How many lines of symmetry does a square have?

A square has 4 lines of symmetry -- 2 diagonals and 2 lines through midpoints of opposite sides.

What is rotational symmetry?

A figure has rotational symmetry if it looks the same after being rotated by less than 360 degrees. The number of times it matches itself in one full turn is its order of rotational symmetry.

What is translational symmetry?

Translational symmetry occurs when an object slides to a new position and the pattern repeats. The object looks the same before and after the slide.

What is the order of rotational symmetry of a circle?

A circle has infinite order of rotational symmetry because it looks the same after any rotation, no matter how small.

Symmetry Exercise Solutions | KalviGavani

Symmetry -- Exercise Solutions

Use this page alongside the official Samacheer Kalvi textbook. Solve each exercise first, then check the methods below.

Exercise 4.1 -- Fill in the Blanks (Q1)

1(i). The reflected image of the letter 'q' is ___

Answerp -- When 'q' is reflected across a vertical line, it becomes 'p'.

1(ii). A rhombus has ___ lines of symmetry.

Answer2 -- The two diagonals of a rhombus are its lines of symmetry.

1(iii). The order of rotational symmetry of the letter 'z' is ___

Answer2 -- Rotating 'z' by 180 degrees maps it onto itself. So the order of rotational symmetry is 2.

1(iv). A figure is said to have rotational symmetry, if the order of rotation is at least ___

Answer2 -- For a figure to have rotational symmetry, it must map onto itself at least twice during a full 360-degree rotation (i.e., order >= 2).

1(v). ___ symmetry occurs when an object slides to a new position.

AnswerTranslational -- In translational symmetry, the object is shifted (slid) along a direction and the pattern repeats.

Exercise 4.1 -- True or False (Q2)

Q2(i). A rectangle has four lines of symmetry.

AnswerFalse -- A rectangle has only 2 lines of symmetry (the lines through midpoints of opposite sides), not 4. Correction: A rectangle has 2 lines of symmetry.

Q2(ii). A shape has reflection symmetry if it has a line of symmetry.

AnswerTrue -- Reflection symmetry and line of symmetry mean the same thing. If a shape has a line of symmetry, it has reflection symmetry.

Q2(iii). The reflection of the name RANI is ИAꓤI.

AnswerTrue -- When reflected across a vertical line: R becomes И, A stays A, N becomes И (reversed), I stays I. The result is ИAꓤI.

Q2(iv). Order of rotation of a circle is infinite.

AnswerTrue -- A circle looks the same after any rotation, so it has infinite order of rotational symmetry.

Q2(v). The number 191 has rotational symmetry.

AnswerFalse -- Rotating 191 by 180 degrees gives 161 (the 9 in the middle becomes 6 when rotated), which is not 191. Correction: 191 does NOT have rotational symmetry. However, it does have reflection symmetry (vertical line through the 9).

Exercise 4.1 -- Match the Following (Q3)

Column A	Column B
(i) Square	(d) Four lines of symmetry
(ii) Parallelogram	(a) No line of symmetry
(iii) Isosceles triangle	(b) One line of symmetry
(iv) Rectangle	(c) Two lines of symmetry

Explanation

A square has 4 lines of symmetry (2 through midpoints of opposite sides + 2 diagonals).
A general parallelogram has no line of symmetry.
An isosceles triangle has 1 line of symmetry (from the apex vertex to the midpoint of the base).
A rectangle has 2 lines of symmetry (through midpoints of opposite sides).

Exercise 4.1 -- Lines of Symmetry (Q4)

Draw the lines of symmetry for each given figure.

Method: For each figure, identify all lines along which you can fold so the two halves overlap perfectly. A regular n-sided polygon has n lines of symmetry.

Key shapes and their lines of symmetry:

Equilateral triangle: 3 lines
Square: 4 lines
Rectangle: 2 lines (through midpoints, NOT diagonals)
Rhombus: 2 lines (along the diagonals)
Circle: Infinite lines (every diameter is a line of symmetry)

Exercise 4.1 -- Complete Each Alphabet (Q5)

Using the given horizontal or vertical line as symmetry, complete each letter to reveal a hidden word.

Method: Reflect each partial letter across the given line of symmetry to complete it. For example, if the left half of 'M' is given with a vertical symmetry line, draw the mirror image on the right to complete 'M'.

Exercise 4.1 -- Draw Line of Symmetry (Q6)

Five sub-parts where a line of symmetry must be drawn so that holes coincide in pairs.

Method: Identify pairs of holes that are mirror images of each other. Draw the line that acts as the mirror between each pair. Each hole must have exactly one matching hole on the other side.

Exercise 4.1 -- Complete the Other Half (Q7)

Four sub-parts where the dotted line is the line of symmetry -- complete the missing half.

Method: For each point in the given half, find its mirror image across the dotted line. If a point is 2 cm to the left of the line, its mirror image is 2 cm to the right. Connect all mirror points to complete the figure.

Exercise 4.1 -- Order of Rotation (Q8)

Find the order of rotational symmetry for each given figure.

Method: Count how many times the figure looks the same during one full 360-degree rotation. This count is the order of rotational symmetry.

Common figures:

Equilateral triangle: order 3 (rotates 120 degrees each time)
Square: order 4 (rotates 90 degrees each time)
Regular pentagon: order 5
Regular hexagon: order 6
Circle: infinite order

Exercise 4.1 -- Standard Die (Q9)

Find the order of rotational symmetry of each face of a standard die.

Face	Order of Rotational Symmetry
1 dot	1 (a single dot at centre has no rotational symmetry beyond full turn)
2 dots	2 (dots at opposite corners; 180-degree rotation maps to itself)
3 dots	3 (dots in diagonal; 120-degree rotation maps to itself)
4 dots	4 (dots at four corners; 90-degree rotation maps to itself)
5 dots	4 (centre + 4 corner dots; 90-degree rotation maps to itself)
6 dots	2 (2 rows of 3 dots; 180-degree rotation maps to itself)

Exercise 4.1 -- Border Kolams (Q10)

Identify the translated pattern in each border kolam.

Method: Find the smallest repeating unit in the kolam. This unit is repeated (translated) along the entire strip. The translational symmetry means the pattern looks the same after sliding it by the length of one unit.

Exercise 4.1 -- Objective Type Questions (Q11-Q15)

Q11. Which of the following letters does NOT have a line of symmetry? Options: A, P, T, U

AnswerP -- A has a vertical line, T has a vertical line, U has a vertical line. P does not have any line of symmetry.

Q12. Which of the following is a symmetrical figure?

AnswerThe figure with a line of symmetry (depends on the images provided). A symmetrical figure can be divided into two identical mirror halves.

Q13. Which word has a vertical line of symmetry? Options: DAD, NUN, MAM, EVE

AnswerMAM -- M, A, and M all have vertical lines of symmetry. When written in uppercase, each letter in "MAM" can be divided by a vertical line into mirror halves.

Note: In some fonts, EVE also has vertical symmetry (E, V, E all have vertical lines). However, the standard answer in the textbook is MAM.

Q14. The order of rotational symmetry of 818 is: Options: 1, 2, 3, 4

Step-by-step:

Rotating 818 by 180 degrees: 8 stays 8, 1 stays 1, 8 stays 8. Result: 818.
It maps onto itself once during a 360-degree rotation (at 180 degrees).
Order = 2

Answer(b) 2

Q15. The order of rotational symmetry of the given figure is:

AnswerCount the number of identical parts in the figure. If it is a regular n-sided polygon, the order is n.

Exercise 4.2 -- Miscellaneous Practice Problems

Q1. Draw the following triangles:

(i) A triangle which has no line of symmetry:

AnswerA scalene triangle (all three sides of different lengths) has no line of symmetry.

(ii) A triangle which has only one line of symmetry:

AnswerAn isosceles triangle has exactly one line of symmetry -- from the apex vertex to the midpoint of the base.

(iii) A triangle which has three lines of symmetry:

AnswerAn equilateral triangle has three lines of symmetry -- one from each vertex to the midpoint of the opposite side.

Exercise 4.2 -- Alphabet Classification (Q2)

Given alphabets: A, M, P, E, D, I, K, O, N, X, S, H, U, V, W, Z

Q2(i). No line of symmetry: P, S, Z

P has no line of symmetry. S and Z have rotational symmetry but no line of symmetry (in their standard printed form).

Q2(ii). Rotational symmetry: N, S, X, Z

These letters look the same after a 180-degree rotation. They have rotational symmetry of order 2.

Q2(iii). Reflection symmetry: A, M, D, I, K, O, X, H, U, V, W

Each of these letters has at least one line of symmetry (vertical or horizontal).

Q2(iv). Both reflection and rotational symmetry: I, O, X, H

These letters have both a line of symmetry AND look the same after a 180-degree rotation.

Exercise 4.2 -- Numbers with Symmetry (Q4)

Give five examples of 3-digit numbers that have both rotational and reflection symmetry.

Step-by-step:

For reflection symmetry (vertical line): the number reads the same forwards and backwards.
For rotational symmetry (180 degrees): when rotated 180 degrees, each digit maps to itself.

Answer181, 808, 888, 818, 111

Digits that are symmetric: 0, 1, 8 (they look the same when reflected and when rotated 180 degrees).

Exercise 4.2 -- Regular Polygon Table (Q9)

Shape	Lines of Symmetry	Order of Rotational Symmetry
Equilateral Triangle	3	3
Square	4	4
Regular Pentagon	5	5
Regular Hexagon	6	6
Regular Octagon	8	8

Q9(i). A regular polygon of 10 sides will have ___ lines of symmetry.

Answer10 -- A regular n-sided polygon has n lines of symmetry.

Q9(ii). If a regular polygon has 10 lines of symmetry, then its order of rotational symmetry is ___

Answer10 -- For regular polygons, lines of symmetry = order of rotational symmetry = number of sides.

Q9(iii). A regular polygon of n sides has ___ lines of symmetry and order of rotational symmetry ___.

Answern lines of symmetry and order of rotational symmetry n.

Exercise 4.2 -- Challenge Problems

Q6. Shade one square so that the figure possesses:

(i) One line of symmetry:

AnswerShade a square that creates exactly one line of symmetry. For example, shade one corner square so that the resulting figure can be folded along a single diagonal line.

(ii) Rotational symmetry of order 2:

AnswerShade a square that is diagonally opposite to an already-shaded corner. The resulting figure will look the same after a 180-degree rotation.

Q7. Join six identical squares with reflection symmetry in three ways.

Answer1. A 1 by 6 rectangle (6 squares in a row) -- horizontal and vertical lines of symmetry 2. A 2 by 3 rectangle (2 rows, 3 columns) -- vertical and horizontal lines of symmetry 3. A T-shape made of 6 squares -- vertical line of symmetry through the centre

Q8. Draw figures with:

(i) Reflection symmetry but no rotational symmetry:

AnswerAn isosceles triangle -- it has one line of symmetry but rotational symmetry of order 1 (no rotational symmetry).

(ii) Rotational symmetry but no reflection symmetry:

AnswerA parallelogram (that is not a rectangle or rhombus) -- it has rotational symmetry of order 2 but no line of symmetry. A swastika symbol also works.

(iii) Both reflection and rotational symmetry:

AnswerA square -- it has 4 lines of symmetry and rotational symmetry of order 4.

Additional Questions

Q1. Which letters of the English alphabet have the same mirror image as themselves when seen in a vertical mirror?

AnswerA, H, I, M, O, T, U, V, W, X -- These letters look the same in a vertical mirror because they have a vertical line of symmetry.

Q2. In the word "SYMMETRY", which letters do NOT look the same as their mirror images?

AnswerS, E, R -- These letters do not have vertical line of symmetry, so their mirror images look different.

Q3. How many lines of symmetry does the Indian National Flag have?

AnswerThe Indian National Flag has 1 line of symmetry -- a horizontal line passing through the centre (between saffron and green, with the Ashoka Chakra on the line). It does not have vertical symmetry because of the Ashoka Chakra's position and design.

Q4. Does a human face have symmetry?

AnswerA human face has approximate vertical symmetry (a vertical line from the forehead to the chin). However, it is not perfectly symmetrical -- the two sides of a face are slightly different.

Common Mistakes to Avoid

1. Confusing reflection and rotational symmetry: Reflection symmetry involves a line (mirror line). Rotational symmetry involves a point (centre of rotation). A shape can have one without the other.

2. Counting diagonal lines for rectangles: A rectangle has 2 lines of symmetry (through midpoints of opposite sides), NOT 4. The diagonals of a rectangle are NOT lines of symmetry.

3. Assuming all triangles have lines of symmetry: Only isosceles (1 line) and equilateral (3 lines) triangles have lines of symmetry. A scalene triangle has none.

4. Forgetting that rotational symmetry requires order >= 2: Every shape has rotational symmetry of order 1 (a full 360-degree turn). But we say a shape "has rotational symmetry" only when its order is at least 2.

5. Confusing rotational symmetry of numbers: When rotating 191 by 180 degrees, the 9 becomes a 6, so 191 becomes 161 (NOT 191). However, 818 stays 818 because 8 is rotationally symmetric.

6. Wrong number of lines for a rhombus: A rhombus has 2 lines of symmetry (its diagonals), not 4 like a square. A general parallelogram has 0 lines of symmetry.

7. Translational vs rotational symmetry: Translational symmetry involves sliding (no rotation). Rotational symmetry involves turning around a point. Do not confuse "sliding" with "turning."

Exam Checklist

[ ] Can you identify lines of symmetry in common shapes (square, rectangle, triangle, circle)?
[ ] Can you distinguish between reflection symmetry, rotational symmetry, and translational symmetry?
[ ] Can you find the order of rotational symmetry of a given figure?
[ ] Can you classify alphabets by their symmetry properties (reflection, rotational, both, none)?
[ ] Do you know the lines of symmetry for: equilateral triangle (3), isosceles triangle (1), scalene triangle (0), square (4), rectangle (2), rhombus (2), parallelogram (0), circle (infinite)?
[ ] Can you complete the missing half of a figure given a line of symmetry?
[ ] Can you give examples of figures with: only reflection symmetry, only rotational symmetry, both, neither?
[ ] Can you state the relationship between lines of symmetry and order of rotational symmetry in regular polygons?
[ ] Can you identify translational symmetry in border patterns and kolams?
[ ] Can you identify 3-digit numbers with both rotational and reflection symmetry?

Frequently Asked Questions

A line of symmetry divides a figure into two identical halves that are mirror images of each other. If you fold along this line, the two halves overlap perfectly.