NCERT Solutions Class 5 Mathematics Mela Chapter 10 Symmetrical Designs

Question 1. Which of the following alphabet cutouts can be made by just drawing half (½) or quarter (¼) of the letter?
Answer: The letters E, N, X, T, K, V, and O can all be made by drawing just half or a quarter of the shape. When you draw lines of symmetry on these letters, you can fold them in half or into quarters and see that each part mirrors the others perfectly. This means you could draw one half or one quarter and then mirror it to complete the full letter.
In simple words: Letters E, N, X, T, K, V, and O have special mirror lines, so you only need to draw part of them and fold the paper to get the whole letter.

Exam Tip: Look for letters where you can fold the paper and both sides match exactly - those are the ones you can make by drawing only half or a quarter.

Question 2. Which of the letters have a horizontal line of symmetry?
Answer: The letters E, X, and O have a horizontal line of symmetry. If you fold these letters along a line going from left to right across the middle, the top half and bottom half will match exactly and line up perfectly with each other.
In simple words: E, X, and O look the same when you fold them across the middle from side to side.

Exam Tip: To check for horizontal symmetry, imagine folding the letter in half sideways - if the top and bottom parts fit together exactly, it has horizontal symmetry.

Question 3. Which of the letters have a vertical line of symmetry?
Answer: The letters X, T, V, and O have a vertical line of symmetry. When you draw a line down the middle of these letters from top to bottom, the left side and right side are mirror images of each other. Fold the letter along that vertical line and both halves will match precisely.
In simple words: X, T, V, and O look the same when you fold them down the middle from top to bottom.

Exam Tip: For vertical symmetry, try folding the letter in half from top to bottom - both sides should be identical mirror images.

Question 4. Which letters have both vertical and horizontal lines of symmetry?
Answer: The letters X and O have both vertical and horizontal lines of symmetry. This means you can fold them down the middle from top to bottom and they match, and you can also fold them from left to right across the middle and they match equally well. These letters have the most symmetry because they work perfectly in both directions.
In simple words: X and O look the same whether you fold them up-and-down or left-and-right.

Exam Tip: If a letter has both types of symmetry, it means you can fold it both ways and get perfect matches in both directions.

Question 5. Does the firki look the same after 1/4, 1/2, 3/4, and a full turn?
Answer: Yes, the firki looks the same after each of these rotations - at 1/4 turn, 1/2 turn, 3/4 turn, and after a full turn. However, the green dot marked inside the firki changes position with each turn because the dot is not at the center. The rest of the design repeats itself perfectly at each of these angles, showing that the firki has rotational symmetry. Only the dot's location shifts as you rotate.
In simple words: The firki shape stays the same when you turn it, but the green dot moves to a new spot each time.

Exam Tip: When checking for rotational symmetry, remember that markers or dots may help you track the rotation, but they move - the actual shape is what matters for symmetry.

Question 6. Do the letters look the same when turned?
Answer: Some letters look the same when turned while others do not. The letter H has rotational symmetry - it looks the same after a 1/2 turn (180-degree rotation). The letter I also looks the same at 1/2 turn because it is symmetrical both ways. However, letters like the green oval shape look different when rotated - it appears sideways after a 1/2 turn. The letter X has rotational symmetry at 1/4 turn, 1/2 turn, and 3/4 turn because it looks identical from all these angles. Some shapes like the yellow letter Y do not have rotational symmetry and look different when turned at most angles.
In simple words: Some letters stay the same when you turn them around, but others look different. Letters like H and X stay the same when rotated.

Exam Tip: Try rotating each letter mentally or on paper - if it looks the same after turning, it has rotational symmetry.

Question 7. Find symmetry in the digits.
Answer: Rotational symmetry means a digit looks the same when turned 180 degrees (upside down). Vertical symmetry means a digit looks the same if you reflect it along a vertical line down the middle (mirror along the center). Horizontal symmetry means a digit looks the same if you reflect it along a horizontal line through the middle (mirror along the middle).

The digits 0, 1, and 8 have all three types of symmetry - rotational, vertical, and horizontal. The digit 3 has only horizontal symmetry but does not have rotational or vertical symmetry. The digits 2, 4, 5, 6, 7, and 9 do not have any of these symmetries.
In simple words: The digits 0, 1, and 8 are special because they look the same in every way - turn them, flip them sideways, or flip them upside down, and they stay the same. The number 3 only looks the same when flipped top-to-bottom.

Exam Tip: Test each digit by rotating it 180 degrees and by folding it vertically and horizontally - this is faster than trying to visualize it.

Question 8. Which digit(s) have reflection symmetry?
Answer: A digit has reflection symmetry if you can fold it along a line (vertical or horizontal) and both halves match exactly, making one side a mirror image of the other. The digits 0, 1, and 8 have reflection symmetry because they can be folded in this way - both their vertical and horizontal halves align perfectly after folding.
In simple words: The digits 0, 1, and 8 can be folded so both sides look exactly alike.

Exam Tip: Reflection symmetry is sometimes called "mirror symmetry" - imagine holding a mirror to the digit and seeing if the mirror image matches the original.

Question 9. Which digit(s) have rotational symmetry?
Answer: The digits 0, 1, and 8 have rotational symmetry. When you rotate these digits by 180 degrees (turn them upside down), they still look identical to their original position. This is why they appear the same after a half turn.
In simple words: The digits 0, 1, and 8 look the same when you turn them upside down.

Exam Tip: To check for rotational symmetry, imagine rotating the digit 180 degrees - if it looks the same, it has this symmetry.

Question 10. Which digit(s) have both rotational and reflection symmetries?
Answer: The digits 0, 1, and 8 have both rotational and reflection symmetries. These special digits can be folded to create mirror images (reflection symmetry) and they also look identical when rotated 180 degrees (rotational symmetry). No other digits have this combination of both types of symmetry.
In simple words: The digits 0, 1, and 8 are special because they look the same when turned around AND when folded in half.

Exam Tip: If a digit has both symmetries, you should be able to both fold it and rotate it and get the same result each time.

Question 11. Do the numbers II and IOOI have rotational symmetry?
Answer: Yes, the numbers II and IOOI have rotational symmetry. After rotating them by 180 degrees, they still read as II and IOOI. This happens because each digit (I and O) individually has rotational symmetry, and when arranged together, the entire number maintains this property. When you turn the number upside down, it appears unchanged.
In simple words: If you turn II and IOOI upside down, they still look and read the same way.

Exam Tip: For numbers made of multiple digits, check if each individual digit has rotational symmetry - if they do, the whole number likely does too.

Question 12. Do the numbers II and IOOI have reflection symmetry?
Answer: Yes, the numbers II and IOOI have reflection symmetry along a vertical axis. A number has reflection symmetry if it can be folded in half (horizontally or vertically) and both halves are mirror images of each other. When you draw a line down the middle of II and IOOI, the left half and right half mirror each other perfectly. This means both numbers read the same when reflected along a vertical line.
In simple words: If you fold II and IOOI down the middle, both sides look exactly alike.

Exam Tip: For vertical reflection symmetry in numbers, imagine a mirror placed down the center - the reflection should match the original exactly.

Question 13. Do the numbers II and IOOI have both rotational and reflection symmetries?
Answer: Yes, the numbers II and IOOI have both rotational and reflection symmetries. They maintain their appearance when rotated 180 degrees and they also remain unchanged when reflected along a vertical line. This dual symmetry makes these numbers special because they work the same in both directions - whether you turn them upside down or fold them in half vertically, they appear identical.
In simple words: The numbers II and IOOI are special because they look the same when turned around and when folded down the middle.

Exam Tip: Numbers or digits with both types of symmetry are rare - II and IOOI are good examples to remember for exams.

Question 14. Give examples of 2-, 3-, and 4-digit numbers which have rotational symmetry, reflection symmetry, or both.
Answer: Examples of 2-digit numbers with reflection symmetry are 11, 22, 44, 55, and 88 - these all read the same when folded down the middle. Examples of 3-digit numbers with reflection symmetry include 101, 121, 131, and 141 - the outer digits match when the number is folded vertically. Examples of 4-digit numbers with reflection symmetry are 1001, 2112, 4444, and 6886 - again, these mirror themselves along the vertical center.

For rotational symmetry, examples of 2-digit numbers are 69 and 96 - these look the same when turned upside down. Examples of 3-digit numbers with rotational symmetry are 101, 181, and 609 - when rotated 180 degrees, they appear unchanged. Examples of 4-digit numbers with rotational symmetry are 1001, 6009, and 8008.

Having both types of symmetry together is extremely rare, though numbers like 88 and 8008 can show both rotational and reflection symmetry because they are made of digits that individually possess both properties.
In simple words: 11, 22, 44 are numbers that look the same folded in half. 69 and 96 look the same turned upside down. Most numbers have only one type of symmetry, but 88 can have both.

Exam Tip: When finding examples, focus on numbers made from digits that are themselves symmetrical - digits like 0, 1, 2, 5, 6, 8, and 9 work best for these examples.

Question 15. Does the design have rotational symmetry?
Answer: No, this design does not have rotational symmetry. The reason is that the small circle on the right side breaks the rotational balance of the pattern. When you rotate the figure by 180 degrees (a half turn) or by 90 degrees (a quarter turn), the circle on the right moves to a new position that does not match the original design. Because of this, the shape fails to look the same after rotation.
In simple words: The extra circle on the side is not in a balanced position, so turning the design makes it look different.

Exam Tip: To test for rotational symmetry quickly, check if all the shapes are arranged equally around a center point - if they are not, there is no rotational symmetry.

Question 16. Try to change the design by adding some shape(s) so that the new design looks the same after a 1/2 turn.
Answer: To make the design look the same after a half turn, you need to balance the extra circle that was breaking the symmetry. This can be done by placing another circle on the left side of the design. By adding a circle on the left, the design becomes symmetric around the center point. Now the top-bottom triangles and the circles on both sides are balanced perfectly around the center, so a 180-degree rotation preserves the entire design. Each part on one side has an exact match on the opposite side.
In simple words: Add a circle on the left side opposite to the one on the right - now when you turn it, both sides match and the design stays the same.

Exam Tip: For rotational symmetry at half turn, every shape must have an identical twin directly opposite it through the center point.

Question 17. Now try to modify or add more shapes so that the new design looks the same after 1/4 turn.
Answer: To make the design look the same after a quarter turn (90-degree rotation), you need to add circles in all four quadrants around the center. Start with the existing circles and place identical circles in the remaining positions so that there is one circle in each quadrant - top, bottom, left, and right. Once you have identical circles positioned equally at all four cardinal points, the design will repeat itself perfectly every time you rotate it by 90 degrees. This creates a pattern that looks identical after each quarter turn.
In simple words: Place the same circles in all four corners around the center - now the design looks the same after turning it a quarter of the way around.

Exam Tip: For 90-degree rotational symmetry, the design must look identical after each quarter turn, meaning it repeats four times in a full rotation.

Question 18. Do the new designs have reflection symmetry? If yes, draw the lines of symmetry.
Answer: Yes, the new designs have reflection symmetry along both vertical and horizontal axes. The vertical line runs down the middle of the design from top to bottom, and if you fold along this line, the left and right sides match perfectly. Similarly, the horizontal line runs across the middle from left to right, and folding along this line makes the top and bottom halves align exactly. The circles placed in all four quadrants and the balanced triangle arrangement create this symmetry in both directions.
In simple words: The design looks the same when you fold it down the middle or fold it sideways - it has both types of reflection symmetry.

Exam Tip: When a design is symmetric about both vertical and horizontal axes, it means the lines of symmetry form a cross through the center.

Question 19. Does this design look the same after 1/2 turn? Does the design look the same after 1/4 turn?
Answer: Yes, the design looks the same after a half turn (180-degree rotation). In this pattern, opposite squares have the same color, which means a 180-degree rotation preserves the pattern perfectly - each colored square aligns with another square of the same color positioned directly across from it through the center.

No, the design does not look the same after a quarter turn (90-degree rotation). The reason is that adjacent squares have different colors. When you rotate the pattern by 90 degrees, the colors shift to new positions that do not match the original arrangement, causing the pattern to look different after a quarter turn.
In simple words: If you turn the square upside down (half turn), it looks the same because opposite corners have the same color. But if you turn it sideways (quarter turn), the colors do not line up because neighboring squares have different colors.

Exam Tip: Check the color arrangement - opposite colors indicate 180-degree symmetry, while alternating colors indicate a lack of 90-degree symmetry.

Question 20. Colour the square given in the adjoining figure using two colours so that the design looks the same after every 1/4 turn.
Answer: To make the design look the same after every quarter turn, you need to color the four squares using two colors in an alternating checkerboard pattern. Color the squares so that diagonally opposite squares have the same color - for example, make the top-left and bottom-right squares one color, and the top-right and bottom-left squares the other color. This creates a pattern where rotating 90 degrees produces an identical looking result each time because all four rotations show the same color arrangement.
In simple words: Color the squares like a checkerboard - opposite corners get the same color, and the other two corners get the other color. Now it looks the same when you turn it.

Exam Tip: For 90-degree rotational symmetry with a 2x2 grid, use a checkerboard color pattern with diagonally opposite squares matching.

Question 21. How many times does this shape look the same during a full turn?
Answer: A full turn is 360 degrees. If the design repeats every quarter turn (90 degrees), then the shape will look the same at 90 degrees, 180 degrees, and 270 degrees. This means the shape appears the same three times during a complete full turn, not counting the starting position.
In simple words: If you turn the shape slowly around completely, it looks the same three times before returning to where you started.

Exam Tip: To count rotational symmetry occurrences, divide 360 by the rotation angle - if it repeats every 90 degrees, then 360 divided by 90 equals 4 total appearances (including the starting position), which means 3 additional times.

Question 22. Do these designs have reflection symmetry also? Draw the line(s) of symmetry.
Answer: No, these designs do not have reflection symmetry. Even though they have rotational symmetry, they cannot be folded along any line to produce mirror images on both sides. The reason is that the colors and shapes do not arrange themselves in a way that creates matching halves when folded vertically or horizontally. Rotational and reflection symmetry are independent properties - a design can have one without the other.
In simple words: These designs look the same when turned around, but they do not look the same when folded in half.

Exam Tip: Remember that rotational and reflection symmetry are different properties - just because a design spins the same does not mean it folds the same.

Question 23. Use lines of symmetry to make paper cutouts of diya, boat, and other designs. Look along the border of the page to find the pictures.
Answer: When making paper cutouts using symmetry, you only need to draw half or a quarter of the design on folded paper, then cut it out. The fold creates the mirror image automatically when you unfold it. For example, to make a diya (lamp) cutout, fold your paper in half, draw half of the diya shape on the folded edge, and cut it out - when you unfold the paper, you get a complete, symmetrical diya. Similarly, for a boat design, fold the paper, draw half the boat, cut, and unfold to reveal a fully symmetrical boat. This method works for any design that has reflection symmetry and saves time because you only draw and cut once instead of twice. The border pictures show various symmetrical shapes you can create using this folding and cutting technique.
In simple words: Fold your paper in half, draw half a shape, cut it out, and unfold it - you get a perfect, balanced design every time.

Exam Tip: When creating symmetrical cutouts, always draw on the folded edge so that the fold line becomes the line of symmetry for your finished design.

Question 24. Cut out squares and equilateral triangles with the same side length. These are provided at the end of the book. Make different symmetrical designs by using these two shapes.
Answer: Using squares and equilateral triangles of equal side length, you can build many different symmetrical designs. Start by arranging shapes around a central point or central line. For example, place a square in the center and arrange triangles around it in a balanced way - you can point all triangles outward, create a star pattern, or alternate triangles and squares in a ring. Color them differently to show the symmetry more clearly. Some possible designs include a pinwheel pattern (triangles radiating from a center), a flower pattern (triangles forming petals around a square center), or striped patterns alternating squares and triangles. Each design you create should have either rotational symmetry, reflection symmetry, or both, depending on how you arrange the pieces. The variety comes from rotating the shapes, changing their colors, and adjusting how densely you pack them together.
In simple words: Arrange triangles and squares to make patterns that look the same when you turn them or fold them - the more balanced you make it, the nicer the symmetry.

Exam Tip: When making symmetrical designs, place your pieces equally around a center point to create rotational symmetry, or along a center line for reflection symmetry.

Question 25. Does this shape have reflection symmetry? If yes, draw its line(s) of symmetry.
Answer: Yes, this shape has reflection symmetry. When you fold the shape along a vertical line (the line that runs down the middle from top to bottom), the left side and right side match exactly, fitting together perfectly. Additionally, if you fold the shape along a horizontal line (from top to bottom, sideways), the top portion and bottom portion also align exactly, creating another perfect match. This means the shape has both vertical and horizontal lines of symmetry, allowing it to be folded in two different directions with both halves becoming mirror images of each other.
In simple words: The shape looks the same when you fold it down the middle or fold it sideways - it has both types of reflection symmetry.

Exam Tip: If a shape has both vertical and horizontal lines of symmetry, you should be able to fold it in both directions and get perfect matches each time.

Question 26. Does it have rotational symmetry? If yes, at which turn?
Answer: No, this shape does not have rotational symmetry. When you rotate it 90 degrees, 180 degrees, or 270 degrees, it does not look the same as the original position. Only after a complete full turn (360 degrees) does it return to its original appearance, but a full turn does not count as rotational symmetry because everything looks the same after a complete rotation. For a shape to have true rotational symmetry, it must match itself at some angle less than a full turn, such as at 90 degrees, 180 degrees, or 270 degrees. This particular shape fails that test.
In simple words: The shape looks different when you turn it sideways or upside down - it only matches itself after turning it all the way around.

Exam Tip: A shape only has rotational symmetry if it looks the same at a rotation angle of less than 360 degrees, such as 90°, 120°, or 180°.

Question 27. Does it have both symmetries?
Answer: No, this shape has only reflection symmetry, not rotational symmetry. Having reflection symmetry means the shape can be folded to create mirror images, which it can - along both vertical and horizontal lines. However, having rotational symmetry means the shape looks identical when rotated, which it does not do at any angle less than a full turn. These are two independent properties, and this shape demonstrates only one of them. A shape would need to satisfy both conditions to have both types of symmetry.
In simple words: The shape is perfectly balanced when folded, but it does not match itself when turned - so it has only reflection symmetry, not rotation symmetry.

Exam Tip: Always test shapes for both types of symmetry separately - a shape might have one, the other, both, or neither.

Question 28. Now, make your designs. Sort your designs in 3 categories - designs with only rotational symmetry, designs with only reflection symmetry, and designs with both rotational and reflection symmetry.
Answer: To complete this activity, create your own designs using the shapes available and then organize them by their symmetry properties. Designs with only rotational symmetry might include patterns where shapes are arranged in a pinwheel or spiral around a center, repeating at 90-degree or 120-degree intervals, but the pattern does not fold into mirror halves. Designs with only reflection symmetry would be patterns with shapes arranged symmetrically along a vertical or horizontal line, where folding produces perfect matches, but rotating them produces a different look. Designs with both types of symmetry would include patterns like regular stars or cross shapes where you can both fold them for mirror matches and rotate them for identical appearances. As you create each design, analyze it carefully to determine which category it belongs to based on whether it matches itself after folding, rotating, or both.
In simple words: Make different patterns and test each one - fold some and turn some to see which ones have folding symmetry, spinning symmetry, or both.

Exam Tip: For any design you create, always test both rotation and reflection before deciding which category it belongs to.

Question 29. Below are images of wooden blocks and a part of their prints. Match each block to its correct print by drawing a line.
Answer: To match each wooden block to its print, examine the carved pattern on each block and find the print that shows what that block would stamp when pressed onto paper. The block labeled (i) with its intricate leaf and swirl pattern matches with the print showing that exact design. Block (ii) with the lotus-shaped carving corresponds to the print displaying the lotus pattern. Block (iii) with the flower design matches the circular flower print. Block (iv) with the semicircular fan design aligns with the fan-shaped print. Block (v) with the vertical leaf carving matches the rectangular striped leaf print. Each block's three-dimensional carved design creates a unique two-dimensional print when applied to paper, so careful observation of both the depth patterns and surface details is needed to make correct matches.
In simple words: Look at each block's carved design and find the matching stamp pattern it would make on paper.

Exam Tip: Pay attention to small details in the carved patterns - unique marks and shape combinations help you match each block to the right print.

Question 30. Observe the pattern made by the wooden block below. We get the final print by using the block 4 times. What type of symmetry does the final print have?
Answer: The design B (the final print created by stamping the block four times) has both rotational and reflection symmetry. The design looks the same after every quarter turn (90 degrees), every half turn (180 degrees), and every three-quarter turn (270 degrees), indicating strong rotational symmetry. Additionally, the design contains vertical and horizontal lines of symmetry - you can fold it down the middle or across the middle and both halves match perfectly. This dual symmetry makes the pattern highly balanced and aesthetically pleasing, showing how repeated stamps of a single block, when positioned symmetrically, can create a design with multiple types of symmetry.
In simple words: The final print looks the same when you turn it around at any angle and also looks the same when you fold it - it has both types of symmetry.

Exam Tip: When a design is made by repeating the same stamp four times in a balanced way, look for both rotational symmetry at 90 degrees and reflection symmetry along multiple axes.

Question 31. Observe the shapes given on the border. Which of the shapes have reflection symmetry? Put a (✓) mark on them. Put a * on the shapes that have rotational symmetry.
Answer: Going through each shape: The pink triangle has reflection symmetry (check mark) and also has rotational symmetry (star). The green star has reflection symmetry and rotational symmetry. The yellow hexagon has reflection symmetry and rotational symmetry. The three circles with lines have reflection symmetry and rotational symmetry. The mixed-color triangle has both types of symmetry. The purple triangle has both types of symmetry. The pink rectangle has reflection symmetry only. The gray diamond has reflection symmetry and rotational symmetry. The light green hexagon has reflection symmetry and rotational symmetry. The light blue square has reflection symmetry and rotational symmetry. The blue cross or plus shape has reflection symmetry and rotational symmetry. The light brown triangle has reflection symmetry only. The composite stars shape has rotational symmetry. The parallelogram has reflection symmetry and rotational symmetry. The pink and brown triangular shapes have reflection symmetry and rotational symmetry respectively.
In simple words: Regular shapes like stars, hexagons, and squares usually have both types of symmetry. Irregular shapes may have only one or neither.

Exam Tip: Regular polygons (equal sides and angles) almost always have both rotational and reflection symmetry, while irregular shapes are less likely to have either.