Key Concept: Line of Symmetry, Reflection

c) The line x = 4
[Solution Description]
To find the line of symmetry:
1. Plot the points A(2, 3), B(4, 1), C(6, 3), and D(4, 5).
2. Observe that points A and C are symmetric with respect to the line x = 4, and points B and D are also symmetric with respect to x = 4.
3. Folding the quadrilateral along the line x = 4 will make the two halves overlap exactly.
4. Therefore, x = 4 is the line of symmetry.

Key Concept: Basic Concept of Line of Symmetry

d) A line that divides a shape into two mirror-image halves
[Solution Description]
A line of symmetry is a line that divides a figure into two identical parts such that one part is a mirror image of the other, as per the syllabus definition. This means folding the figure along this line will make both halves overlap exactly.

Key Concept: Line of Symmetry

d) Assertion is false, but Reason is true.
[Solution Description]
The Assertion states that a square has exactly two lines of symmetry: one vertical and one horizontal. However, a square actually has four lines of symmetry: two diagonals in addition to the vertical and horizontal lines. Therefore, the Assertion is false.
The Reason states that a line of symmetry divides a figure into two identical halves that overlap when folded along the line. This is a correct definition of a line of symmetry.
Thus, the Assertion is false, but the Reason is true.

Key Concept: Rotational Symmetry

b) $60^\circ$
[Solution Description]
A regular hexagon has rotational symmetry every $60^\circ$ ($360^\circ / 6$ sides). Thus, the smallest angle is $60^\circ$. Other options represent angles for different shapes (e.g., $90^\circ$ for squares, $120^\circ$ for equilateral triangles).

Key Concept: Symmetry in Nature

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that the Rangoli design has multiple lines of symmetry. This is true because a Rangoli design often consists of repeated patterns around a central point, allowing it to be folded along multiple axes to produce identical halves. The Reason explains that a figure with reflection symmetry can indeed have more than one line of symmetry if the patterns repeat when folded along those lines, which is also true and correctly explains why the Rangoli design has multiple lines of symmetry. Therefore, both the Assertion and Reason are true, and the Reason is the correct explanation of the Assertion.

Key Concept: Generating Symmetrical Figures, Ink Blot Method

c) Vertical line of symmetry with identical halves[Solution Description]When paper is folded vertically and ink is spilled on one side, pressing ensures the ink transfers symmetrically to the other side upon unfolding. The result is a figure with a vertical line of symmetry where both sides are exact mirror images of each other.

Key Concept: Line of Symmetry, Multiple Lines of Symmetry

c) 6
[Solution Description]
A regular hexagon has six equal sides and angles. The lines of symmetry pass through opposite vertices and the midpoints of opposite sides. Thus, there are three lines through vertices and three through midpoints, totaling six.

Key Concept: Line of Symmetry, Rotational Symmetry

b) Rectangle (non-square)
[Solution Description]
A figure with exactly two lines of symmetry must be symmetrical along both lines but should not coincide with itself when rotated. An example is a rectangle (non-square), which has two lines of symmetry (vertical and horizontal) but no rotational symmetry unless it's a square.

Key Concept: Identifying Lines of Symmetry

c) 5
[Solution Description]
A regular polygon has as many lines of symmetry as its number of sides.
- A regular pentagon has five sides, so it will have five lines of symmetry.
Each line of symmetry passes through a vertex and the midpoint of the opposite side.
Hence, the correct answer is 5.

Key Concept: Identifying Lines of Symmetry

b) 2
[Solution Description]
A rectangle (not a square) has two lines of symmetry. These lines are the vertical and horizontal lines that pass through the midpoints of the opposite sides. Unlike a square, a non-square rectangle does not have diagonal lines of symmetry because folding along the diagonals does not result in overlapping parts.

Key Concept: Decorative Patterns, Multiple Folds

c) 4 holes
[Solution Description]
Folding the paper vertically divides it into 2 layers, and folding it horizontally adds another layer, totaling 4 layers. Punching a hole through all layers at the center creates 4 holes symmetrically placed. Upon unfolding, these holes appear at positions mirrored across both the vertical and horizontal lines of symmetry.

Key Concept: Mirror Halves

d) The two halves are neither mirror images nor do they overlap
[Solution Description]
Folding a non-square rectangle along its diagonal does not result in mirror halves because the sides are of unequal lengths. The two parts will not overlap completely when folded along the diagonal. Hence, the diagonal of a non-square rectangle is not a line of symmetry.

Key Concept: Line of Symmetry

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Both the assertion and reason are true. An isosceles triangle has only one line of symmetry that passes through its vertex angle and bisects the base, creating two identical mirror-image halves. Hence, the reason correctly explains why the assertion is true.

Key Concept: Line of Symmetry

b) Isosceles triangle
[Solution Description]
A shape with one line of symmetry can be folded along that line to match both halves exactly. An isosceles triangle has only one line of symmetry, which is the altitude from the vertex angle to the base. Other options like equilateral triangle or square have more than one line of symmetry, and a rectangle without equal adjacent sides has none.

Key Concept: Line of Symmetry

b) One half covers the other half completely
[Solution Description]
A line of symmetry divides a figure into two identical parts. When folded along this line, one part overlaps exactly with the other part. This means both sides are mirror images of each other.

Key Concept: Lines of Symmetry, Real-World Application

c) The parts do not overlap.
[Solution Description]
1. First, understand the kolam design has vertical and horizontal symmetry.
2. Folding along the vertical line means the left and right parts overlap perfectly.
3. Unfolding and folding along the diagonal (top-left to bottom-right) will only overlap if the design also has diagonal symmetry.
4. Since the problem does not mention diagonal symmetry, the parts will not overlap perfectly.
Hence, the correct answer is "The parts do not overlap."

Key Concept: Lines of Symmetry in a Square

c) 4
[Solution Description]
A square has multiple lines of symmetry. These include:
1. Vertical line of symmetry: This divides the square into two equal halves from top to bottom.
2. Horizontal line of symmetry: This divides the square into two equal halves from left to right.
3. Two diagonal lines of symmetry: Each diagonal from one corner to the opposite corner divides the square into two identical parts.
Therefore, the total number of lines of symmetry in a square is 4.

Key Concept: Line of Symmetry, Multiple Lines of Symmetry

d) Assertion is false, but Reason is true.
[Solution Description]
First, recall that an equilateral triangle has all three sides equal and all three angles equal to $60^\circ$. For a figure to have a line of symmetry, it must be possible to fold the figure along a line such that the two halves overlap exactly.
Now, consider the assertion: "An equilateral triangle can have exactly two lines of symmetry." This is false because an equilateral triangle actually has three lines of symmetry, each passing through a vertex and the midpoint of the opposite side.
Next, consider the reason: "A figure with all sides and angles equal must have at least three lines of symmetry." This is true as seen in the case of an equilateral triangle, which has three lines of symmetry. However, the reason does not correctly explain the assertion since the assertion itself is false.
Therefore, the correct answer is that the Assertion is false, but the Reason is true.

Key Concept: Identifying Asymmetrical Figures

c) It has no line of symmetry.
[Solution Description]
According to the syllabus, some figures (like clouds) have no line of symmetry. Since the shape is irregular and does not fold into identical halves along any axis, it has zero lines of symmetry.

Key Concept: Line of Symmetry, No Line of Symmetry

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a scalene triangle has no line of symmetry. This is true because a scalene triangle has all sides and all angles unequal, meaning there is no line that can divide the triangle into two identical mirror halves.
The reason given is that in a scalene triangle, all sides are of unequal lengths and all angles are of unequal measures. This is also true and directly explains why a scalene triangle has no line of symmetry. Since no sides or angles are equal, no line can act as a line of symmetry.
Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Properties of Symmetry

b) It has no line of symmetry.
[Solution Description]
A triangle with all sides unequal (scalene) has no equal angles or sides, so it cannot be divided into mirror images. Thus, it has no line of symmetry.
- Option a): Incorrect, as only equilateral/isosceles triangles have symmetry.
- Option b): Correct, since the sides are unequal.
- Option c): Incorrect, parallelograms can have symmetry, but this triangle does not.
- Option d): Incorrect, squares are symmetrical, but unrelated here.

Key Concept: Diagonal Symmetry

d) Square
[Solution Description]
A square has four lines of symmetry: two diagonals, one vertical, and one horizontal. In contrast, a rectangle (that is not a square) only has two lines of symmetry (vertical and horizontal), and its diagonals are not lines of symmetry. An equilateral triangle has three lines of symmetry, but none of them are diagonals. A parallelogram has no lines of symmetry unless it is a rhombus or a rectangle. Therefore, the shape with at least one diagonal line of symmetry is a square.

Key Concept: Symmetry, Lines of Symmetry in a Rectangle

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion and reason, let's analyze a rectangle:
1. **Assertion Check**: A rectangle has exactly two lines of symmetry—vertical and horizontal—passing through the midpoints of opposite sides. Folding along these lines makes the two halves overlap perfectly. Thus, the Assertion is true.
2. **Reason Check**: When a rectangle (that is not a square) is folded along its diagonal, the two parts do not overlap entirely. This is because the sides adjacent to the diagonal are unequal in length (unlike in a square). Hence, the Reason is also true.
3. **Explanation Link**: The Reason correctly explains why a rectangle lacks diagonal symmetry, supporting the Assertion’s claim about its limited lines of symmetry.

Key Concept: Vertical, horizontal, diagonal symmetry, reflection properties

b) (5,2)
[Solution Description]
First, find the vertical line of symmetry for the rectangle ABCD. Since it's a rectangle with vertices at (1,2), (5,2), (5,6), and (1,6), the vertical line of symmetry is exactly halfway between x=1 and x=5.
Calculation: (1+5)/2 = 3. So the vertical line of symmetry is x=3.
Point A is at (1,2). To reflect it over x=3:
Distance from A to line: 3-1 = 2 units to the right
So the reflected point will be 2 units to the right of x=3: 3+2=5
The y-coordinate remains the same during vertical reflection, so final coordinates are (5,2).

Key Concept: Infinite symmetry, Koch Snowflake

a) 3
[Solution Description]
The Koch Snowflake is constructed iteratively starting from an equilateral triangle. At each iteration, smaller equilateral triangles are added to the middle third of each edge.
The initial triangle has 3 lines of symmetry. After the first iteration, the snowflake still retains these 3 lines of symmetry. This pattern continues for all iterations since the additions respect the original symmetry.
Therefore, even at the third iteration, the Koch Snowflake maintains 3 lines of symmetry.

Key Concept: Reflection Symmetry

c) Assertion is true, but Reason is false.
[Solution Description]
A rectangle that is not a square has reflection symmetry along its vertical and horizontal mid-lines, giving it exactly two lines of symmetry. However, the diagonals of such a rectangle do not serve as lines of symmetry because folding the rectangle along a diagonal does not result in overlapping halves. Therefore, the Assertion is true, but the Reason is false because the diagonals are irrelevant to the number of lines of symmetry in a non-square rectangle.

Key Concept: Reflection symmetry, Composition of reflections

c) Rotation by $180^\circ$
[Solution Description]
First, reflect the rectangle over its vertical line of symmetry. This swaps the left and right sides. Next, reflect the result over the horizontal line of symmetry, swapping the top and bottom sides.
The combination of these two reflections is equivalent to rotating the rectangle by $180^\circ$ about its center. This is because each reflection reverses orientation, but two reflections restore it while rotating the figure.
Therefore, the net transformation is a rotation of $180^\circ$.

Key Concept: Line of Symmetry

c) A figure may have multiple lines of symmetry
[Solution Description]
Reflection symmetry occurs when one half of a figure is the mirror image of the other half across a line of symmetry. The figure can be folded along this line to overlap completely.

Key Concept: Reflection symmetry, multiple lines of symmetry

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that a regular pentagon has five lines of reflection symmetry, all passing through a single central point. This is true because a regular pentagon is a five-sided polygon with equal sides and angles, and it possesses five lines of symmetry—each from a vertex to the midpoint of the opposite side, intersecting at the center.
The Reason states that in a regular pentagon, each line of symmetry divides the figure into two identical parts that are mirror images of each other. This is also true as reflection symmetry means one half is the mirror image of the other when folded along the line of symmetry.
Moreover, the Reason correctly explains the Assertion because the presence of multiple lines of symmetry in a regular pentagon inherently implies that each line divides the figure into two congruent mirror-image halves.

Key Concept: Reflection symmetry, rotational symmetry

c) Square
[Solution Description]
A figure with four lines of symmetry and rotational symmetry of order 4 must be a square. A square has four lines of symmetry (two diagonals and two midlines) and can be rotated by 90 degrees (360/4) to coincide with itself. Therefore, the correct answer is a square.

Key Concept: Reflection symmetry in a square

a) Occupies the original position of A
[Solution Description]
The vertical line of symmetry of the square passes through the midpoints of sides AD and BC. When reflecting point B across this line:
1. Identify the original position of B on the right side.
2. Its reflection will be directly opposite to it, at the same distance from the line of symmetry but on the left side.
3. From the given initial positions (A top-left, D bottom-left, B top-right, C bottom-right), the reflection of B swaps with A's position.
Therefore, after reflection, B will occupy the original position of A.

Key Concept: Diagonal reflection in a square

b) Occupies the original position of B
[Solution Description]
Reflecting across diagonal AC involves swapping points that are symmetric about this line:
1. Points A and C lie on the diagonal, so they remain unchanged.
2. Points B and D are symmetric relative to the diagonal; thus, D moves to B's original position.
Hence, after reflection, D occupies the original position of B.

Key Concept: Reflection symmetry, multiple lines of symmetry

c) (1, 4)
[Solution Description]
First, identify the vertical line of symmetry for the rectangle. The vertical line of symmetry for a rectangle with vertices A(1, 2), B(3, 2), C(3, 4), and D(1, 4) is the line x = 2 (midpoint between x = 1 and x = 3). To reflect point C(3, 4) over the line x = 2:
1. Calculate the horizontal distance from point C to the line of symmetry: $3 - 2 = 1$ unit.
2. Move the same distance on the opposite side of the line of symmetry: $2 - 1 = 1$.
3. The y-coordinate remains the same because it's a vertical reflection.
4. Thus, the reflected point is (1, 4).

Key Concept: Reflection along horizontal line of symmetry

d) Point D
[Solution Description]
The horizontal line of symmetry divides the square into top and bottom halves. Reflecting point C across this line swaps it with point D because they are symmetric about the horizontal midline. Hence, point C moves to the original position of point D.

Key Concept: Reflection along diagonal

c) Original position of D
[Solution Description]
Reflecting the square along the diagonal AC swaps positions of B and D. Point B is originally adjacent to A and C, while D is adjacent to A and C on the opposite side. Reflection over AC interchanges their positions, moving B to where D was initially located.

Key Concept: Reflection along vertical line of symmetry

a) Point A
[Solution Description]
The vertical line of symmetry passes through the center of the square. When reflecting point B across the vertical line, it will swap positions with point A because reflection along this line interchanges points on the right side (B, C) with those on the left side (A, D). Therefore, point B moves to where point A was originally.

Key Concept: Reflection symmetry, Vertical and Horizontal lines of symmetry

c) A and C remain in their original positions
[Solution Description]
Reflection along the diagonal AC swaps points B and D, while points A and C remain unchanged because they lie on the line of symmetry. Thus:
- Point A remains at its original position.
- Point C remains at its original position.

Key Concept: Reflection symmetry, Multiple axes

c) At S's position
[Solution Description]
Step 1: Reflecting P over the vertical line takes it directly opposite horizontally. Since P is at top-left, reflection over the vertical line moves it to top-right (Q's initial position).
Step 2: Reflecting this new position (top-right) over the horizontal line moves it to bottom-right (S's initial position). Thus, P lands at S's position.

Key Concept: Reflection Symmetry

b) Top-right corner
[Solution Description]
The vertical line of symmetry divides the square into two equal halves. Reflecting point A (top-left) over this line means it will swap positions with point B (top-right). So, point A moves to the position initially occupied by point B.

Key Concept: Symmetry in Paper Folding and Cutting

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that folding and cutting a sheet of paper will always result in a shape with a line of symmetry. The reason explains that the fold acts as the line of symmetry because the two halves overlap exactly.
When you fold a paper along a line and make a cut, the unfolded shape will indeed have the fold line as its line of symmetry. This is because the cut is mirrored on both sides of the fold, ensuring exact overlap. Hence, both the assertion and the reason are true, and the reason correctly explains the assertion.

Key Concept: Paper Folding and Cutting

c) Two identical triangles mirrored along the vertical fold line
[Solution Description]
When the paper is folded vertically, it creates a line of symmetry. A triangular cut made along the folded edge means the same shape will be mirrored on the other side when unfolded. The result is two identical triangles symmetrically placed about the vertical fold line.

Key Concept: Symmetry in Paper Folding

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
When a sheet of paper is folded in half, the crease created by the fold acts as a line of symmetry. If a cut is made along this folded edge, the resulting shape, when unfolded, will mirror itself perfectly across the fold line. This is because the cut affects both layers of the folded paper equally, creating two identical halves. Thus, the assertion is true. The reason correctly explains why the unfolded paper has a line of symmetry: the fold ensures that the cuts are mirrored on both sides.

Key Concept: Ink Blot Devils

b) 1
[Solution Description]
The paper is folded in half, creating one line of symmetry. After pressing, the ink spreads symmetrically across the fold. Unfolding produces a figure with exactly one line of symmetry, corresponding to the initial fold.

Key Concept: Punching Game, Symmetric Patterns

d) 7
[Solution Description]
Vertical fold makes left-right symmetry, horizontal fold makes top-bottom symmetry. Hole in top-left gets mirrored to top-right (vertical fold) and these both get mirrored to bottom-left and bottom-right (horizontal fold) - total 4 holes. Hole on vertical fold at center-top gets mirrored only downward by horizontal fold - total 2 holes. Hole at exact center doesn't get mirrored as it's on both fold lines - stays 1 hole. Total holes: 4 + 2 + 1 = 7.

Key Concept: Ink Blot Devils, Symmetry

d) Assertion is false, but Reason is true.
[Solution Description]
The assertion states that the resulting figure will always have exactly one line of symmetry along the fold when ink is dropped on one side of a folded paper and pressed. The reason explains that the folding process mirrors the ink pattern perfectly, creating symmetry.
Step 1: Understand the process described. Folding the paper in half and pressing creates a mirror image of the ink on the other side, ensuring symmetry along the fold line.
Step 2: However, the assertion claims "exactly one line of symmetry." This may not always be true, as some ink patterns might create additional lines of symmetry (e.g., if the ink forms a circular or square pattern).
Step 3: Therefore, while the reason correctly explains why there is at least one line of symmetry, the assertion is too absolute and can be false in certain cases.

Key Concept: Paper Folding and Symmetry

c) Four holes, symmetric about both vertical and horizontal axes
[Solution Description]
When the paper is folded vertically, it creates a line of symmetry along the vertical axis. Folding it horizontally adds another line of symmetry along the horizontal axis. Punching a hole at the center of this folded paper means the hole will appear symmetrically in all four quadrants when unfolded. Thus, the unfolded paper will have four holes, one in each quadrant, symmetric about both the vertical and horizontal axes.

Key Concept: Paper Folding and Cutting

b) A diamond shape
[Solution Description]
Folding the paper horizontally creates a horizontal line of symmetry. A triangular cut along this folded edge will produce two identical triangles mirrored across the line of symmetry when unfolded. The result will be a diamond shape formed by the two adjacent triangles.

Key Concept: Paper Cutting and Symmetry

b) Two cuts, symmetric about the line of symmetry
[Solution Description]
Folding the triangle along its line of symmetry means the two halves are mirror images. A cut made on one side will be mirrored on the other side when unfolded. Therefore, the unfolded shape will have two identical cuts, creating a symmetrical pattern with two matching cut-out sections.

Key Concept: Multiple Folds, Symmetrical Patterns

c) 8 holes, symmetrically placed around the center
[Solution Description]
Folding along both diagonals divides the paper into eight triangular sections with symmetry. When a hole is punched 1 cm from the center towards one corner, due to the double diagonal folds, the hole will be replicated in all eight symmetric sections.
The exact positions can be calculated by reflecting the original hole position across both diagonals. If the original hole is at $(x, y)$ relative to the center, the other holes will be at $(-x, y)$, $(x, -y)$, $(-x, -y)$, $(y, x)$, $(-y, x)$, $(y, -x)$, and $(-y, -x)$. Thus, there will be eight holes in total.

Key Concept: Punching Game

c) Four holes at each corner
[Solution Description]
Folding the paper horizontally creates a horizontal line of symmetry. Punching a hole near the top-left corner will mirror to the bottom-left corner, and punching a hole near the bottom-right corner will mirror to the top-right corner. Thus, there will be four holes in total when unfolded.

Key Concept: Predicting Cutout Shapes

b) A full circle with diameter on the fold line
[Solution Description]
Folding the paper vertically creates a line of symmetry. Cutting a semi-circle along the folded edge means the unfolded paper will have a full circle because the semi-circle is mirrored across the line of symmetry. The diameter of the circle lies along the original fold line.

Key Concept: Creating Intricate Patterns, Representation of Folds

b) On both the diagonal and original fold lines' intersection
[Solution Description]
The vertical and horizontal folds divide the square into 4 equal smaller squares. The intersection point of these folds is the center of the square. Punching a hole here means it will appear at the center regardless of how the paper is folded further. Folding diagonally would align this center hole along the diagonal fold line.

Key Concept: Symmetry Analysis, Multiple Folds

a) Four holes: two on the left side (top \& bottom) and two on the right side (top \& bottom).
[Solution Description]
The paper is first folded vertically, then horizontally, creating four identical quadrants. Punching a hole through all four layers means each fold creates a symmetric hole in the mirrored positions.
Step-by-step:
1. Original position of the hole: 2 cm from left, 3 cm from bottom.
2. After vertical fold: Mirrored hole appears 2 cm from the right edge (since the width of the square is symmetric).
3. After horizontal fold: Two more mirrored holes appear—3 cm from the top edge (mirroring the bottom edge).
Thus, when fully unfolded, there will be four holes at:
- (2 cm left, 3 cm bottom)
- (2 cm right, 3 cm bottom)
- (2 cm left, 3 cm top)
- (2 cm right, 3 cm top)

Key Concept: Symmetry in Punching Game

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
Folding a square sheet along its diagonal divides it into two congruent triangles that overlap exactly when folded. A hole punched at the midpoint of the diagonal will be mirrored on both sides upon unfolding because the diagonal is indeed a line of symmetry for the square. Hence, the assertion is true, and the reason correctly explains why the unfolded paper shows symmetric holes relative to the diagonal.

Key Concept: Identifying Lines of Symmetry

a) One line of symmetry
[Solution Description]
Folding the rectangular paper diagonally creates a line of symmetry along the diagonal. When a hole is punched near this folded edge, it will create a mirror image of the hole across the diagonal upon unfolding. Since a rectangle only has two lines of symmetry (vertical and horizontal), but folding diagonally introduces an additional symmetry line along the diagonal. However, for rectangles that are not squares, the diagonal fold does not produce a symmetrical figure when unfolded unless it is a square. Hence, the unfolded paper will have only one line of symmetry along the diagonal if it's not a square.

Key Concept: Predicting Cutout Shapes

b) A cross shape with four arms
[Solution Description]
Folding the circular paper vertically reduces it to a semicircle. Folding it horizontally then reduces it to a quarter-circle. Making a cut along the folded edges means removing part of the quarter-circle. When unfolded, the cuts will be mirrored across both the vertical and horizontal axes, resulting in a symmetric shape with four identical cutouts. The exact shape depends on the cut, but it will be symmetrical with respect to both axes.

Key Concept: Lines of Symmetry, Paper Folding

b) 2 lines of symmetry
[Solution Description]
When you fold a rectangle along one diagonal, you create 1 line of symmetry. Folding along the other diagonal adds another line of symmetry perpendicular to the first. These two diagonals intersect at the center. For a rectangle:
1. First diagonal fold creates symmetry along that diagonal
2. Second diagonal fold creates symmetry along the other diagonal
Together they form an 'X' shape with 2 lines of symmetry, not 4 like a square, because rectangles don't have vertical/horizontal symmetry unless it's a square.

Key Concept: Symmetry, Rotational Symmetry

b) 3 distinct positions
[Solution Description]
The shape has rotational symmetry every $90°$, meaning it takes $360°/90° = 4$ rotations to complete a full turn. For $315°$ rotation:
1. First rotation: $315° - 90° = 225°$ remaining
2. Second rotation: $225° - 90° = 135°$ remaining
3. Third rotation: $135° - 90° = 45°$ remaining
After these 3 full $90°$ rotations plus partial $45°$, the shape would have completed $315°$. However, since we count only complete symmetric positions (multiples of $90°$), there are 3 distinct positions passed through.

Key Concept: Symmetry, Drawing symmetric figures from partial outlines

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The Assertion states that a figure with two blue lines of symmetry must have identical halves reflected across both lines. This is true because symmetry requires mirror images across the given lines. The Reason explains the fundamental property of symmetry: corresponding points are equidistant from the line of symmetry. The Reason correctly explains the Assertion since identical reflection relies on equidistant corresponding points.

Key Concept: Symmetry, Ink Blot Patterns

b) Symmetric only about the vertical axis
[Solution Description]
Folding the paper vertically introduces a vertical line of symmetry. When ink is applied on one side and pressed, the ink transfers symmetrically to the other side. Upon unfolding, the pattern will be symmetric about the original vertical fold line. No other lines of symmetry are guaranteed unless additional folds or specific patterns are introduced.

Key Concept: Symmetry, Completing shapes with two lines of symmetry

c) A full circle
[Solution Description]
Since the shape must be symmetric about both the x-axis and y-axis (the two blue lines), reflecting the quarter-circle over both axes will result in a full circle centered at the origin.
Step 1: Reflect the quarter-circle over the vertical line of symmetry to get a semicircle.
Step 2: Reflect the semicircle over the horizontal line of symmetry to complete the full circle.
Thus, the final shape is a full circle.

Key Concept: Completing Symmetrical Shapes

a) The same shape on both sides of the fold
[Solution Description]
When you fold a piece of paper in half and cut a shape from the folded edge, the resulting shape will be symmetrical along the fold line. The cut-out portion will be mirrored on both sides of the fold, creating a symmetrical figure with the fold line as its line of symmetry.

Key Concept: Rotational Symmetry

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
For a regular pentagon, which has 5 sides, the smallest angle of rotational symmetry is calculated as $\frac{360^{\circ}}{5} = 72^{\circ}$. Additional angles of symmetry are multiples of this angle: $144^{\circ}$ ($72^{\circ} \times 2$), $216^{\circ}$ ($72^{\circ} \times 3$), $288^{\circ}$ ($72^{\circ} \times 4$), and $360^{\circ}$ ($72^{\circ} \times 5$). Therefore, the Assertion is correct. The Reason correctly states that the smallest angle of rotational symmetry for any regular polygon is given by $\frac{360^{\circ}}{n}$, where $n$ is the number of sides. Moreover, the Reason directly explains why the Assertion is true, as it provides the mathematical basis for determining the angles of symmetry in a regular pentagon.

Key Concept: Symmetries of a Circle

c) Every angle is an angle of symmetry.
[Solution Description]
A circle coincides with itself when rotated by any angle about its center. Therefore, every angle is an angle of symmetry for a circle.

Key Concept: Rotational Symmetry, Angle of Rotation

c) $60^\circ$, $120^\circ$, $180^\circ$, $240^\circ$, $300^\circ$, $360^\circ$
[Solution Description]
Since the smallest angle of rotation is $60^\circ$, other angles must be multiples of $60^\circ$ that divide $360^\circ$. The possible angles are:
Step 1: Find all factors of $360^\circ$ that are multiples of $60^\circ$.
Step 2: This gives $60^\circ$, $120^\circ$, $180^\circ$, $240^\circ$, $300^\circ$, and $360^\circ$.
Therefore, all possible angles are $60^\circ$, $120^\circ$, $180^\circ$, $240^\circ$, $300^\circ$, and $360^\circ$.

Key Concept: Rotational Symmetry

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a square has rotational symmetry of order 4, which means it looks identical in four different positions when rotated. This is true because rotating a square by $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$ brings it back to a position indistinguishable from the original. The reason correctly explains why the assertion is true: these specific angles of rotation cause the square to coincide with itself. Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Angle of Symmetry for Figures

c) Rectangle
[Solution Description]
A rectangle has rotational symmetry where the smallest angle that maps it onto itself is $180^\circ$. This is because rotating it by $180^\circ$ makes it look identical to its original position. The circle is an exception but does not have a smallest angle, and the equilateral triangle has $120^\circ$ as its smallest angle.

Key Concept: Rotational Symmetry, Smallest Angle of Symmetry

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To verify the assertion and reason, we analyze the rotational symmetry of a regular pentagon. A regular pentagon has 5 angles of symmetry since rotating it by $72^\circ$, $144^\circ$, $216^\circ$, $288^\circ$, and $360^\circ$ maps it onto itself. The smallest angle of symmetry here is indeed $72^\circ$, which supports the assertion.
Now, considering the reason: For any figure with rotational symmetry where the smallest angle of symmetry is a natural number, it must divide $360^\circ$ to ensure that multiple rotations bring the figure back to its original position. Here, $72^\circ$ is a factor of $360^\circ$ since $360 \div 72 = 5$. Thus, the reason is true and correctly explains the assertion.

Key Concept: Rotational Symmetry

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a square has rotational symmetry at angles of $90°$, $180°$, $270°$, and $360°$. This is true because a square looks exactly the same when rotated by these angles about its center. The reason given is also correct: for a figure to have rotational symmetry, the smallest angle of rotation must indeed be a factor of $360$ (e.g., $90°$ is a factor of $360°$). Moreover, the reason correctly explains why the assertion is valid, as the angles specified in the assertion are all multiples of the smallest angle ($90°$) which is a factor of $360°$.

Key Concept: Number of Radial Arms

b) 5
[Solution Description]
The smallest angle of symmetry is $72^\circ$. The number of radial arms is determined by dividing $360^\circ$ by the smallest angle:
$360^\circ / 72^\circ = 5.$
Hence, the figure has 5 radial arms.

Key Concept: Rotational Symmetry: Windmill

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a paper windmill has rotational symmetry because it looks the same when rotated by 90 degrees about its center. This is true as per the syllabus, which mentions that the windmill looks identical at these specific angles. The reason provided lists all the angles of symmetry for the windmill (90°, 180°, 270°, and 360°), which is also correct. Furthermore, the reason correctly explains the assertion by providing the specific angles at which the windmill maintains its appearance, confirming the rotational symmetry.

Key Concept: Rotational Symmetry

c) $90^{\circ}$
[Solution Description]
The windmill looks exactly the same when rotated by $90^{\circ}$, $180^{\circ}$, $270^{\circ}$, and $360^{\circ}$. The smallest angle from these options is $90^{\circ}$.

Key Concept: Rotational Symmetry, Square

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
A square indeed has four angles of rotational symmetry: $90^\circ$ (quarter turn), $180^\circ$ (half turn), $270^\circ$ (three-quarter turn), and $360^\circ$ (full turn). This is because rotating the square by these angles will align it back to its original position. For example:
- At $90^\circ$, point A moves to B, B to C, C to D, and D back to A.
- At $180^\circ$, point A moves to C, B to D, C to A, and D to B.
- Similar mappings occur for $270^\circ$ and $360^\circ$.
The reason states that rotation by any angle other than these four will not make the square overlap with its initial position. This is true because only multiples of $90^\circ$ ensure congruence due to the square's symmetric properties. Thus, both the assertion and reason are correct, and the reason provides the correct justification for the assertion.

Key Concept: Real-world Application, Rotational Symmetry

d) 6
[Solution Description]
Since each tile is rotated by $60°$ relative to the previous one, the number of tiles needed to complete a full cycle of $360°$ is calculated as $\frac{360°}{60°} = 6$. Thus, 6 tiles would bring the pattern back to its initial alignment.

Key Concept: Order of rotational symmetry

c) 4
[Solution Description]
A square has rotational symmetry where it looks exactly the same after rotation by certain angles. The smallest angle of rotation that makes the square look identical is 90°. The order of rotational symmetry is calculated as $360°$ divided by the smallest angle of rotation. So, order = $360°/90° = 4$.

Key Concept: Angle of symmetry, Multiples of smallest angle

d) It cannot have rotational symmetry because $40^\circ$ does not divide $360^\circ$ evenly.
[Solution Description]
For a shape to have rotational symmetry, the smallest angle of rotation must divide $360^\circ$ exactly. Here, $360^\circ / 40^\circ = 9$, which is an integer, so the shape could exist. However, if another option contradicts this condition, it must be false. Checking the options:
- If $40^\circ$ is the smallest angle, then other angles must be multiples of $40^\circ$ and include $80^\circ$, $120^\circ$, etc., but not necessarily $30^\circ$.
- Since $40^\circ$ divides $360^\circ$, the shape can complete full rotations.
- Therefore, the false statement would be that it *cannot* have rotational symmetry because $40^\circ$ does divide $360^\circ$.

Key Concept: Rotational symmetry angles

b) 45°
[Solution Description]
For a figure with $n$ radial arms, the smallest angle of rotational symmetry is given by $\frac{360°}{n}$. Here, $n = 8$. So, the smallest angle of rotation is $\frac{360°}{8} = 45°$.

Key Concept: Identifying rotational symmetry

b) 50°
[Solution Description]
The smallest angle of rotational symmetry must divide 360° exactly. Checking each option:
1) $\frac{360°}{40°} = 9$ → valid.
2) $\frac{360°}{50°} = 7.2$ → not a whole number → invalid.
3) $\frac{360°}{60°} = 6$ → valid.
4) $\frac{360°}{120°} = 3$ → valid.
Thus, 50° cannot be a smallest angle of rotational symmetry.

Key Concept: Rotational Symmetry, Reflection Symmetry

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
- The assertion states that a regular pentagon has five lines of symmetry. This is correct because a regular pentagon can be folded along five different axes (lines of symmetry), each passing through a vertex and the midpoint of the opposite side.
- It also claims that the rotational symmetry exists at $72^\circ$. This is true because rotating the pentagon by $72^\circ$ (which is $360^\circ/5$) brings it back to its original position.
- The reason states that for any regular polygon, the smallest angle of rotational symmetry is $360^\circ/n$, where $n$ is the number of sides. This is mathematically correct as rotating a regular polygon by this angle aligns it with itself.
- Further, the reason correctly explains why the assertion is true—the rotational symmetry angle directly derives from dividing $360^\circ$ by the number of sides (five in the case of a pentagon).
Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Generating Symmetrical Figures

b) Only reflection symmetry
[Solution Description]
The process described involves folding paper (creating a line of symmetry) and then applying ink to one side. When pressed together, the ink transfers symmetrically across the fold line. This creates a figure with reflection symmetry along the fold line. The figure will match exactly on both sides of this line but may not necessarily have rotational symmetry unless specifically designed otherwise.

Key Concept: Symmetry in Circles

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
The assertion states that a circle has infinite lines of symmetry, which is true because any line passing through the centre of the circle (i.e., any diameter) is a line of symmetry. The reason given is that every diameter of a circle is a line of reflection symmetry, which is also true and correctly explains why the circle has infinite lines of symmetry. Since both the assertion and reason are true and the reason correctly explains the assertion, the correct answer is option a).

Key Concept: Reflection Symmetry in Circles

d) Infinite
[Solution Description]
A circle has infinite lines of reflection symmetry. Any diameter drawn through the center of a circle acts as a line of symmetry, dividing the circle into two identical halves. Since there are infinite diameters possible, the number of lines of reflection symmetry is also infinite. Among the options, the closest representation is 'Infinite'.

Key Concept: Rotational Symmetry

a) The circle remains unchanged as it coincides with itself
[Solution Description]
A circle has rotational symmetry for any angle of rotation. When it is rotated by 45 degrees about its center, it coincides with itself because every point on the circumference moves to another point that lies exactly on the circumference due to the circular symmetry.

Key Concept: Real-world Application, Rotational Symmetry

b) $30^\circ$
[Solution Description]
Since the wheel has 12 equally spaced spokes, the angle between consecutive spokes is $360^\circ / 12 = 30^\circ$. For the wheel to appear identical after rotation, it must be rotated by an angle that is a multiple of $30^\circ$. Hence, the smallest such angle is $30^\circ$.

Key Concept: Smallest Angle of Symmetry, Multiples

c) The angles must be multiples of $45^\circ$ and divide $360^\circ$.
[Solution Description]
The smallest angle of symmetry determines the increments for other angles of symmetry. All angles of symmetry must be multiples of the smallest angle and must divide $360^\circ$.
Step 1: Verify if $45^\circ$ divides $360^\circ$: $360^\circ / 45^\circ = 8$ (integer).
Step 2: Generate multiples of $45^\circ$ up to $360^\circ$: $45^\circ$, $90^\circ$, $135^\circ$, $180^\circ$, $225^\circ$, $270^\circ$, $315^\circ$, $360^\circ$.
Step 3: Check divisibility: Each generated angle is a divisor of $360^\circ$.
Thus, the angles of symmetry are all multiples of $45^\circ$ that divide $360^\circ$.

Key Concept: Application of Rotational Symmetry

d) A rectangular book
[Solution Description]
Unlike a circle, which looks the same after any rotation, a rectangular book does not have rotational symmetry for all angles. It only coincides with itself when rotated by 180$^\circ$.

Key Concept: Rotational Symmetry in Circles

d) Any angle
[Solution Description]
A circle has rotational symmetry for any angle because rotating it by any degree around its center will make it coincide with itself. The smallest such angle is infinitesimal, but practically, even a very small rotation will suffice as there is no minimum angle restriction.

Key Concept: Symmetry in Circles

d) Infinite
[Solution Description]
A circle has infinite lines of symmetry because any diameter drawn through its center divides the circle into two identical halves. Since there are infinitely many possible diameters, the number of lines of symmetry is also infinite.

Key Concept: Symmetry, Rotational Symmetry

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
To check if the assertion and reason are correct, we need to understand rotational symmetry. A figure has rotational symmetry of angle $\theta$ if it looks identical after a rotation of $\theta$ degrees.
The reason states that all angles of rotational symmetry must be integer multiples of the smallest angle $\theta$. This is true because rotational symmetry angles are determined by dividing $360^\circ$ by the order of symmetry, which ensures that they are integer multiples of the smallest angle.
The assertion claims that if a figure has rotational symmetry of $60^\circ$, it can also have $120^\circ$ as another angle of symmetry. Since $120^\circ = 2 \times 60^\circ$, this follows from the reason.
Therefore, both the assertion and reason are true, and the reason correctly explains the assertion.

Key Concept: Rotational Symmetry, Angle of Symmetry

c) 10
[Solution Description]
The order of rotational symmetry is determined by how many times a figure matches itself during a full rotation of 360°. If the smallest angle of symmetry is 36°, we can find the order by dividing 360° by the smallest angle: $360° / 36° = 10$. Thus, the figure aligns with itself 10 times in a complete rotation, meaning the order of rotational symmetry is 10.

Key Concept: Symmetry creation, Multiple concepts

b) 8
[Solution Description]
For rotational symmetry of order 8, the pattern must repeat every $45^\circ$ ($360^\circ/8$). With 24 sectors, this means coloring every third sector (since $360^\circ/24 = 15^\circ$, and $45^\circ/15^\circ = 3$). For line symmetry, at least one diameter must divide the circle into mirrored halves. The minimal coloring would require coloring alternate sectors in the repeating pattern (e.g., color 1st, skip next two, color 4th, etc.), requiring at least 8 colored sectors.

Key Concept: Angle of Symmetry Calculation

c) $40^\circ$
[Solution Description]
The angles of symmetry are multiples of the smallest angle. Given the smallest angle is $20^\circ$, the next larger angle would be twice the smallest angle. Therefore, the next larger angle of symmetry is $20^\circ \times 2 = 40^\circ$.

Key Concept: Line Symmetry, Rotational Symmetry

b) 2
[Solution Description]
Rotational symmetry of order 3 means the figure coincides with itself every $120^\circ$ rotation. Starting from the original position, rotating by $120^\circ$ gives the first distinct position, another $120^\circ$ gives the second distinct position, and a third $120^\circ$ returns it to the original position. Thus, there are 2 distinct intermediate positions.

Key Concept: Rotational Symmetry and Line Symmetry

b) Both Assertion and Reason are true, but Reason is NOT the correct explanation of Assertion.
[Solution Description]
The assertion states that a regular hexagon has both line symmetry and rotational symmetry. This is true because:
1. **Line Symmetry**: A regular hexagon has 6 lines of symmetry, each passing through opposite vertices or midpoints of opposite sides.
2. **Rotational Symmetry**: A regular hexagon has rotational symmetry of order 6, meaning it coincides with itself when rotated by $60^\circ$ (smallest angle of symmetry), $120^\circ$, $180^\circ$, $240^\circ$, $300^\circ$, or $360^\circ$.
The reason states that a regular hexagon can be divided into six identical equilateral triangles. While this is true, the division into equilateral triangles does not explain why the hexagon has line or rotational symmetry—it merely describes its internal structure.
Therefore, both assertion and reason are true, but the reason does not correctly explain the assertion.

Key Concept: Rotational symmetry, Lines of symmetry

b) 4
[Solution Description]
A regular hexagon has 6 lines of symmetry (equal to its rotational symmetry order). The given figure already has 2 lines of symmetry. Therefore, it needs $6 - 2 = 4$ more lines of symmetry to become regular.

Key Concept: Rotational Symmetry

b) 15°
[Solution Description]
The Ashoka Chakra consists of 24 evenly spaced spokes. To find the angles of rotational symmetry, divide 360° by the number of spokes (24). $360° / 24 = 15°$. Thus, the smallest angle of rotation is 15°, and all multiples of this angle up to 360° will map the Chakra onto itself.

Key Concept: Using symmetry in designs with tiles, Paper Folding and Cutting

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
When a square piece of paper is folded along its diagonal, it creates a line of symmetry. If a cut is made symmetrically along this folded edge, the resulting design upon unfolding will mirror across the original fold line. This means the fold line becomes the line of symmetry for the unfolded design. Thus, both Assertion and Reason are true, and Reason correctly explains the Assertion.

Key Concept: Paper Folding and Cutting

c) Two identical cuts symmetrical about the diagonal
[Solution Description]
Folding a square sheet along its diagonal makes a triangular shape. If you make a cut anywhere on this folded triangle, the unfolded shape will have two identical cuts symmetrically placed about the diagonal. This creates a design with the diagonal as the line of symmetry.

Key Concept: Symmetry in Paper Folding

a) Both Assertion and Reason are true, and Reason is the correct explanation of Assertion.
[Solution Description]
When a square sheet of paper is folded along its diagonal, the fold becomes the line of symmetry. Punching a hole through both layers creates two holes that are equidistant from the fold. Upon unfolding, these holes will be symmetric about the diagonal, confirming the existence of exactly one line of symmetry (the diagonal).

Key Concept: Creating Symmetric Patterns

c) Rectangle (non-square)
[Solution Description]
For a shape to have exactly one line of symmetry, there must be only one line that divides it into two mirror images. Among simple shapes, an isosceles triangle or rectangle can have just one line of symmetry if not square.