Symmetrical Designs — NCERT Solutions

Given: We need to find multi-digit numbers with various types of symmetry.

Concept:

• For a number to have reflection symmetry (vertical): it must read the same forwards and backwards (palindrome), using only digits that themselves have vertical symmetry (0, 1, 8).
• For a number to have rotational symmetry ($\frac{1}{2}$ turn): it must look the same upside down, using digits that look the same or swap correctly when rotated (0→0, 1→1, 8→8, 6→9, 9→6).

2-digit numbers:

• Reflection symmetry only: 11 (vertical symmetry, reads same both ways)
• Rotational symmetry only: 69 (6 becomes 9 and 9 becomes 6 when rotated $\frac{1}{2}$ turn)
• Both symmetries: 11, 88, 00 (trivial)

3-digit numbers:

• Reflection symmetry: 101, 111, 181, 808, 818, 888
• Rotational symmetry: 609 (rotated $\frac{1}{2}$ turn → 609), 619, 689, 906, 916, 986, 808, 818, 888, 111
• Both symmetries: 111, 808, 818, 888

4-digit numbers:

• Reflection symmetry: 1001, 1111, 1881, 8008, 8118, 8888
• Rotational symmetry: 1001, 1691, 1961, 6009, 6119, 6699, 9006, 9116, 9669, 8008, 8118, 8888
• Both symmetries: 1001, 8008, 8118, 8888, 1111

Summary Table:

Given: A design is shown (image not visible, but we solve based on the concept).

(a) Does the design have rotational symmetry?

Concept: A design has rotational symmetry if it looks the same after rotating it by some angle less than 360° about its centre.

Working: Look at the design and rotate it mentally (or using tracing paper) by $\frac{1}{4}$, $\frac{1}{2}$, $\frac{3}{4}$ turns.

Answer: Students should observe the given design carefully. If the design does NOT look the same at any partial turn → No. If it does → Yes (state at which turn).

(b) Modify the design to have $\frac{1}{2}$ turn symmetry:

Concept: For $\frac{1}{2}$ turn (180°) symmetry, every part of the design must have a matching part exactly opposite to it through the centre.

Method:

• Take the original design.
• Add a copy of each shape, rotated 180° about the centre of the design.
• The new design will look the same after a $\frac{1}{2}$ turn.

Answer: Draw the original design, then add its 180°-rotated copy on the opposite side. (Students draw this in their notebook.)

(c) Modify the design to have $\frac{1}{4}$ turn symmetry:

Concept: For $\frac{1}{4}$ turn (90°) symmetry, the design must look the same after every 90° rotation. This means the design must have 4-fold rotational symmetry.

Method:

• Take the design from part (b).
• Add copies rotated by 90° and 270° about the centre.
• Now the design has 4 identical parts, one in each quadrant.

Answer: Draw the design with 4-fold symmetry. (Students draw this in their notebook.)

(d) Do the new designs have reflection symmetry?

Concept: A design with $\frac{1}{2}$ turn symmetry may or may not have reflection symmetry. A design with $\frac{1}{4}$ turn symmetry often has 4 lines of symmetry (vertical, horizontal, and two diagonal).

Answer:

• The design with $\frac{1}{2}$ turn symmetry: Check if a mirror line exists. If shapes are symmetric about a line → Yes, draw that line.
• The design with $\frac{1}{4}$ turn symmetry: Typically has 4 lines of symmetry — one vertical, one horizontal, and two diagonal lines passing through the centre.

Students should draw the lines of symmetry on their designs in their notebooks.

Given: Designs are shown in the textbook (images not fully visible; solving conceptually).

Concept:

• $\frac{1}{2}$ turn = 180° rotation about the centre.
• $\frac{1}{4}$ turn = 90° rotation about the centre.

For the first design (appears to be a simple shape or pattern):

Working: Rotate the design 180° about its centre. If every part of the design maps onto another part of the design → it has $\frac{1}{2}$ turn symmetry.

Answer: Students should use tracing paper or mentally rotate the design.

• If the design has 2-fold symmetry: Yes, it looks the same after $\frac{1}{2}$ turn.
• If the design has 4-fold symmetry: Yes, it looks the same after both $\frac{1}{4}$ and $\frac{1}{2}$ turns.

General guidance:

• A design with $\frac{1}{4}$ turn symmetry automatically also has $\frac{1}{2}$ turn symmetry.
• A design with $\frac{1}{2}$ turn symmetry does NOT necessarily have $\frac{1}{4}$ turn symmetry.

Given: A square divided into smaller parts (grid). We need to colour it with two colours so it has $\frac{1}{4}$ turn (90°) rotational symmetry.

Concept: For a design to look the same after every $\frac{1}{4}$ turn, it must have 4-fold rotational symmetry. This means if we rotate the design by 90°, it maps onto itself.

How to colour:

• Divide the square into 4 quadrants.
• Whatever pattern you colour in one quadrant, rotate it 90° for the next quadrant, and so on.
• Use Colour 1 and Colour 2 in a pattern that repeats every 90°.

Example (for a 4×4 grid):
$$\begin{array}{|c|c|c|c|}\hline A & B & B & A \\ \hline B & A & A & B \\ \hline B & A & A & B \\ \hline A & B & B & A \\ \hline\end{array}$$
where A = Colour 1, B = Colour 2. This pattern looks the same after every $\frac{1}{4}$ turn.

How many times does it look the same during a full turn?

Working: If the design has $\frac{1}{4}$ turn symmetry, it looks the same at:

• $\frac{1}{4}$ turn (90°) → 1st time
• $\frac{1}{2}$ turn (180°) → 2nd time
• $\frac{3}{4}$ turn (270°) → 3rd time
• Full turn (360°) → 4th time

Answer: The shape looks the same 4 times during a full turn.

Reflection symmetry:

A design with 4-fold rotational symmetry often also has reflection symmetry.

Answer: Yes, such designs typically have 4 lines of symmetry:

1. Vertical line through the centre
2. Horizontal line through the centre
3. Diagonal line from top-left to bottom-right
4. Diagonal line from top-right to bottom-left

Students should draw all 4 lines of symmetry on their coloured design.

Given: A shape made by combining squares and equilateral triangles (image shown in textbook).

Concept:

• Reflection symmetry: A line exists such that one half is the mirror image of the other.
• Rotational symmetry: The shape looks the same after rotating by some angle less than 360°.

Reflection Symmetry:

Working: Look at the shape and check if a mirror line can be drawn such that both halves match.

Answer: (Based on the typical shape shown — a square with triangles on sides)

• If triangles are placed symmetrically → Yes, the shape has reflection symmetry.
• Lines of symmetry: Typically vertical and/or horizontal lines through the centre, depending on the arrangement.
• Students should draw the line(s) on their shape.

Rotational Symmetry:

Working: Rotate the shape and check at which turn it looks the same.

Answer:

• If the shape has triangles on all 4 sides of a square → it has 4-fold rotational symmetry (looks the same at $\frac{1}{4}$, $\frac{1}{2}$, $\frac{3}{4}$, and full turn).
• If triangles are only on 2 opposite sides → it has 2-fold rotational symmetry (looks the same at $\frac{1}{2}$ turn).

Both symmetries:

• If the shape has both a line of symmetry AND rotational symmetry → Yes, it has both.

Sorting designs into 3 categories:

1. Only rotational symmetry: Designs that look the same when rotated but have no mirror line (e.g., a pinwheel/firki shape).
2. Only reflection symmetry: Designs that have a mirror line but do not look the same when rotated (e.g., letter A).
3. Both symmetries: Designs like a square with triangles on all sides, letter H, digit 8, etc.

Given: Images of wooden blocks and their corresponding prints are shown. One matching is already done as an example.

Concept: When a wooden block is pressed onto fabric, it creates a print. The print is the mirror image (reflection) of the block design, because the block is pressed face-down.

Working:

• Observe the design on each wooden block.
• The print will be the mirror/reflected image of the block.
• Match each block to the print that is its mirror image.

Method for students:

1. Look at the shape/pattern on the wooden block.
2. Mentally flip it (as if pressing it onto paper).
3. Find the print that matches this flipped image.
4. Draw a line connecting the block to its matching print.

Answer: Students should draw lines matching each block to its correct print based on the reflection relationship. (Since the actual images are not visible in the OCR, students must refer to their textbook and match accordingly, keeping in mind that the print is the mirror image of the block.)

Given: Design A has $\frac{1}{4}$ turn rotational symmetry. We need to determine the rotational symmetry of Design B.

Concept:

• A design made by using a block 4 times (rotating 90° each time) → $\frac{1}{4}$ turn symmetry.
• A design made by using a block 2 times (rotating 180°) → $\frac{1}{2}$ turn symmetry.
• The number of times the block is used determines the order of rotational symmetry.

Working: Looking at Design B (made from a wooden block used in a specific arrangement):

• If the block is used 2 times (placed at 0° and 180°) → the design looks the same after every $\frac{1}{2}$ turn.
• If the block is used 4 times → $\frac{1}{4}$ turn symmetry.

Answer:

The design B looks the same after every $\boxed{\dfrac{1}{2}}$ turn. This design has rotational symmetry.

(Note: The exact answer depends on the image of Design B. If Design B uses the block twice in opposite orientations, it has $\frac{1}{2}$ turn symmetry. Students should verify with the textbook image.)

Given: Various shapes are shown on the border of the page (images not fully visible in OCR).

Concept:

• Reflection symmetry: A shape has reflection symmetry if a line can be drawn through it such that both halves are mirror images.
• Rotational symmetry: A shape has rotational symmetry if it looks the same after rotating by some angle less than 360°.

General guidelines for students:

For each shape on the border:

Step 1 — Check reflection symmetry:

• Try to draw a vertical, horizontal, or diagonal line through the shape.
• If one half is the mirror image of the other → put a ✓ mark.

Step 2 — Check rotational symmetry:

• Try rotating the shape by $\frac{1}{4}$, $\frac{1}{2}$, or $\frac{3}{4}$ turn.
• If it looks the same at any of these turns → put a * mark.

Common shapes and their symmetry:

Answer: Students should look at each specific shape on the border of their textbook page and mark ✓ for reflection symmetry and * for rotational symmetry based on the above guidelines.