Figure it Out
Q1. Do you see any line of symmetry in the figures at the start of the chapter? What about in the picture of the cloud?
Answer:
The figures at the start of the chapter include a butterfly, flower, pinwheel, and rangoli. The butterfly and flower have a vertical line of symmetry. The rangoli has multiple lines of symmetry (e.g., vertical, horizontal, and diagonal). The pinwheel does not have a line of symmetry but has rotational symmetry. The picture of the cloud has no line of symmetry because it is irregular and has no repetitive pattern.
Q2. For each of the following figures, identify the line(s) of symmetry if it exists. (Figures not provided in text, but general solution given)
Answer:
For any given figure, a line of symmetry is a line that divides the figure into two identical mirror halves. To identify it, look for a line along which the figure can be folded so that both sides match exactly. For example, a square has 4 lines of symmetry (vertical, horizontal, and two diagonals). A rectangle (not square) has 2 lines (vertical and horizontal). A regular hexagon has 6 lines. An irregular shape may have none.
Q1. In each of the following figures, a hole was punched in a folded square sheet of paper and then the paper was unfolded. Identify the line along which the paper was folded.
Answer:
For figure (a): The fold line is vertical through the middle. For figure (b): The fold line is horizontal through the middle. For figure (c): The fold line is diagonal. For figure (d): The paper was folded along two lines (vertical and horizontal) or one diagonal, but since only one hole is punched, the single fold line is the line of symmetry that places the hole at the same distance from it on the unfolded sheet. In figure (d), the hole appears four times, indicating two folds (vertical and horizontal) or a diagonal fold. However, since it is created by a single hole, the paper was folded twice (e.g., first vertical fold, then horizontal fold) before punching.
Q2. Given the line(s) of symmetry, find the other hole(s): (a) (b) (c) (d) (e)
Answer:
For (a): The line of symmetry is vertical. The given hole on the left should have a corresponding hole on the right at the same distance from the line. For (b): Two lines of symmetry (vertical and horizontal). The given hole in one quadrant should have holes in the other three quadrants. For (c): Horizontal line of symmetry. Given hole above should have a corresponding hole below at same distance. For (d): Diagonal line of symmetry. Given hole on one side of diagonal should have hole on the other side. For (e): Multiple lines of symmetry. The holes should be repeated in all symmetric positions.
Q3. Here are some questions on paper cutting. Consider a vertical fold. We represent it this way: Vertical Fold. Similarly, a horizontal fold is represented as follows: Horizontal Fold.
Answer:
This is an introduction to the representation of folds. No specific question to solve here, but understanding that a vertical fold creates symmetry about a vertical line, and a horizontal fold about a horizontal line.
Q4. After each of the following cuts, predict the shape of the hole when the paper is opened. After you have made your prediction, make the cutouts and verify your answer. (a) (b) (c) (d)
Answer:
(a) If the paper is folded in half vertically and a cut is made along a curve starting from the fold line, the opened shape will be symmetric about the vertical fold line, forming a shape like a heart or leaf. (b) If folded horizontally and cut along a straight line from the fold, the opened shape will be symmetric horizontally. (c) If folded twice (vertical and horizontal) and a cut is made at a corner, the opened shape will have four identical symmetric parts. (d) If folded diagonally and cut, the opened shape will have symmetry along the diagonal. Verify by actual paper cutting.
Q5. Suppose you have to get each of these shapes with some folds and a single straight cut. How will you do it? a. The hole in the centre is a square. b. The hole in the centre is a square. Note: For the above two questions, check if the 4-sided figures in the centre satisfy both the properties of a square.
Answer:
a. To get a square hole in the centre: Fold the paper in half vertically, then horizontally, then along both diagonals (or just vertical and horizontal). Make a single straight cut along a line perpendicular to the folds at the centre. When unfolded, the cut will form a square hole. b. Similarly, for a different orientation, fold the paper along its diagonals and make a straight cut perpendicular to the folds at the centre. The square hole properties: all sides equal and all angles 90°.
Q6. How many lines of symmetry do these shapes have? a. b. A triangle with equal sides and equal angles. c. A hexagon with equal sides and equal angles.
Answer:
a. The shape is a regular pentagon (5 equal sides). It has 5 lines of symmetry (each from a vertex to the midpoint of the opposite side). b. A triangle with equal sides and equal angles is an equilateral triangle. It has 3 lines of symmetry (each from a vertex to the midpoint of the opposite side). c. A hexagon with equal sides and equal angles is a regular hexagon. It has 6 lines of symmetry (3 through opposite vertices and 3 through midpoints of opposite sides).
Q7. Trace each figure and draw the lines of symmetry, if any: (Figures not provided in text, but general solution given)
Answer:
For each figure, trace it on paper. Fold the traced figure along potential lines to check if both halves match. Alternatively, use a ruler to draw possible lines. For a circle, infinite lines through the centre. For a rectangle, 2 lines (vertical and horizontal). For a regular polygon, number of lines equals number of sides. For an irregular shape, there may be zero or one line of symmetry.
Q8. Find the lines of symmetry for the kolam below. (Figure not provided but typical kolam has multiple lines of symmetry)
Answer:
A typical kolam (rangoli) pattern often has 4 lines of symmetry: vertical, horizontal, and two diagonals. Look for repeating patterns and fold along these lines to check if both sides match exactly.
Q9. Draw the following. a. A triangle with exactly one line of symmetry. b. A triangle with exactly three lines of symmetry. c. A triangle with no line of symmetry. Is it possible to draw a triangle with exactly two lines of symmetry?
Answer:
a. An isosceles triangle (two equal sides) has exactly one line of symmetry (the altitude from the vertex where the equal sides meet to the base). b. An equilateral triangle has three lines of symmetry (all three altitudes). c. A scalene triangle (all sides different) has no line of symmetry. It is not possible to draw a triangle with exactly two lines of symmetry because if it has two, it must also have a third (proved by symmetry principles).
Q10. Draw the following. In each case, the figure should contain at least one curved boundary. a. A figure with exactly one line of symmetry. b. A figure with exactly two lines of symmetry. c. A figure with exactly four lines of symmetry.
Answer:
a. Draw a heart shape (two semicircles on top meeting at a point at the bottom). It has one vertical line of symmetry. b. Draw a shape like a rectangle with semicircular ends (like a sports field). It has two lines of symmetry: vertical and horizontal. c. Draw a circle with four identical curved lobes (like a four-leaf clover). It has four lines of symmetry: vertical, horizontal, and two diagonals.
Q11. Copy the following on squared paper. Complete them so that the blue line is a line of symmetry. Problem (a) has been done for you. (a) (b) (c) (d) (e) (f) Hint: For (c) and (f), see if rotating the book helps!
Answer:
For each figure, reflect the given part across the blue line (axis of symmetry). Use a ruler or mirror to ensure each point is the same distance from the line on the opposite side. For (c) and (f), rotating the book helps to see the reflection clearly. Complete the figure to make it symmetric.
Q12. Copy the following drawing on squared paper. Complete each one of them so that the resulting figure has the two blue lines as lines of symmetry. (a) (b) (c) (d) (e) (f)
Answer:
For each figure, the two blue lines are perpendicular (vertical and horizontal). Reflect the given part across both lines. First reflect across one line, then across the other to fill all four quadrants. Ensure each point is equidistant from both axes.
Q13. Copy the following on a dot grid. For each figure draw two more lines to make a shape that has a line of symmetry.
Answer:
For each given figure, identify a possible line of symmetry. Then draw additional lines (e.g., arcs, straight lines) that mirror the existing figure across that line. Ensure the new lines create a symmetric shape.
Q1. Find the angles of symmetry for the given figures about the point marked •. (a) (b) (c)
Answer:
(a) If the figure has 2 radial arms (like a 180° rotation symmetry), the angles of symmetry are 180° and 360°. (b) If the figure has 3 radial arms equally spaced, the angles are 120°, 240°, and 360°. (c) If the figure has 4 radial arms equally spaced, the angles are 90°, 180°, 270°, and 360°. The centre of rotation is the point marked •.
Q2. Which of the following figures have more than one angle of symmetry? (Figures not provided in text, but general solution given)
Answer:
Figures that have rotational symmetry of order greater than 1 (i.e., return to original shape after less than 360° rotation). Examples: a square (angles 90°, 180°, 270°, 360°); a rectangle that is not square (angles 180°, 360°); a regular hexagon (60°, 120°, 180°, 240°, 300°, 360°). Figures with only one angle of symmetry (360°) have no rotational symmetry.
Q3. Give the order of rotational symmetry for each figure: (Figures not provided, but general solution given)
Answer:
Order of rotational symmetry is the number of times the figure fits onto itself in a full 360° rotation. For a regular polygon with n sides, order = n. For a rectangle (not square), order = 2. For a windmill with 4 blades, order = 4. For an irregular shape with no rotational symmetry, order = 1.
Q1 (page 238). Colour the sectors of the circle below so that the figure has i) 3 angles of symmetry, ii) 4 angles of symmetry, iii) what are the possible numbers of angles of symmetry you can obtain by colouring the sectors in different ways?
Answer:
(i) For 3 angles of symmetry, colour every 4th sector with the same pattern (since 12÷3=4). (ii) For 4 angles of symmetry, colour every 3rd sector with the same pattern (since 12÷4=3). (iii) By colouring the 12 sectors in different ways, the possible numbers of angles of symmetry are the divisors of 12: 1, 2, 3, 4, 6, and 12.
Q2. Draw two figures other than a circle and a square that have both reflection symmetry and rotational symmetry.
Answer:
Draw an equilateral triangle (3 lines of symmetry, rotational symmetry order 3) and a regular hexagon (6 lines of symmetry, rotational symmetry order 6).
Q3a. Draw, wherever possible, a rough sketch of: a. A triangle with at least two lines of symmetry and at least two angles of symmetry.
Answer:
Draw an equilateral triangle. It has 3 lines of symmetry and 3 angles of symmetry (120°, 240°, 360°).
Q3b. b. A triangle with only one line of symmetry but not having rotational symmetry.
Answer:
Draw an isosceles triangle that is not equilateral. It has exactly 1 line of symmetry (from vertex to base midpoint) and only 1 angle of symmetry (360°), so it does not have rotational symmetry (except 360°).
Q3c. c. A quadrilateral with rotational symmetry but no reflection symmetry.
Answer:
Draw a parallelogram that is not a rectangle or rhombus. It has rotational symmetry of order 2 (180° and 360°) but no line of symmetry.
Q3d. d. A quadrilateral with reflection symmetry but not having rotational symmetry.
Answer:
Draw an isosceles trapezoid. It has one line of symmetry (through the midpoints of the parallel sides) but only one angle of symmetry (360°), so it does not have rotational symmetry (other than 360°).
Q4. In a figure, 60° is the smallest angle of symmetry. What are the other angles of symmetry of this figure?
Answer:
The other angles of symmetry are multiples of 60° up to 360°: 120°, 180°, 240°, 300°, and 360°.
Q5. In a figure, 60° is an angle of symmetry. The figure has two angles of symmetry less than 60°. What is its smallest angle of symmetry?
Answer:
Since 60° is an angle of symmetry and there are two smaller angles, the angles must be divisors of 60°. The two smaller angles less than 60° could be 30° and 20° (but 360÷20=18, so angles: 20°, 40°, 60°... works). So the smallest angle of symmetry is 20°.
Q6a. Can we have a figure with rotational symmetry whose smallest angle of symmetry is: a. 45°?
Answer:
Yes, because 360° is a multiple of 45° (360÷45=8). So the other angles would be 90°, 135°, 180°, etc.
Q6b. b. 17°?
Answer:
No, because 360° is not a multiple of 17° (360÷17≈21.176, not a whole number). So a figure cannot have 17° as the smallest angle of symmetry.
Q7a. This is a picture of the new Parliament Building in Delhi. a. Does the outer boundary of the picture have reflection symmetry? If so, draw the lines of symmetries. How many are they?
Answer:
Yes, the outer boundary of the Parliament Building has reflection symmetry. There are 2 lines of symmetry: one vertical line through the centre, and one horizontal line through the centre.
Q7b. b. Does it have rotational symmetry around its centre? If so, find the angles of rotational symmetry.
Answer:
Yes, it has rotational symmetry. The angles of rotational symmetry are 180° and 360° (order 2).
Q8. How many lines of symmetry do the shapes in the first shape sequence in Chapter 1, Table 3, the Regular Polygons, have? What number sequence do you get?
Answer:
Regular polygons: triangle (3 sides) has 3 lines, square (4 sides) has 4 lines, pentagon (5 sides) has 5 lines, etc. The number sequence is: 3, 4, 5, 6, 7, ... (each number equals the number of sides).
Q9. How many angles of symmetry do the shapes in the first shape sequence in Chapter 1, Table 3, the Regular Polygons, have? What number sequence do you get?
Answer:
Regular polygons have angles of symmetry equal to the number of sides. For example, triangle has 3 angles (120°, 240°, 360°), square has 4 (90°, 180°, 270°, 360°), etc. The number sequence is: 3, 4, 5, 6, 7, ... (same as the number of sides).
Q10. How many lines of symmetry do the shapes in the last shape sequence in Chapter 1, Table 3, the Koch Snowflake sequence, have? How many angles of symmetry?
Answer:
The Koch Snowflake has 6 lines of symmetry and 6 angles of symmetry (60°, 120°, 180°, 240°, 300°, 360°).
Q11. How many lines of symmetry and angles of symmetry does Ashoka Chakra have?
Answer:
The Ashoka Chakra has 24 spokes. It has 24 lines of symmetry (each line through opposite spokes) and 24 angles of symmetry (each angle = 360° ÷ 24 = 15°, so angles: 15°, 30°, 45°, ..., 360°).
Revise Reflect Refine
Q1. Can you draw a figure with radial arms that has a) exactly 5 angles of symmetry, b) 6 angles of symmetry? Also find the angles of symmetry in each case.
Answer:
a) For 5 angles of symmetry, use 5 radial arms with angle between adjacent arms = 360° ÷ 5 = 72°. The angles of symmetry are: 72°, 144°, 216°, 288°, 360°. b) For 6 angles of symmetry, use 6 radial arms with angle between adjacent arms = 360° ÷ 6 = 60°. The angles of symmetry are: 60°, 120°, 180°, 240°, 300°, 360°.
Q2. Consider a figure with radial arms having exactly 7 angles of symmetry. What will be its smallest angle of symmetry? Is the number of degrees a whole number in this case? If not, express it as a mixed fraction.
Answer:
Smallest angle = 360° ÷ 7 = 51 3/7°. This is not a whole number, so it is 51 3/7 degrees.