Samacheer Kalvi Class 6 Maths Solutions Term 3 Chapter 4 Symmetry Exercise 4.2

Miscellaneous Practice Problems

 

Question 1. Draw and answer the following.
(i) A triangle which has no line of symmetry.
(ii) A triangle which has only one line of symmetry
(iii) A triangle which has three lines of symmetry.
Answer:
(i) **Scalene triangle:** This type of triangle has all sides of different lengths and all angles of different sizes. Because of this, it cannot be folded along any line to make both halves match perfectly. It has no line of symmetry.

(ii) **Isosceles triangle:** This triangle has two sides that are exactly the same length. The angles opposite these equal sides are also equal. This means you can fold it down the middle, from the top corner to the base, and both sides will be a perfect mirror image. It has exactly one line of symmetry.

(iii) **Equilateral triangle:** All three sides of this triangle are equal in length, and all three angles are equal (each is 60 degrees). Because of its perfect balance, it has three lines of symmetry. You can fold it from each corner to the middle of the opposite side, and it will always match perfectly. It also has rotational symmetry of order 3.

In simple words: Triangles can have different numbers of symmetry lines. A scalene triangle has no symmetry, an isosceles has one, and an equilateral triangle has three.

🎯 Exam Tip: Remember the basic properties of each triangle type. Drawing small sketches helps visualize their lines of symmetry.

 

Question 2. Find the alphabets in the box which have

(i) No line of symmetry
(ii) Rotational symmetry
(iii) Reflection symmetry
(iv) Reflection and rotational symmetry.
Answer:
(i) The alphabets that have no line of symmetry are P, N, S, Z. These letters cannot be folded in half to create a mirror image.
(ii) The alphabets that have Rotational symmetry are I, O, N, X, S, H, Z. This means they look the same after being turned around a central point, by less than a full circle.
(iii) The alphabets that have reflection symmetry are A, M, E, D, I, K, O, X, H, U, V, W. These letters can be split by a line (vertical, horizontal, or both) where one side mirrors the other.
(iv) The alphabets that have both reflection and rotational symmetry are I, O, X, H. These letters have both properties: they can be reflected and they look the same after being rotated by a specific angle.
In simple words: Some letters can be folded to match (reflection symmetry), some look the same when turned (rotational symmetry), some have both, and some have neither.

🎯 Exam Tip: When checking for symmetry, imagine folding the letter or spinning it. Always double-check each letter against both types of symmetry.

 

Question 3. For the following pictures, find the number of lines of symmetry and also find the order of rotation.
Answer:
(i) For the L-shaped figure (a tetromimo): 0 lines of symmetry, order of rotation 2. This means it looks the same if you turn it by 180 degrees.
(ii) For the isosceles triangle: 1 line of symmetry, order of rotation 1. It only looks the same when rotated a full 360 degrees.
(iii) For the 1x5 rectangular strip: 2 lines of symmetry, order of rotation 2. It has both horizontal and vertical mirror lines and looks the same when turned 180 degrees.
(iv) For the regular octagon: 8 lines of symmetry, order of rotation 8. It has many lines of symmetry and looks identical after small turns.
(v) For the irregular 5-square shape (like a T-shape but with a wider top): 1 line of symmetry, order of rotation 1. It has a vertical mirror line but only looks the same after a full 360-degree rotation.
In simple words: Look at each shape to see how many ways you can fold it to get a mirror image (lines of symmetry) and how many times it looks the same if you spin it around before it makes a full circle (order of rotation).

🎯 Exam Tip: The order of rotational symmetry is always at least 1 (for a 360-degree rotation). For regular polygons, the number of sides equals both the lines of symmetry and the order of rotational symmetry.

 

Question 4. The three-digit number 101 has rotational and reflection symmetry. Give five more examples of three-digit numbers that have both rotational and reflection symmetry
Answer: The digits 0, 1, and 8 are special because they have both rotational and reflection symmetry themselves. When we combine these digits to form numbers, they can also show these symmetries. This happens because the shape of these digits remains symmetric when reflected or rotated. Therefore, the three-digit numbers 181, 111, 808, 818, 888 have both rotational and reflection symmetry.
In simple words: Numbers made only from the digits 0, 1, and 8 will look the same when turned around or seen in a mirror. Examples are 181, 111, 808, 818, and 888.

🎯 Exam Tip: To find numbers with both symmetries, focus on digits that are individually symmetric (like 0, 1, 8) and arrange them symmetrically.

 

Question 5. Translate the given pattern and complete the design in a rectangular strips?
Answer: To complete the design with translation symmetry, the original pattern must be repeated exactly as it is, moving it along the strip without turning or flipping it. This means the arrangement of fish-like shapes and diamond shapes will repeat side-by-side to fill the rectangular strip. Below is how the pattern would be completed, with the shapes repeating horizontally across the strip.

In simple words: Translation symmetry means sliding a pattern along a line to repeat it exactly. The strips are filled by simply copying and pasting the original design many times next to each other.

🎯 Exam Tip: For translation symmetry, ensure the pattern is copied without any rotation, reflection, or change in size.

Challenge Problems

 

Question 6. Shade one square so that it possesses

(i) One line of symmetry
(ii) Rotational symmetry of order 2
Answer:
(i) To have one line of symmetry, we can shade a single square. If you shade a corner square, a diagonal line passing through it and the opposite corner of the 2x2 grid will be a line of symmetry.

(ii) To have rotational symmetry of order 2, we need to shade two squares that are diagonally opposite each other. When this shape is rotated 180 degrees, it will look exactly the same.

In simple words: For one line of symmetry, shade a single corner square. For rotational symmetry of order 2, shade two squares opposite each other.

🎯 Exam Tip: For small grids, test shading patterns by drawing lines of symmetry or mentally rotating to see if the shape looks the same.

 

Question 7. Join six identical squares so that atleast one side of a square fits exactly with any other side of the square and has reflection symmetry (any three ways).
Answer: To create a shape with six squares that has reflection symmetry, we need to arrange them so that at least one line can divide the shape into two perfect mirror halves. A common way to achieve this is to form a rectangle or a symmetric 'L' or 'T' shape. A 3x2 rectangle is a simple shape that uses six squares and has two lines of symmetry.
(i) A 3x2 rectangle (two lines of symmetry):

(ii) Another way to form a 3x2 rectangle, which is identical to the first. This also has two lines of symmetry.

(iii) Yet another arrangement forming a 3x2 rectangle, providing two lines of symmetry.

In simple words: To make a shape from six squares with reflection symmetry, you can arrange them into a 3x2 rectangle. This shape can be folded in half in two different ways to make perfect mirror images.

🎯 Exam Tip: When joining squares for symmetry, try to create shapes like rectangles, crosses, or T-shapes where you can easily spot a folding line that makes both sides match.

 

Question 8. Draw the following
(i) A figure which has reflection symmetry but no rotational symmetry.
(ii) A figure which has rotational symmetry but no reflection symmetry.
(iii) A figure which has both reflection and rotational symmetry.
Answer:
(i) A figure with reflection symmetry but no rotational symmetry (order greater than 1) can be an isosceles triangle or an arrowhead shape. These shapes have a line you can fold them along, but they don't look the same if you turn them less than a full circle.

(ii) A figure with rotational symmetry but no reflection symmetry is like a pinwheel or the letter 'S' in some fonts. The figure shown is a skewed star or a specific propeller shape that looks the same after rotation (e.g., 90 or 180 degrees) but cannot be folded to get a mirror image.

(iii) A figure with both reflection and rotational symmetry is very common, such as a regular star, a square, or a regular hexagon. The figure shown is a regular 6-pointed star, which has many lines of symmetry and looks the same when rotated by equal angles.

In simple words: Figures can have different kinds of symmetry. Some can be folded (reflection), some look the same when turned (rotation), and some have both.

🎯 Exam Tip: Draw clear diagrams. For rotational symmetry, try drawing a small arrow on the figure to track its rotation. For reflection symmetry, imagine folding along a line.

 

Question 9. Find the line of symmetry and the order of rotational symmetry' of the given regular polygons and complete the following table and answer the questions given below.

(i) A regular polygon of 10 sides will have lines of symmetry.
(ii) If a regular polygon has 10 lines of symmetry then its order of rotational symmetry is
(iii) A regular polygon of 'n' sides has lines of symmetry and the order of rotational symmetry is
Answer:
(i) A regular polygon with 10 sides will have **10** lines of symmetry. This is a property of all regular polygons: the number of sides equals the number of lines of symmetry.
(ii) If a regular polygon has 10 lines of symmetry, then its order of rotational symmetry is **10**. For regular polygons, the order of rotational symmetry is also equal to the number of sides (and lines of symmetry).
(iii) A regular polygon of 'n' sides has **n** lines of symmetry and the order of rotational symmetry is **n**. This is the general rule that applies to all regular polygons.
In simple words: For any shape that is a regular polygon (like a square or a regular hexagon), the number of sides is the same as how many lines of symmetry it has, and also the same as its rotational symmetry order. So, if it has 'n' sides, it has 'n' lines of symmetry and an 'n' order of rotation.

🎯 Exam Tip: Remember the golden rule for regular polygons: Number of sides = Number of lines of symmetry = Order of rotational symmetry.

 

Question 10. Colour the boxes in such a way that it possesses translation symmetry.
(i)
(ii)
Answer: Translation symmetry means the pattern repeats exactly when shifted in a certain direction. To achieve this, we copy a basic repeating unit and place it side-by-side to fill the area. The colored boxes below show patterns that exhibit translation symmetry.
(i) For the square grid, a simple checkerboard pattern shows translation symmetry, where a 2x2 block can be repeated.

(ii) For the triangular grid, an alternating pattern of colored and uncolored triangles can show translation symmetry. Here, a block of two triangles (one up, one down, or vice versa) can be repeated.

In simple words: To make a pattern with translation symmetry, just take a small part of the design and copy it exactly again and again, moving it across the page. This creates a repeating pattern.

🎯 Exam Tip: Identify the smallest repeating unit (the "motif") in a pattern. Once you have this, you can just slide and repeat it to show translation symmetry.