Question 1

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?

Accepted Answer

Analysis of Figures:    Flower: The image of the flower has six petals arranged symmetrically around the center. It has 6 lines of symmetry. Each line passes through the center and the tip of a petal, and also through the center and the midpoint between two petals.    Rangoli: The rangoli design is...

Question 2

Q2: For each of the following figures, identify the line(s) of symmetry if it exists.

Accepted Answer

Analysis of Each Figure: 1.  First Figure (Keyhole shape): This figure has one vertical line of symmetry passing through the center of the circle and the middle of the rectangular part. 2.  Second Figure (Vase shape): This figure has one vertical line of symmetry that divides it into two identical l...

Question 3

Q1: Find the angles of symmetry for the given figures about the point marked •.

Accepted Answer

Angles of Symmetry:    a. The figure has rotational symmetry of order 4. The angles of symmetry are multiples of $\frac{360^\circ}{4} = 90^\circ$. The angles are $90^\circ, 180^\circ, 270^\circ, 360^\circ$.    b. The figure is a vertical line segment. Rotating it by $180^\circ$ about its center maps...

Question 4

Q2: Which of the following figures have more than one angle of symmetry?

Accepted Answer

A figure has more than one angle of symmetry if it has rotational symmetry (i.e., an angle of symmetry between $0^\circ$ and $360^\circ$).    Equilateral triangle: Yes, order 3. (Angles: $120^\circ, 240^\circ, 360^\circ$)    Circle with a sector removed: No, only $360^\circ$.    Square: Yes, order 4...

Question 5

Q3: Give the order of rotational symmetry for each figure:

Accepted Answer

The order of rotational symmetry is the number of times a figure maps onto itself during a full $360^\circ$ rotation.    (a) Rhombus: Order 2.    (b) Square: Order 4.    (c) Equilateral Triangle: Order 3.    (d) Cross shape: Order 4.    (e) Regular Hexagon: Order 6.    (f) Five-pointed Star in a Pen...

Question 6

Q1: 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...

Accepted Answer

Given: A circle divided into 12 equal sectors.    i) 3 angles of symmetry: To get an order of 3, the pattern must repeat every $\frac{12}{3} = 4$ sectors. For example, color sectors 1, 5, 9 in red and the rest in white.    ii) 4 angles of symmetry: To get an order of 4, the pattern must repeat every...

Question 7

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 symme...

Accepted Answer

Analysis of Koch Snowflake Sequence:    Stage 0 (Initial shape): An equilateral triangle. It has 3 lines of symmetry and 3 angles of symmetry.    Stage 1: This shape is a regular hexagram (Star of David). It has 6 lines of symmetry and 6 angles of symmetry. *   Subsequent Stages: Each subsequent ite...

Question 8

Q11: How many lines of symmetry and angles of symmetry does Ashoka Chakra have?

Accepted Answer

Analysis of Ashoka Chakra: The Ashoka Chakra is a wheel with 24 spokes, all equally spaced.    Lines of Symmetry: It has 24 lines of symmetry. 12 lines pass through the centers of opposite spokes, and 12 lines pass through the midpoints of the gaps between opposite spokes.    Angles of Symmetry: It...

Question 9

Q2: Draw two figures other than a circle and a square that have both reflection symmetry and rotational symmetry.

Accepted Answer

Examples: 1.  A Regular Hexagon: It has 6 lines of reflection symmetry and rotational symmetry of order 6. 2.  An Equilateral Triangle: It has 3 lines of reflection symmetry and rotational symmetry of order 3.

Question 10

Q3: Draw, wherever possible, a rough sketch of: a. A triangle with at least two lines of symmetry and at least two angles of symmetry. b. A triangle with...

Accepted Answer

Sketches:    a. An equilateral triangle. It has 3 lines of symmetry and 3 angles of symmetry ($120^\circ, 240^\circ, 360^\circ$).    b. An isosceles triangle. It has one line of symmetry. Its only angle of symmetry is $360^\circ$, so it does not have rotational symmetry (which requires an angle stri...

Question 11

Q4: In a figure, $60^{\circ}$ is the smallest angle of symmetry. What are the other angles of symmetry of the figure?

Accepted Answer

Given: The smallest angle of symmetry is $60^\circ$. Solution: All other angles of symmetry will be multiples of the smallest angle, up to $360^\circ$. The angles are: $2 	imes 60^\circ = 120^\circ$ $3 	imes 60^\circ = 180^\circ$ $4 	imes 60^\circ = 240^\circ$ $5 	imes 60^\circ = 300^\circ$ $6 \...

Question 12

Q5: In a figure, $60^{\circ}$ is an angle of symmetry. The figure has two angles of symmetry less than $60^{\circ}$. What is its smallest angle of symmetr...

Accepted Answer

Given: $60^\circ$ is an angle of symmetry. There are two angles of symmetry smaller than $60^\circ$. Solution: Let the smallest angle of symmetry be $\alpha$. Then all other angles of symmetry must be multiples of $\alpha$. Since $60^\circ$ is an angle of symmetry, $\alpha$ must be a factor of 60. T...

Question 13

Q6: Can we have a figure with rotational symmetry whose smallest angle of symmetry is: a. $45^{\circ}$ ? b. $17^{\circ}$ ?

Accepted Answer

Condition: For a figure to have rotational symmetry with the smallest angle $\alpha$, $\alpha$ must be a factor of $360^\circ$.    a. $45^\circ$:     We check if 45 is a factor of 360.      $360 \div 45 = 8$.      Since the result is an integer, yes, it is possible. An example is a regular octagon....

Question 14

Q7: 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 o...

Accepted Answer

Analysis: The outer boundary of the new Parliament Building is shaped like an equilateral triangle.    a. Reflection Symmetry: Yes, an equilateral triangle has reflection symmetry. It has 3 lines of symmetry. Each line of symmetry is an altitude of the triangle, passing from a vertex to the midpoint...

Question 15

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 ge...

Accepted Answer

Rule: A regular polygon with $n$ sides has $n$ lines of symmetry. Sequence:    Equilateral Triangle ($n=3$): 3 lines of symmetry    Square ($n=4$): 4 lines of symmetry    Regular Pentagon ($n=5$): 5 lines of symmetry    Regular Hexagon ($n=6$): 6 lines of symmetry *   ... and so on. Final Answer: Th...

Question 16

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 g...

Accepted Answer

Rule: A regular polygon with $n$ sides has rotational symmetry of order $n$, which means it has $n$ angles of symmetry. Sequence:    Equilateral Triangle ($n=3$): 3 angles of symmetry    Square ($n=4$): 4 angles of symmetry    Regular Pentagon ($n=5$): 5 angles of symmetry    Regular Hexagon ($n=6$)...

Question 17

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...

Accepted Answer

Analysis of Folds:    Figure (a): The two holes are arranged vertically. This pattern is created by folding the paper along a horizontal line of symmetry and punching one hole. When unfolded, the hole is mirrored on the other half.    Figure (b): The two holes are arranged horizontally. This is crea...

Question 18

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 w...

Accepted Answer

Sketches of Figures with Curved Boundaries:    a. A figure with exactly one line of symmetry: A semicircle or a heart shape.    b. A figure with exactly two lines of symmetry: An ellipse or the shape formed by the intersection of two equal circles. *   c. A figure with exactly four lines of symmetry...

Question 19

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.

Accepted Answer

Description of Completed Figures:    (b): The given shape is in the top-left quadrant defined by a diagonal line. The completed figure will be a shape reflected across this diagonal into the bottom-right quadrant, forming a symmetric figure along the diagonal.    (c): The given L-shape is on one sid...

Question 20

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.

Accepted Answer

For each figure, the given drawing is in one of the four quadrants created by the vertical and horizontal lines of symmetry. To complete the figure, the drawing must be reflected into the other three quadrants. 1.  Reflect the given shape across the vertical line of symmetry. 2.  Reflect the given s...

NCERT Solutions