Symmetry — NCERT Solutions, AI Tutor & Practice

A line of symmetry is a line that divides a figure into two identical halves that are mirror images of each other.

Steps to understand using a butterfly:

1. Imagine folding the butterfly along a vertical line down its middle (through the body).
2. The left wing falls exactly on top of the right wing.
3. Since both halves match perfectly, the fold line is a line of symmetry.

Thus a butterfly has a vertical line of symmetry through its body.

A square has 4 lines of symmetry.

The four lines are:

1. One vertical line through the midpoints of the top and bottom sides.
2. One horizontal line through the midpoints of the left and right sides.
3. One diagonal from the top-left corner to the bottom-right corner.
4. One diagonal from the top-right corner to the bottom-left corner.

Each of these folds makes the two halves match exactly, so a square has $4$ lines of symmetry.

(a) Equilateral triangle: It has 3 lines of symmetry — one from each vertex to the midpoint of the opposite side.

(b) Rectangle: It has 2 lines of symmetry — one vertical (through midpoints of shorter sides) and one horizontal (through midpoints of longer sides). Note: the diagonals are NOT lines of symmetry.

(c) Circle: A circle has infinitely many lines of symmetry, because every diameter divides it into two identical halves.

Let us test each letter:

• A — has a vertical line of symmetry. ✓
• B — has a horizontal line of symmetry. ✓
• F — has no line of symmetry. ✗
• H — has both a vertical and a horizontal line of symmetry. ✓
• O — has a vertical, a horizontal (and more) line of symmetry. ✓
• P — has no line of symmetry. ✗

Letters with a line of symmetry: A, B, H, O.

Steps to complete a symmetric figure:

1. Identify the line of symmetry (here, the vertical line).
2. Take each point on the given (left) half and measure its perpendicular distance from the line.
3. Mark a matching point on the right side at the same distance on the opposite side of the line.
4. Join these mirror points smoothly.

Since the left half of the heart is a curve, its mirror image is drawn on the right so that both curves meet at the top and bottom, forming a complete symmetric heart shape.

A scalene triangle has all three sides of different lengths and all three angles different.

Reasoning:

1. For a line of symmetry, folding must make two halves coincide.
2. Since no two sides are equal, there is no fold that maps the triangle onto itself.

Therefore, a scalene triangle has 0 lines of symmetry (no line of symmetry).

A regular hexagon has 6 lines of symmetry.

They are:

1. 3 lines joining opposite vertices (corners).
2. 3 lines joining the midpoints of opposite sides.

In general, a regular polygon with $n$ sides has $n$ lines of symmetry. For a hexagon, $n = 6$, so it has $6$ lines of symmetry.

(a) Exactly one line of symmetry: A human face (front view) or a spoon — folding along the single vertical middle line makes both halves match, but no other fold works.

(b) More than one line of symmetry: A square carrom board — it has $4$ lines of symmetry. Another example is a circular clock face, which has infinitely many lines of symmetry.

These show that different objects can have one, many, or infinitely many lines of symmetry.

Vertical mirror image of 3: When you reflect the digit 3 across a vertical line (mirror on the right/left), it flips to look like a backwards 3 (facing the opposite way), similar to the shape 'Ɛ'.

Does 3 have a line of symmetry?

1. A vertical fold does NOT match the two halves (the curves open to one side).
2. A horizontal fold, however, DOES make the top and bottom halves of the digit 3 coincide.

So the digit 3 has a horizontal line of symmetry, but not a vertical one.

Using symmetry to complete the rangoli:

1. The square rangoli has a vertical line of symmetry and a horizontal line of symmetry.
2. Reflect the drawn quarter across the vertical line to get the adjacent quarter (this fills half the square).
3. Now reflect that half across the horizontal line to fill the bottom half.
4. The four quarters together form the complete symmetric rangoli.

By applying reflection twice (about two perpendicular lines), one-fourth of the design generates the whole pattern.

Rhombus:

• A rhombus has all four sides equal.
• Its two diagonals act as lines of symmetry, because folding along either diagonal makes the halves coincide.
• So a rhombus has 2 lines of symmetry.

Parallelogram:

• A general parallelogram (non-rectangle, non-rhombus) has opposite sides equal but slanted.
• No fold — neither along a diagonal nor through midpoints — makes the two halves match.
• So a parallelogram has 0 lines of symmetry.

Conclusion: The rhombus has lines of symmetry; the parallelogram does not.

General rule: A regular polygon with $n$ sides has exactly $n$ lines of symmetry.

Applying the rule:

• Regular pentagon: $n = 5$, so it has $5$ lines of symmetry.
• Regular octagon: $n = 8$, so it has $8$ lines of symmetry.

So the answers are 5 and 8 respectively.