Question 1

If one half of a figure fits exactly over the other half then it is called a ______

Accepted Answer

Symmetric figure — [{"id": "2b26a805-d37b-4cfa-992a-18336fe51883", "type": "html", "value": " Figure in balanced proportion is called a symmetric figure. "}]

Question 2

While playing a game, Rohan asked Rhea to identify all the symmetric alphabets among the following: A, B, C, D, E, F, G. Which option Rhea should pick?

Accepted Answer

A, C, D, E — [{"id": "bac99edc-b51d-4927-820d-3cbb9158b5f7", "type": "html", "value": " "}]

Question 3

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

Accepted Answer

a. Angles of symmetry = 90°, 180°, 270°, 360°
b. Angle of symmetry = 360°
c. Angles of symmetry = 180°, 360° — The angles of symmetry are found by determining the rotations about the marked point that map the figure onto itself. For example, a square has angles of symmetry at every 90°, so 90°, 180°, 270°, and 360°. A figure with no rotational symmetry except the full rotation has only 360°. A figure symmetric after half rotation has 180° and 360°.

Question 4

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

Accepted Answer

The figures that have more than one angle of symmetry are those which can be rotated by more than one angle less than 360° to map onto themselves. The textbook shows such figures with multiple rotational symmetries. — A figure has more than one angle of symmetry if it can be rotated by multiple angles less than 360° and still look the same. For example, regular polygons like squares, hexagons have multiple angles of symmetry.

Question 5

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

Accepted Answer

Orders of rotational symmetry:
- Figure 1: order = 2
- Figure 2: order = 4
- Figure 3: order = 6
- Figure 4: order = 3
- Figure 5: order = 4
- Figure 6: order = 5 — The order of rotational symmetry is the number of times a figure maps onto itself during a full 360° rotation. For example, a figure with order 4 symmetry looks the same every 90° rotation.

Question 6

True or False
- Every figure will have 360 degrees as an angle of symmetry.
- If the smallest angle of symmetry of a figure is a natural number in degrees, then it is a factor of 360.

Accepted Answer

Both statements are True.
- Every figure has 360° as an angle of symmetry because a full rotation maps the figure onto itself.
- If the smallest angle of symmetry is a natural number, it divides 360° exactly, as angles of symmetry are multiples of the smallest angle. — 360° rotation always maps a figure onto itself. The smallest angle of symmetry divides 360° evenly, so it must be a factor of 360 if it is a natural number.

Question 7

Q1. Color 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 coloring the sectors in different ways?

Accepted Answer

(i) Color the circle in 3 equal sectors with the same color pattern to get 3 angles of symmetry.
(ii) Color the circle in 4 equal sectors with the same color pattern to get 4 angles of symmetry.
(iii) By coloring the sectors in different ways, possible numbers of angles of symmetry include 3, 4, and 12. — The number of angles of symmetry corresponds to the number of equal sectors with repeating color patterns. For example, 3 equal colored sectors give 3 angles of symmetry, 4 equal colored sectors give 4 angles, and the full 12 sectors colored symmetrically can give 12 angles.

Question 8

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

Accepted Answer

Two figures drawn have:
- Number of line symmetry = 4
- Order of rotational symmetry = 4
These figures are shown in the textbook with 4 lines of symmetry and order 4 rotational symmetry. — Figures with both reflection and rotational symmetry have multiple lines of symmetry and rotational symmetry orders greater than 1. Examples include certain regular polygons like a regular rhombus or kite with symmetrical properties.

Question 9

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 only one line of symmetry but not having rotational symmetry.
c. A quadrilateral with rotational symmetry but no reflection symmetry.
d. A quadrilateral with reflection symmetry but not having rotational symmetry.

Accepted Answer

a. Equilateral triangle with 3 lines of symmetry and 3 angles of symmetry.
b. Isosceles triangle with 1 line of symmetry and no rotational symmetry.
c. Quadrilateral with no line symmetry but rotational symmetry of order 2 (angles 180°, 360°).
d. Quadrilateral with reflection symmetry but no rotational symmetry (e.g., kite shape). — a. Equilateral triangle is symmetric about all medians.
b. Isosceles triangle has one axis of symmetry but does not map onto itself on rotation except 360°.
c. Some quadrilaterals can have rotational symmetry without reflection symmetry.
d. Some quadrilaterals have reflection symmetry but no rotational symmetry.

Question 10

Q4. In a figure, 60° is the smallest angle of symmetry. What are the other angles of symmetry of the figure?

Accepted Answer

Other angles of symmetry = 120°, 180°, 240°, 300°, 360°. — Angles of symmetry are multiples of the smallest angle. Multiplying 60° by 2, 3, 4, 5, and 6 gives the other angles.

Question 11

Q5. In the figure, 60° is an angle of symmetry. The figure has two angles of symmetry less than 60°. What is its smallest angle of symmetry?

Accepted Answer

The smallest angle of symmetry = 20°. — Since 60° is an angle of symmetry and there are two smaller angles of symmetry less than 60°, dividing 60° by 3 gives 20°, which is the smallest angle of symmetry.

Question 12

Q6. Can we have a figure with rotational symmetry whose smallest angle of symmetry is
a. 45°?
b. 17°?

Accepted Answer

a. Yes, because 360° is a multiple of 45°.
b. No, because 360° is not a multiple of 17°. — The smallest angle of symmetry must divide 360° exactly for rotational symmetry. 45 divides 360 evenly (360/45=8), but 17 does not.

Question 13

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 of symmetries. How many are they?
b. Does it have rotational symmetry around its centre? If so, find the angles of rotational symmetry.

Accepted Answer

a. Yes, the outer boundary has 3 lines of symmetry.
b. Yes, it has rotational symmetry with angles 120°, 240°, and 360°. — The building's outer boundary is symmetric about 3 axes (lines of symmetry). Rotating by 120° multiples maps the figure onto itself, giving rotational symmetry.

Question 14

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?

Accepted Answer

Number of lines of symmetry for regular polygons:
Triangle: 3
Quadrilateral: 4
Pentagon: 5
Hexagon: 6
Heptagon: 7
Octagon: 8
Nonagon: 9
Decagon: 10
This forms a counting number sequence. — Regular polygons have lines of symmetry equal to the number of sides. The sequence is 3,4,5,... which is a counting sequence.

Question 15

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?

Accepted Answer

Number of lines of symmetry: 3, 6, 6, 6, 6
Number of angles of symmetry: 3, 6, 6, 6, 6 — The Koch Snowflake sequence shapes have increasing lines and angles of symmetry as the figure evolves, stabilizing at 6.

Symmetry

Symmetry — Study Notes

Introduction to Symmetry

Line Symmetry

Symmetry in Letters and Numbers

Practice Questions — Symmetry

All 10 Chapters in Ganita Prakash