MathsICSE Class 8 · Chapter 18

Symmetry ICSE Class 8: Reflection Symmetry and Lines of Symmetry

Updated 15 October 2026 · Fact-checked

A figure has reflection symmetry when a line divides it into two halves that are mirror images, so folding along the line makes the halves match exactly. ICSE Class 8 counts lines of symmetry of triangles, quadrilaterals, regular polygons and letters, and reflects points and shapes in a mirror line.

What Symmetry is about

A shape has reflection symmetry (also called line symmetry or mirror symmetry) if a straight line divides it into two parts that are mirror images of each other. That line is a line of symmetry or axis of symmetry. If you fold the shape along it, the two halves fit exactly on top of each other.

The CISCE Class VIII curriculum asks for the idea of reflection symmetry and symmetrical shapes. In a school exam you will:

find and draw the lines of symmetry of triangles, quadrilaterals, regular polygons, circles and letters,
reflect a point or shape in a mirror line (often on squared paper),
complete a figure so that a given line is a line of symmetry,
reason about shapes from their symmetry (for example, a triangle with one line of symmetry is isosceles).

The key rule of reflection: each point and its image are the same distance from the mirror line, on opposite sides, and the line joining them is perpendicular to the mirror.

Weightage: CISCE publishes no question paper, marking scheme or chapter weightage for Class VIII Mathematics: your school sets and marks the exam. Symmetry sits in the Geometry theme of the CISCE Curriculum for Upper Primary Classes (VI-VIII).Source: CISCE Curriculum for Upper Primary Classes (VI-VIII), RDCD 2016

Key concepts

Line of symmetry

A line of symmetry splits a figure into two halves that are mirror images.

Fold test: trace the shape, fold along the line; if the two halves match exactly, the line is a line of symmetry.

Mirror test: stand a mirror on the line; the half you see plus its reflection should make the whole shape.

A shape can have no line, one line, several lines, or (for a circle) infinitely many.

Worked example

How many lines of symmetry does an isosceles triangle (two equal sides) have, and where?

Try it first, then show the solutionSolution

The line from the vertex between the equal sides to the mid-point of the base folds one equal side onto the other.
No other fold works (unless all three sides are equal).

Answer: One line: from the top vertex to the mid-point of the base.

Triangles and quadrilaterals

Shape	Lines of symmetry
Scalene triangle	0
Isosceles triangle	1
Equilateral triangle	3
Parallelogram (not a rectangle or rhombus)	0
Kite	1 (the diagonal between the unequal pairs of sides)
Isosceles trapezium	1 (through the mid-points of the parallel sides)
Rectangle	2 (through the mid-points of opposite sides)
Rhombus	2 (its diagonals)
Square	4 (2 diagonals and 2 mid-point lines)

Note that a rectangle's diagonals are not lines of symmetry: folding along a diagonal does not match the halves (unless it is a square).

Worked example

Name the lines of symmetry of a rectangle 8 cm by 6 cm, and explain why a diagonal is not one.

Try it first, then show the solutionSolution

Two lines: one joining the mid-points of the 8 cm sides, one joining the mid-points of the 6 cm sides.
Folding along a diagonal puts an 8 cm side onto a 6 cm side, so the halves do not match.

Answer: 2 lines (mid-point lines); diagonals are not lines of symmetry.

Regular polygons and the circle

A regular polygon has all sides equal and all angles equal. A regular polygon with $n$ sides has exactly ** $n$ lines of symmetry**.

If $n$ is odd, each line joins a vertex to the mid-point of the opposite side.
If $n$ is even, half the lines join opposite vertices and half join mid-points of opposite sides.

A circle has infinitely many lines of symmetry: every diameter is one.

Worked example

How many lines of symmetry does a regular hexagon have, and what do they join?

Try it first, then show the solutionSolution

Regular, $n = 6$ : 6 lines.
$n$ is even: 3 lines join opposite vertices and 3 join mid-points of opposite sides.

Answer: 6 lines (3 through opposite vertices, 3 through mid-points of opposite sides)

Symmetry of letters

Capital letters written in a simple plain style have these lines of symmetry:

vertical line only: A, M, T, U, V, W, Y;
horizontal line only: B, C, D, E, K;
both (2 lines): H, I, O, X;
none: F, G, J, L, N, P, Q, R, S, Z.

The exact answer can depend on how a letter is written, so draw it plainly and test the fold.

Worked example

Which letters of the word MATHS have a line of symmetry, and how many lines in total?

Try it first, then show the solutionSolution

M: 1 (vertical), A: 1 (vertical), T: 1 (vertical), H: 2, S: 0.
Total $= 1 + 1 + 1 + 2 + 0 = 5$ .

Answer: M, A, T, H; 5 lines in total

Reflection in a mirror line

When a point $P$ is reflected in a line $l$ to give the image $P^{'}$ :
1. $P$ and $P^{'}$ are the same distance from $l$ , on opposite sides;
2. the segment $P P^{'}$ is perpendicular to $l$ ;
3. so $l$ is the perpendicular bisector of $P P^{'}$ .

A point on the mirror line is its own image. A reflected shape is the same size and shape but laterally inverted (left and right swap), which is why words look reversed in a mirror.

Draw-and-label checklist (squared paper): draw the mirror line; for each vertex, count squares perpendicular to the line and count the same number on the other side; join the image points; label them $A^{'}$ , $B^{'}$ , ...

Worked example

Point $P$ is 3 cm from mirror line $l$ . Find the distance from $P$ to its image $P^{'}$ , and the angle between $P P^{'}$ and $l$ .

Try it first, then show the solutionSolution

$P^{'}$ is 3 cm from $l$ on the other side.
$P P^{'} = 3 + 3 = 6$ cm, and $P P^{'}$ meets $l$ at $90°$ .

Answer: $P P^{'} = 6$ cm; $90°$

Using symmetry to reason about shapes

Symmetry tells you about sides and angles:

matching sides across a line of symmetry are equal, and matching angles are equal;
a triangle with exactly one line of symmetry is isosceles; with three, it is equilateral; a triangle cannot have exactly two;
a quadrilateral with four lines is a square.

To complete a symmetric figure, reflect every given vertex in the line and join the images in the same order.

Worked example

A triangle has exactly one line of symmetry, and the angle at the vertex on that line is $52°$ . Find the other two angles.

Try it first, then show the solutionSolution

One line of symmetry: isosceles, and the base angles are equal.
Each base angle $= \frac{180° - 52°}{2} = 64°$ .

Answer: $64°$ and $64°$

Maths in real life: Symmetry

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Mirror distance

Suppose you stand 2 m in front of a flat mirror on a wall.

How far behind the mirror does your image appear, and how far are you from your image?

Try it first, then show the solutionSolution

The image is the same distance behind the mirror: 2 m.
Distance from you to your image $= 2 + 2 = 4$ m.

Answer: 2 m behind the mirror; 4 m from you

Memory hook: Object and image are equally far from the mirror.

Lateral inversion

Imagine a car driver sees a vehicle behind in the rear-view mirror, and the word on its front is painted in reversed letters.

How does the word appear in the mirror, and why?

Try it first, then show the solutionSolution

A mirror swaps left and right (lateral inversion).
Reversed letters are swapped back, so the word reads correctly in the mirror.

Answer: It reads the right way round.

Memory hook: Two reversals make the right way round.

Line of symmetry in design

Suppose you fold a sheet of paper in half, cut out half a butterfly along the fold, and open it.

How many lines of symmetry does the butterfly have, and where?

Try it first, then show the solutionSolution

The two wings are mirror images across the fold.
So there is 1 line of symmetry: the fold line down the middle.

Answer: 1 line, along the fold

Memory hook: The fold line is the line of symmetry.

Topics in this chapter

Lines of Symmetry and Reflection

Symmetry formula sheet

Regular polygon: $lines of symmetry = n$; n = number of sides.
Reflection distance: $P P^{'} = 2 d$; d = distance of P from the mirror.
Reflection angle: $P P^{'} ⊥ l$; The mirror is the perpendicular bisector of PP′.

How the school exam asks Symmetry

The question patterns to expect, and the method that works for each.

Question type	How it is asked	Method
Count lines	How many lines of symmetry does a rhombus, a kite or a regular pentagon have?	Use the table, then sketch and fold-test.
Draw lines	Draw all the lines of symmetry of a given figure.	Dotted lines through vertices or mid-points, extended beyond the figure.
Letters	Which letters of a word have a horizontal line of symmetry?	Write each letter plainly and test vertical and horizontal folds.
Reflect	Reflect a point or a triangle in a mirror line on squared paper.	Same distance, opposite side, perpendicular to the mirror.
Reasoning	A triangle has one line of symmetry and one angle of 40°; find the others.	Equal parts across the line of symmetry.

Tricks to remember Symmetry

Short memory aids. Say each one aloud once, then use it on a question the same day.

n sides, n lines
A regular polygon has as many lines of symmetry as it has sides.
Use it when: Regular polygons.
Square = 4, rectangle and rhombus = 2
Rectangles use the mid-point lines, rhombuses use the diagonals, and the square uses both.
Use it when: Quadrilaterals.
Same distance, other side, at 90°
Every reflection follows these three rules.
Use it when: Reflecting points and shapes.
Fold it
If you are unsure, imagine folding a paper copy along the line.
Use it when: Checking any line of symmetry.

Symmetry: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Example 1: Square · 1 mark · MCQExam level

The number of lines of symmetry of a square is:
(a) 1
(b) 2
(c) 4
(d) infinitely many

Try it first, then show the solutionSolution

Two diagonals and two lines through mid-points of opposite sides.
Total 4.

Answer: (c)

Example 2: Three shapes · 3 marks · Short answerExam level

State the number of lines of symmetry of (i) an equilateral triangle (ii) a rectangle (iii) a rhombus, and describe where they are.

Try it first, then show the solutionSolution

(i) 3: each from a vertex to the mid-point of the opposite side.
(ii) 2: each joining the mid-points of a pair of opposite sides.
(iii) 2: the two diagonals.

Answer: 3, 2, 2

Example 3: Letters with a vertical line · 3 marks · Short answerExam level

In the word SYMMETRY, which letters have a vertical line of symmetry, and which have a horizontal one?

Try it first, then show the solutionSolution

S: none; R: none.
Vertical: Y, M, M, T, Y (5 letters).
Horizontal: E.

Answer: Vertical: Y, M, M, T, Y; horizontal: E

Example 4: Reflecting a point · 3 marks · Short answerHarder exam level

Point $P$ is 2.5 cm from mirror line $l$ , and $Q$ is a point on $l$ . If $QP = 4$ cm, find $P P^{'}$ and $Q P^{'}$ , where $P^{'}$ is the image of $P$ .

Try it first, then show the solutionSolution

$P^{'}$ is 2.5 cm on the other side, so $P P^{'} = 5$ cm.
$l$ is the perpendicular bisector of $P P^{'}$ , so any point on $l$ is equally far from $P$ and $P^{'}$ .
$Q P^{'} = QP = 4$ cm.

Answer: $P P^{'} = 5$ cm; $Q P^{'} = 4$ cm

Example 5: Regular polygons · 4 marks · Long answerHarder exam level

(i) A regular polygon has 9 lines of symmetry. How many sides does it have? (ii) Do its lines pass through vertices, mid-points of sides, or both? (iii) Name a quadrilateral with no line of symmetry.

Try it first, then show the solutionSolution

(i) A regular $n$ -gon has $n$ lines, so $n = 9$ (a regular nonagon).
(ii) $n$ is odd, so each line passes through a vertex and the mid-point of the opposite side.
(iii) A parallelogram that is not a rectangle or a rhombus.
Fold test: no fold of such a parallelogram makes its halves match.

Answer: (i) 9 (ii) both: vertex to opposite mid-point (iii) a general parallelogram

Example 6: Rectangle and its diagonals · 4 marks · Long answerHarder exam level

Rectangle $A B C D$ has $A B = 8$ cm and $B C = 6$ cm. Draw its lines of symmetry and explain, using a fold, why diagonal $A C$ is not a line of symmetry. Which special rectangle does have its diagonals as lines of symmetry?

Try it first, then show the solutionSolution

Lines of symmetry: the line joining the mid-points of $A B$ and $D C$ , and the line joining the mid-points of $A D$ and $B C$ .
Folding along $A C$ would send $A B$ (8 cm) onto $A D$ (6 cm).
Different lengths cannot match, so $A C$ is not a line of symmetry.
In a square $A B = A D$ , so the diagonals are lines of symmetry.

Answer: 2 lines; diagonals fail because $8 \neq = 6$ ; a square

Example 7: Triangles and symmetry · 5 marks · Long answerHOTS / competency

(i) A triangle has exactly one line of symmetry and its angle at the vertex on that line is $40°$ . Find the other angles. (ii) What are the angles of a triangle with three lines of symmetry? (iii) Explain why no triangle has exactly two lines of symmetry.

Try it first, then show the solutionSolution

(i) One line: isosceles; base angles equal: $\frac{180° - 40°}{2} = 70°$ each.
(ii) Three lines: equilateral, so each angle is $60°$ .
(iii) Each line of symmetry through a vertex makes the two sides at that vertex equal.
Two lines would give two pairs of equal sides, which means all three sides are equal.
Then the triangle is equilateral and has three lines, not two.

Answer: (i) $70°, 70°$ (ii) $60°$ each (iii) two lines force a third

Case-study practice (Section B style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 10-mark question split into 3 + 3 + 4 marks.

A rangoli design

Suppose a rangoli is drawn on a regular hexagon. The designer wants the finished pattern to keep every line of symmetry of the hexagon.

(i) · 3 marks
How many lines of symmetry does the hexagon have? Describe them.
Answer
Answer: 6
A regular hexagon has 6: three through opposite vertices and three through mid-points of opposite sides.
(ii) · 3 marks
She draws a petal that is symmetric about the line through the mid-point of one side. How many petals are needed so that the pattern keeps all 6 lines? Explain.
Answer
Answer: 6 petals, one on each side
Reflecting the petal in the lines of symmetry places a copy on every side, so the pattern needs 6 petals (one per side) to look the same in every line.
(iii) · 4 marks
A dot is placed on the line joining the centre to one vertex, 2 cm from the centre. How many dots must there be in all, and how far is each from the centre?
Answer
Answer: 6 dots, each 2 cm from the centre
Reflections keep distances from the centre, so every image is 2 cm from the centre. Reflecting in the 6 lines of symmetry puts a copy on the ray from the centre towards each of the 6 vertices, giving 6 dots.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

State the number of lines of symmetry of a rectangle, a rhombus and a square. (3 marks)

Step	Marks
Rectangle: 2 (mid-point lines)	1
Rhombus: 2 (diagonals)	1
Square: 4 (both kinds)	1

Point $P$ is 3 cm from line $l$ . $P^{'}$ is its reflection in $l$ , and $Q$ is any point on $l$ . Find $P P^{'}$ , the angle between $P P^{'}$ and $l$ , and show that $QP = Q P^{'}$ . (4 marks)

Step	Marks
$P^{'}$ is 3 cm from $l$ on the opposite side	1
$P P^{'} = 6$ cm	1
$P P^{'} ⊥ l$ ( $90°$ ), and $l$ bisects $P P^{'}$	1
Every point on the perpendicular bisector is equidistant from $P$ and $P^{'}$ , so $QP = Q P^{'}$	1

An isosceles triangle has a base angle of $65°$ . Find the vertex angle, state how many lines of symmetry it has, and describe the line. (5 marks)

Step	Marks
Base angles are equal: both $65°$	1
Vertex angle $= 180° - 130° = 50°$	1
Not all angles are equal, so it is not equilateral	1
Exactly 1 line of symmetry	1
It joins the vertex to the mid-point of the base, at right angles to the base	1

Where students lose marks in Symmetry

The slips examiners see most often, and what to write instead.

Mistake: Counting a rectangle's diagonals
Folding along a diagonal does not match the halves.
Wrong working
Rectangle: 4 lines
Right working
Rectangle: 2 lines
Do this: Fold-test every line you draw.
Usually costs: 1 mark.
Mistake: Reflecting at a slant
The image must be straight across, perpendicular to the mirror line.
Do this: Count squares at right angles to the mirror, not along a diagonal (unless the mirror itself is diagonal).
Usually costs: The reflection.
Mistake: Wrong distance behind the mirror
The image is as far behind the mirror as the object is in front.
Wrong working
Object 3 cm from $l$ ; image 3 cm from the object
Right working
Image 3 cm from $l$ , so 6 cm from the object
Do this: Measure from the mirror line, not from the point.
Usually costs: 1 mark.
Mistake: Saying a parallelogram has 2 lines
A general parallelogram has no line of symmetry.
Do this: Only rectangles and rhombuses (special parallelograms) have lines of symmetry.
Usually costs: 1 mark.
Mistake: Drawing lines that stop inside the figure
A line of symmetry should be drawn right across the figure.
Do this: Use dotted lines extended beyond the shape.
Usually costs: Presentation mark.

Practice questions on Symmetry

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQExam level
A circle has:
(a) 1 line of symmetry
(b) 2 lines of symmetry
(c) 4 lines of symmetry
(d) infinitely many lines of symmetry
Hint
Think of diameters.
Answer
(d)
Q2 · 1 mark · Very short answerExam level
How many lines of symmetry does a regular pentagon have?
Hint
n sides, n lines.
Answer
5.
Q3 · 2 marks · Short answerExam level
State the number of lines of symmetry of (i) a parallelogram that is not a rectangle or rhombus (ii) a kite.
Hint
Fold-test.
Answer
(i) 0 (ii) 1.
Q4 · 2 marks · Short answerHarder exam level
An isosceles triangle has a base of 10 cm. Its line of symmetry meets the base at $M$ . Find $B M$ and the angle at $M$ .
Hint
The line bisects the base at right angles.
Answer
$B M = 5$ cm; the angle is $90°$ .
Q5 · 1 mark · Assertion-reasonHarder exam level
Assertion (A): The diagonals of a rectangle are its lines of symmetry.
Reason (R): The diagonals of a rectangle bisect each other.
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
Fold a long thin rectangle along a diagonal.
Answer
(b)
Q6 · 2 marks · Short answerHarder exam level
In the word HOME (plain capitals), how many lines of symmetry are there in total?
Hint
H 2, O 2, M 1, E 1.
Answer
$2 + 2 + 1 + 1 = 6$ .
Q7 · 2 marks · Short answerHarder exam level
On squared paper, a vertical mirror line is drawn. Point $A$ is 4 squares to the left of it and 1 square above a reference line. Where is its image, and how many squares apart are $A$ and its image?
Hint
Same distance, opposite side, same height.
Answer
4 squares to the right of the mirror, 1 square above the reference line; 8 squares apart.
Q8 · 3 marks · Short answerHarder exam level
Name a quadrilateral with (i) exactly one line of symmetry (ii) exactly two (iii) exactly four.
Hint
Use the table.
Answer
(i) kite or isosceles trapezium (ii) rectangle or rhombus (iii) square.
Q9 · 3 marks · Short answerHarder exam level
A regular hexagon has how many lines of symmetry? How many pass through two vertices and how many through mid-points of two sides?
Hint
Even number of sides.
Answer
6 lines: 3 through opposite vertices and 3 through mid-points of opposite sides.
Q10 · 2 marks · Short answerHarder exam level
Describe the line of symmetry of an isosceles trapezium.
Hint
It meets both parallel sides.
Answer
One line, joining the mid-points of the two parallel sides, perpendicular to both.
Q11 · 3 marks · Short answerHOTS / competency
A regular polygon has 12 sides. How many lines of symmetry does it have, how many pass through vertices only, and how many through mid-points of sides only?
Hint
n is even.
Answer
12 lines: 6 through pairs of opposite vertices and 6 through mid-points of opposite sides.
Q12 · 3 marks · Short answerHOTS / competency
Triangle $P QR$ has $P Q = P R$ and $∠ Q = 65°$ . Find $∠ P$ and $∠ R$ . How many lines of symmetry does the triangle have?
Hint
Equal sides face equal angles.
Answer
$∠ R = 65°$ , $∠ P = 50°$ . One line of symmetry, through $P$ and the mid-point of $QR$ .

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.What is a line of symmetry?Show answer
A line that divides a figure into two mirror-image halves.
2.How many lines of symmetry does a regular octagon have?Show answer
8.
3.How many lines of symmetry does a circle have?Show answer
Infinitely many (every diameter).
4.How many lines of symmetry does a scalene triangle have?Show answer
None.
5.Which quadrilateral has exactly 4 lines of symmetry?Show answer
A square.
6.Does the letter H have a horizontal line of symmetry?Show answer
Yes; it has both a vertical and a horizontal line.
7.What is lateral inversion?Show answer
The left–right reversal of an image in a plane mirror.
8.A point is 4 cm from a mirror line. How far is it from its image?Show answer
8 cm.

Best order to study it

Learn what a line of symmetry is and use the fold test.
Learn the table for triangles and quadrilaterals.
Learn the n-lines rule for regular polygons and the circle.
Practise letters and simple designs.
Finish with reflecting points and shapes, and reasoning with symmetry.

Quick revision checklist

The night before the exam: can you do each of these without looking?

Did you fold-test each line?
Did you remember that a rectangle's diagonals are not lines of symmetry?
For a regular polygon, did you use n lines?
Is each image point the same distance behind the mirror?
Is the line joining a point and its image perpendicular to the mirror?
Are your lines of symmetry drawn dotted and right across the figure?

Symmetry: frequently asked questions

Is reflection symmetry the same as line symmetry?

Yes. Reflection symmetry, line symmetry and mirror symmetry all mean that a line divides the figure into mirror-image halves.

Is rotational symmetry in the Class 8 syllabus?

No. The CISCE Class VIII curriculum lists only the idea of reflection symmetry and symmetrical shapes, so this page covers reflection (line) symmetry.

Why does a rhombus have its diagonals as lines of symmetry but a rectangle does not?

In a rhombus all sides are equal, so folding along a diagonal matches equal sides. In a rectangle, folding along a diagonal would match a long side with a short side.

Do letters always have the same symmetry?

It depends on how they are written. In a plain capital style, H has two lines, A has one, and S has none.

Why are words reversed in a mirror?

A plane mirror swaps left and right (lateral inversion), so letters appear in reverse.

Based on: CISCE Curriculum for Upper Primary Classes (VI-VIII), Mathematics (Class VIII); CISCE RDCD: Curriculum Pre-School to Class VIII