetry (3 through opposite vertices, 3 through midpoints of opposite sides).
  2. When a square is inscribed such that its vertices are at the midpoints of the hexagon's sides, only the 3 lines of symmetry of the hexagon that pass through the midpoints of its opposite sides (and thus through the midpoints of the square's opposite sides) remain as lines of symmetry for the combined hexagon-square figure. The other 3 lines of symmetry of the hexagon (passing through its vertices) are 'destroyed' by the square's orientation.
  3. Similarly, when an equilateral triangle is inscribed such that its vertices are at the midpoints of the square's sides, only the 3 lines of symmetry of the square that pass through the midpoints of its opposite sides (and thus through the vertices of the triangle) remain as lines of symmetry for the combined square-triangle figure. These are precisely the same 3 lines identified in the previous step.
  4. Therefore, the entire combined figure possesses exactly 3 lines of symmetry.

Answer: 3

Practice questions on Symmetry

  1. Q1.easy

    A peculiar alien symbol is found to have rotational symmetry. If its smallest angle of rotation is 72°, what is the order of its rotational symmetry?
    1. A)6
    2. B)3
    3. C)5
    4. D)4
    Show answer

    Answer: 5

    Hint: The order of rotational symmetry is related to how many times the figure maps onto itself during a full 360° rotation.

  2. Q2.easy

    Points P(4, 7) and Q(-3, 7) are two vertices of a symmetrical figure. If the figure has the x-axis as a line of symmetry, which of the following points MUST also be a vertex?
    1. A)(7, -3)
    2. B)(4, -7)
    3. C)(-3, -7)
    4. D)(7, 4)
    Show answer

    Answer: (4, -7)

    Hint: Reflection across the x-axis changes the sign of the y-coordinate while keeping the x-coordinate the same.

  3. Q3.easy

    Which of the following words, when written in capital letters, will appear exactly the same when viewed in a vertical mirror?
    1. A)SUN
    2. B)TOT
    3. C)BOX
    4. D)MOM
    Show answer

    Answer: TOT

    Hint: Check each letter individually for vertical mirror symmetry and consider if the word is a palindrome after reflection.

  4. Q4.medium

    Consider a regular pentagon with an equilateral triangle attached to the outside of each of its sides. What is the order of rotational symmetry of the resulting complex figure?
    1. A)A) 1
    2. B)B) 5
    3. C)C) 10
    4. D)D) 15
    Show answer

    Answer: B) 5

    Hint: Think about the properties of a regular pentagon and how attaching identical shapes affects its rotational symmetry.

  5. Q5.medium

    A 4×4 grid contains shaded and unshaded squares. The top-left 2×2 subgrid is given as:
    [Shaded, Unshaded]
    [Unshaded, Shaded]
    If the entire 4×4 grid is symmetrical about both its horizontal and vertical midlines, how many shaded squares will be in the complete 4×4 grid?
    1. A)A) 4
    2. B)B) 6
    3. C)C) 8
    4. D)D) 10
    Show answer

    Answer: C) 8

    Hint: Use the given 2×2 subgrid and apply the reflectional symmetry across both the horizontal and vertical midlines to fill the rest of the 4×4 grid.

  6. Q6.medium

    A figure is formed by shading 3 cells in a 3×3 grid: (1,1), (1,2), (2,1) (row, column). If this 3×3 grid has rotational symmetry of order 4 about its center, how many cells must be shaded in total in the complete symmetrical grid?
    1. A)A) 4
    2. B)B) 6
    3. C)C) 8
    4. D)D) 9
    Show answer

    Answer: C) 8

    Hint: For order 4 rotational symmetry (90° rotations), each shaded cell must have 3 other corresponding shaded cells unless it's the center or part of a rotationally symmetrical group of 2.

  7. Q7.hard

    A complex design has rotational symmetry of order N. If rotating the design by 210° clockwise results in a configuration that is identical to rotating it by 150° counter-clockwise from its original position, what is the smallest possible value of N?
    1. A)6
    2. B)10
    3. C)12
    4. D)15
    Show answer

    Answer: 12

    Hint: The given rotations must both be multiples of the smallest angle of rotational symmetry. Find the greatest common divisor of these angles.

  8. Q8.hard

    A square with vertices at P(1,1), Q(3,1), R(3,3), S(1,3) is reflected across the line y = x. The reflected image is then reflected across the x-axis. What are the coordinates of the final image of vertex P?
    1. A)(1,1)
    2. B)(1,-1)
    3. C)(-1,1)
    4. D)(-1,-1)
    Show answer

    Answer: (1,-1)

    Hint: Apply the reflections sequentially. Remember the rules for reflection across y=x and then across the x-axis.

  9. Q9.hard

    A sequence of three figures is shown below. Each figure is a square with one quadrant shaded. Figure 1 has the top-left quadrant shaded. Figure 2 has the bottom-left quadrant shaded. Figure 3 has the bottom-right quadrant shaded. Which of the following figures correctly completes the sequence as the fourth figure?
    1. A)Square with Top-Left quadrant shaded.
    2. B)Square with Bottom-Left quadrant shaded.
    3. C)Square with Bottom-Right quadrant shaded.
    4. D)Square with Top-Right quadrant shaded.
    Show answer

    Answer: Square with Top-Right quadrant shaded.

    Hint: Observe the position of the shaded quadrant in each step. Determine the consistent geometric transformation applied from one figure to the next.

These are 9 of the 60 questions available for Symmetry. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.