Exercise 8.2 — Symmetry
Point symmetry, line symmetry, rotational symmetry and tessellations.
Exercise 8.2 – Symmetry: Line Symmetry, Rotational Symmetry & Point Symmetry
Exercise 8.2 from Chapter 8, "Exploration of Geometric Figures," of Class 8 Mathematics (CBSE, Telangana & Andhra Pradesh syllabus) builds on the basic idea of symmetry that students first meet visually — in butterflies, leaves, and everyday shapes — and turns it into a precise mathematical tool. This exercise covers three closely related ideas: line symmetry, rotational symmetry, and point symmetry, and then applies all three to the 26 capital letters of the English alphabet and to common geometric figures.
Understanding symmetry is not just useful for exam questions — it is the foundation for topics like tessellations, reflection and rotation transformations, and even later coordinate geometry work in higher classes. A solid grip on this exercise also makes the introductory symmetry concepts much easier to revise before exams.
What Is Line Symmetry?
A figure is said to have line symmetry (also called reflection symmetry) if it can be divided into two halves by a straight line such that, when the figure is folded along this line, one half coincides exactly with the other half. The line along which the figure is folded is called the line of symmetry or axis of symmetry.
- A figure can have more than one line of symmetry — for example, a square has 4 lines of symmetry, while a circle has infinitely many.
- Not every figure is symmetric — many irregular shapes have zero lines of symmetry.
- Natural examples of line symmetry include a butterfly (one vertical line of symmetry), a banana (no clean line of symmetry), and a four-leaf clover (multiple lines of symmetry).
Number of Lines of Symmetry in Common Shapes
The textbook illustrates how the number of lines of symmetry increases with the regularity of a shape:
| Shape | Number of Lines of Symmetry |
|---|---|
| Isosceles triangle | 1 |
| Rectangle | 2 |
| Equilateral triangle | 3 |
| Square | 4 |
| Circle | Infinity |
What Is Rotational Symmetry?
A figure has rotational symmetry (also known as radial symmetry) if, when it is rotated by some angle around a fixed central point, the rotated figure looks exactly the same as the original. Shapes like squares, rhombuses, and circles all show rotational symmetry because turning them by certain angles produces a figure identical to the starting position.
Order of Rotational Symmetry
The order of rotational symmetry of a figure is defined as the number of times the figure looks exactly like its original position during one complete 360° rotation. A very useful pattern emerges from this definition:
Order of rotational symmetry = Number of lines of symmetry = Number of sides of a regular polygon
| Geometrical Figure | No. of Axes of Symmetry | No. of Times It Resumes Original Position | Order of Rotation |
|---|---|---|---|
| Isosceles triangle | 1 | 1 | 1 |
| Rectangle | 2 | 2 | 2 |
| Equilateral triangle | 3 | 3 | 3 |
| Square | 4 | 4 | 4 |
| Circle | Infinity | Infinity | Infinity |
What Is Point Symmetry?
A figure has point symmetry if it looks exactly the same when viewed from two opposite directions — that is, when it is rotated 180° about a central point, it appears unchanged. Point symmetry is closely linked to rotational symmetry of order 2: any figure with point symmetry will look identical after a half-turn.
Good examples of point symmetry include certain geometric patterns (like an eight-pointed star design), a set of three curved arrows arranged in a circular flow, and several capital letters such as H, I, N, O, S, X, and Z — each of these letters looks the same when rotated 180°.
Exercise 8.2 – Solved Questions, Step by Step
This question asks students to physically cut out bold capital letters, paste them in a notebook, and draw every possible line of symmetry for each one. Based on this hands-on activity, six follow-up questions are answered.
Working through all 26 letters carefully shows that letters behave very differently depending on their shape. Letters like A, M, T, U, V, W, Y have a single vertical line of symmetry. Letters like B, C, D, E, K have a single horizontal line of symmetry. Letters like H, I, O, X have more than one line of symmetry (both horizontal and vertical, plus sometimes diagonals for O and X), while letters such as F, G, J, L, N, P, Q, R, S, Z have no line of symmetry at all.
Answers to the Six Sub-Questions
- (i) Letters with no line of symmetry: There are 10 such letters.
- (ii) Letters with exactly one line of symmetry: There are 12 such letters.
- (iii) Letters with exactly two lines of symmetry: There are 3 such letters.
- (iv) Letters with more than two lines of symmetry: There is 1 such letter (the letter O, which behaves like a circle with infinite lines of symmetry, or X depending on the font style used).
- (v) Letters with rotational symmetry: 7 letters — H, I, N, O, S, X, Z. Each of these looks identical after being rotated by 180°.
- (vi) Letters with point symmetry: The same 7 letters — H, I, N, O, S, X, Z — also have point symmetry, because point symmetry and rotational symmetry of order 2 go hand in hand.
This question asks students to draw all lines of symmetry for a set of polygons — a rectangle, a triangle, a square, a regular pentagon, a regular hexagon, a regular heptagon (7 sides), and a regular octagon — and then determine which of these figures also have point symmetry.
Solution Summary
- Among the figures studied — rectangle, triangle, square, pentagon, hexagon, heptagon, and octagon — the rectangle, square, hexagon, and octagon have point symmetry.
- The triangle, pentagon, and heptagon do not have point symmetry.
Relation Between Line Symmetry and Point Symmetry
The key relationship discovered here is: a figure has point symmetry only if it has an even number of lines of symmetry. The rectangle (2 lines), square (4 lines), hexagon (6 lines), and octagon (8 lines) all have an even count and show point symmetry. The triangle (3 lines), pentagon (5 lines), and heptagon (7 lines) have an odd count and do not show point symmetry.
Question 3: Natural Objects with at Least One Line of Symmetry
This question asks students to identify natural objects whose faces show at least one line of symmetry. Common, easily observable answers include:
- Butterfly — symmetric along the line where its wings meet
- Star fish — symmetric along multiple lines through its centre
- Apple — symmetric when cut vertically through the stem
- Human face — approximately symmetric along a vertical line down the centre
- Hibiscus flower — symmetric petal arrangement around the centre
- Faces of animals, such as cats and dogs — generally symmetric along a vertical line
Question 4: Drawing Tessellations
A tessellation is a repeating pattern of shapes that covers a flat surface completely, without any gaps or overlaps. This question asks students to draw three different tessellations and name the basic shapes used to build each one.
- Tessellation 1: Built from squares arranged with rotated star-like motifs inside each square, repeated in a grid pattern.
- Tessellation 2: Built from rhombuses (diamonds) arranged in interlocking rows to create a continuous diamond-and-cross pattern.
- Tessellation 3: Built from hexagons arranged edge-to-edge, forming a honeycomb-style pattern — one of the most efficient and common tessellations found in nature (for example, in beehives).
Common Mistakes to Avoid in Exercise 8.2
- Confusing line symmetry with rotational symmetry: A figure can have lines of symmetry without having rotational symmetry of a high order, and vice versa — always check both separately.
- Drawing diagonal lines of symmetry incorrectly for letters: Letters like X have diagonal symmetry lines that must pass exactly through the centre — a slightly off-centre line is not a valid axis of symmetry.
- Assuming all regular-looking shapes have point symmetry: Remember the rule — point symmetry only occurs when the number of lines of symmetry is even. A regular pentagon (5 lines) does not have point symmetry, even though it looks "regular."
- Forgetting that order of rotation can be 1: Every figure technically returns to its original position after a full 360° turn, so the minimum order of rotational symmetry is 1 — even for figures with no other symmetry.
- Mixing up "axis of symmetry" with "axis of rotation": The axis (line) of symmetry is used for folding/reflection, while rotational symmetry is about turning the figure around a central point — these are related but different ideas.
What Exercise 8.2 Prepares You For
Mastering line symmetry, rotational symmetry, and point symmetry in this exercise builds the visual and logical foundation for several upcoming topics. These ideas reappear when studying construction of quadrilaterals, where understanding the symmetry of squares, rectangles, and rhombuses helps you verify that a construction is correct.
Symmetry concepts also connect to coordinate geometry in later classes, where reflections and rotations of points and shapes on a graph are described using exactly the same vocabulary — line of symmetry, order of rotation, and point symmetry — but applied to coordinates instead of physical cut-outs. Students preparing for CBSE, Telangana, or Andhra Pradesh board exams should revise this exercise thoroughly, since questions on identifying lines of symmetry and classifying letters/shapes by symmetry type are common in both objective and short-answer formats.