Chapter 13 Symmetry (Additional Questions)

A figure is said to have line symmetry if it can be divided into two identical halves by a line, such that if the figure is folded along that line, the two halves fit exactly on top of each other.

In other words, the two halves must coincide exactly when folded along the line of symmetry.

Therefore, the correct option is (B) Coincide exactly.

The line along which a figure can be folded so that the two halves exactly coincide is known as the line of symmetry.

This line is also commonly referred to as the Axis of symmetry or the Mirror line because it acts like a mirror, reflecting one half of the figure onto the other.

Among the given options, Axis of symmetry is a standard term for this line.

Therefore, the correct option is (C) Axis of symmetry.

A square is a quadrilateral with four equal sides and four right angles. It possesses multiple lines of symmetry.

The lines of symmetry in a square are:

1. The line joining the midpoints of the opposite horizontal sides.

2. The line joining the midpoints of the opposite vertical sides.

3. The diagonal joining the top-left vertex to the bottom-right vertex.

4. The diagonal joining the top-right vertex to the bottom-left vertex.

Thus, a square has a total of 4 lines of symmetry.

Therefore, the correct option is (D) 4.

A letter has horizontal line symmetry if it can be folded along a horizontal line such that the top and bottom halves coincide exactly.

Let's examine the symmetry of each given letter:

(A) A: Has a vertical line of symmetry passing through the middle. It does not have horizontal line symmetry.

(B) B: Has a horizontal line of symmetry passing through the middle. It does not have vertical line symmetry.

(C) H: Has both a horizontal line of symmetry and a vertical line of symmetry.

(D) X: Has both a horizontal line of symmetry and a vertical line of symmetry.

The question asks for the letter that has only horizontal line symmetry.

Based on our analysis, the letter 'B' has horizontal symmetry but not vertical symmetry.

Therefore, the letter of the English alphabet that has only horizontal line symmetry is B.

The correct option is (B).

When an object is placed in front of a mirror, the image formed is a mirror image.

A mirror image is created by flipping the object across the plane of the mirror. This type of transformation, where every point of an object is mapped to a corresponding point on the other side of a line (or plane) such that the line (or plane) is the perpendicular bisector of the segment joining the two points, is called reflection.

Therefore, an object and its image in a mirror are examples of Reflection.

The correct option is (C) Reflection.

A circle is a perfectly round shape where all points on the circumference are equidistant from the center.

Any line passing through the center of the circle will divide the circle into two identical halves (two semicircles).

Since there are infinitely many lines that can pass through the center of a circle, a circle has an infinite number of lines of symmetry.

Therefore, the correct option is (D) Infinite number.

A figure has line symmetry if it can be divided into two identical halves by a line.

If a figure cannot be divided into two identical halves by any line, it does not have line symmetry.

Such a figure is described as asymmetric.

Therefore, a figure with no line of symmetry is called asymmetric.

The correct option is (B) Asymmetric.

Let's examine the number of lines of symmetry for each shape:

(A) Square: Has 4 lines of symmetry (two through opposite midpoints of sides and two through opposite vertices).

(B) Rectangle: Has 2 lines of symmetry (through the midpoints of opposite sides).

(C) Equilateral triangle: An equilateral triangle has three equal sides and three equal angles. The lines of symmetry are the angle bisectors, which are also the medians and altitudes from each vertex. Each line passes through a vertex and the midpoint of the opposite side. There are 3 such lines.

(D) Isosceles triangle: An isosceles triangle has at least two equal sides. It has exactly 1 line of symmetry, which passes through the vertex between the two equal sides and the midpoint of the opposite side.

The question asks for the shape with exactly 3 lines of symmetry.

Based on our analysis, the Equilateral triangle has exactly 3 lines of symmetry.

Therefore, the correct option is (C) Equilateral triangle.

When an object is placed in front of a plane mirror, the image formed has the following characteristics:

1. The image is always virtual. A virtual image is one that cannot be projected onto a screen, as the light rays do not actually converge at the image location; they only appear to diverge from it.

2. The image is always erect (or upright) relative to the object. This means the top of the object corresponds to the top of the image, and the bottom corresponds to the bottom.

3. The image is laterally inverted. This means the image is flipped left-to-right.

4. The image is the same size as the object.

5. The image is located behind the mirror at the same distance as the object is in front of the mirror.

Based on these properties, the image formed in a plane mirror is virtual and erect.

Therefore, the correct option is (D) Virtual and erect.

A rectangle (that is not a square) is a quadrilateral with four right angles and opposite sides equal in length. Since it is not a square, its adjacent sides have unequal lengths.

Let the sides of the rectangle be length $l$ and width $w$, where $l \neq w$.

The lines of symmetry for such a rectangle are:

1. The line joining the midpoints of the two opposite sides of length $l$. This line is parallel to the sides of length $w$.

2. The line joining the midpoints of the two opposite sides of length $w$. This line is parallel to the sides of length $l$.

Folding the rectangle along either of these lines will result in the two halves coinciding exactly.

The diagonals of a rectangle that is not a square are not lines of symmetry, because folding along a diagonal would not make the sides or angles coincide.

Thus, a rectangle (that is not a square) has a total of 2 lines of symmetry.

Therefore, the correct option is (C) 2.

A figure has vertical line symmetry if it can be folded along a vertical line to make the left and right halves coincide.

Let's check the given numbers:

The numeral '0' has vertical symmetry.

The numeral '1', when written as a simple vertical stroke, has vertical symmetry.

The numeral '8' has vertical symmetry.

Since 0, 1, and 8 all have vertical line symmetry, the option "All of the above" is correct.

Therefore, the correct option is (D) All of the above.

Reflection is a transformation where a figure is flipped across a line (the line of reflection) to create a new figure.

This process results in an image that is a mirror image of the original object. Therefore, the image produced by reflection is commonly called a mirrored image.

The correct option is (B) Mirrored.

Line symmetry exists if a figure can be folded along a line such that the two halves coincide exactly.

Let's consider each object:

(A) A butterfly: A butterfly typically has a vertical line of symmetry passing through its body and between its wings.

(B) A human face: A human face is generally considered to have approximate line symmetry along a vertical line down the middle, dividing it into left and right halves.

(C) A pair of scissors: A standard pair of scissors has line symmetry along the line passing through the pivot screw and extending between the handles.

(D) A teacup (with handle): A teacup with a handle does not have line symmetry because the presence of the handle on one side breaks any potential symmetry line that could divide the cup into identical halves.

The objects that show line symmetry among the options are a butterfly, a human face, and a pair of scissors.

Therefore, the correct options are (A), (B), and (C).

A letter has horizontal line symmetry if it can be folded along a horizontal line such that the top and bottom halves coincide. A letter has vertical line symmetry if it can be folded along a vertical line such that the left and right halves coincide.

Let's check the symmetry of each given letter (assuming standard block capital letters):

(A) A: Has vertical line symmetry, but not horizontal.

(B) O: Has both horizontal line symmetry (along a line through the middle horizontally) and vertical line symmetry (along a line through the middle vertically).

(C) L: Does not have horizontal or vertical line symmetry.

(D) T: Has vertical line symmetry, but not horizontal.

The letter that has both horizontal and vertical line symmetry is 'O'.

Therefore, the correct option is (B) O.

The case study describes a rangoli pattern that is symmetric in two ways:

1. Symmetric about a central point: Symmetry about a point is a type of rotational symmetry. If a figure is symmetric about a point, it looks the same after being rotated by $180^\circ$ around that point (point symmetry). In general, symmetry about a central point implies rotational symmetry.

2. Symmetric about several lines passing through the point: This refers to line symmetry, where the lines of symmetry all intersect at the central point.

A figure having multiple lines of symmetry that intersect at a single point also possesses rotational symmetry about that point.

Therefore, the rangoli pattern exhibits both line symmetry and rotational symmetry.

The option that correctly describes this is (C) Both line and potentially rotational symmetry.

Assertion (A): An isosceles triangle always has at least one line of symmetry.

An isosceles triangle is defined by having at least two equal sides. The line segment from the vertex angle (the angle between the two equal sides) to the midpoint of the opposite side serves as a line of symmetry, dividing the triangle into two congruent halves. Thus, Assertion (A) is True.

Reason (R): An isosceles triangle has two equal sides and two equal angles, allowing for a fold along the angle bisector of the vertex angle.

The first part of Reason (R) correctly states a property of isosceles triangles (two equal sides and two equal angles opposite those sides). The second part correctly identifies the angle bisector of the vertex angle as a line along which the triangle can be folded to show symmetry. This line is indeed the line of symmetry for an isosceles triangle. Thus, Reason (R) is True.

Reason (R) explains the property of an isosceles triangle (having equal sides and angles) that leads to the existence of a line of symmetry, specifically by mentioning the fold along the angle bisector of the vertex angle. This line acts as the line of symmetry, thereby correctly explaining why the assertion is true.

Therefore, both Assertion (A) and Reason (R) are true, and Reason (R) is the correct explanation of Assertion (A).

The correct option is (A) Both A and R are true and R is the correct explanation of A.

Reflection is a transformation that flips a point or figure across a given line, known as the line of reflection.

The result of this transformation is an image that is the same distance from the line of reflection as the original point or figure, but on the opposite side.

This image is often referred to as the symmetric image or the mirror image because it exhibits symmetry with the original figure with respect to the line of reflection, similar to how an object relates to its image in a mirror.

Among the given options, "Symmetric" is the most appropriate term to describe the image produced by reflection in the context of geometric transformations and symmetry.

Therefore, the correct option is (B) Symmetric.

Let's determine the number of lines of symmetry for each shape listed:

(i) A Rhombus (not a square) has exactly 2 lines of symmetry. These are its two diagonals.

(ii) A Parallelogram (not a rhombus or rectangle) has 0 lines of symmetry.

(iii) A Kite (not a rhombus) has exactly 1 line of symmetry. This line is the diagonal that connects the vertices where the equal sides meet.

(iv) A Scalene Triangle, having all sides of different lengths, has 0 lines of symmetry.

Now, we match the shapes with the corresponding number of lines of symmetry:

(i) Rhombus (not a square) $\to$ 2 (c)

(ii) Parallelogram (not a rhombus or rectangle) $\to$ 0 (b)

(iii) Kite (not a rhombus) $\to$ 1 (a)

(iv) Scalene Triangle $\to$ 0 (b)

Comparing this with the given options, the correct matching is (i)-(c), (ii)-(b), (iii)-(a), (iv)-(b).

This matches option (A).

Therefore, the correct option is (A) (i)-(c), (ii)-(b), (iii)-(a), (iv)-(b).

Let's determine the number of lines of symmetry for each figure:

(A) Square: A square has 4 lines of symmetry (two diagonals and two lines joining the midpoints of opposite sides).

(B) Rectangle: A rectangle (that is not a square) has exactly 2 lines of symmetry (the lines joining the midpoints of opposite sides).

(C) Equilateral triangle: An equilateral triangle has 3 lines of symmetry (the medians from each vertex).

(D) Circle: A circle has an infinite number of lines of symmetry (any line passing through its center).

Comparing the number of lines of symmetry, the figure with exactly 2 lines of symmetry is a Rectangle (assuming it's not a square, which is the standard interpretation in such questions where 'Square' is listed separately).

Therefore, the correct option is (B) Rectangle.

To determine the symmetry of the letter 'M', we consider if it can be folded along a line such that the two halves coincide exactly.

Horizontal line symmetry exists if a figure can be folded along a horizontal line so that the top and bottom portions are mirror images.

For the letter 'M', if we draw a horizontal line through the middle, the upper part is different from the lower part. So, 'M' does not have horizontal line symmetry.

Vertical line symmetry exists if a figure can be folded along a vertical line so that the left and right portions are mirror images.

For the letter 'M', if we draw a vertical line down the center, the left side is an exact mirror image of the right side. So, 'M' has vertical line symmetry.

Since 'M' has vertical line symmetry but not horizontal line symmetry, it possesses only vertical symmetry.

Therefore, the correct option is (B) Vertical.

We need to identify the shape that has rotational symmetry but lacks line symmetry.

Let's analyze each option:

(A) Square: Has 4 lines of symmetry and rotational symmetry of order 4 (rotates onto itself at $90^\circ$, $180^\circ$, $270^\circ$, $360^\circ$).

(B) Equilateral Triangle: Has 3 lines of symmetry and rotational symmetry of order 3 (rotates onto itself at $120^\circ$, $240^\circ$, $360^\circ$).

(C) Parallelogram (not a rhombus or rectangle): A parallelogram has rotational symmetry of order 2 about the intersection point of its diagonals (rotates onto itself at $180^\circ$). However, a parallelogram that is not a rhombus or a rectangle does not have any line of symmetry.

(D) Circle: Has infinite lines of symmetry and rotational symmetry of infinite order.

The shape that fits the criteria of having rotational symmetry but no line symmetry is a Parallelogram (not a rhombus or rectangle).

Therefore, the correct option is (C) Parallelogram (not a rhombus or rectangle).

Symmetry refers to a property of a figure or object that remains unchanged under certain transformations, such as reflection or rotation.

In design and architecture, symmetry is widely used to achieve aesthetic qualities.

When a design is symmetric, it often appears visually pleasing and stable because the parts are arranged in a balanced way around a line or a point.

This balanced arrangement contributes to a sense of balance and harmony.

Asymmetry creates imbalance, complexity can be achieved with or without symmetry, and irregularity is the opposite of symmetry.

Therefore, symmetry is important in design and architecture because it creates balance and harmony.

The correct option is (C) Balance and harmony.

When a point is reflected across the x-axis in a coordinate plane, the x-coordinate of the point remains unchanged, and the y-coordinate changes its sign.

If the original point is $(x, y)$, its reflection across the x-axis is $(x, -y)$.

The given point is $(2, 3)$. Here, $x=2$ and $y=3$.

Applying the reflection rule across the x-axis, the reflected point will be $(2, -3)$.

Therefore, the reflection of the point $(2, 3)$ across the x-axis is $(2, -3)$.

This matches option (A).

The correct option is (A) (2, -3).

A number has horizontal line symmetry if it can be folded along a horizontal line such that the top and bottom halves coincide exactly.

Let's examine the symmetry of each given number (assuming standard digital representations):

(A) 0: The digit '0' has a horizontal line of symmetry passing through its center. If folded along this line, the two halves coincide.

(B) 1: The digit '1' typically does not have horizontal line symmetry.

(C) 3: The digit '3' has a horizontal line of symmetry passing through the middle. If folded along this line, the top curve matches the bottom curve.

Based on this analysis, both '0' and '3' have horizontal line symmetry. The question asks which of the following numbers has horizontal line symmetry, and option (D) is "Both (A) and (C)", which refers to numbers 0 and 3.

Since both 0 and 3 have horizontal line symmetry, option (D) is the correct choice.

Therefore, the correct option is (D) Both (A) and (C).

Let's trace the number of triangles at each step:

1. We start with the original triangle. At this point, we have 1 triangle.

2. The original triangle is reflected across a vertical line. This creates a first image. Now we have the original triangle + the first image = 2 triangles.

3. Both the original triangle and its first image are reflected across a horizontal line.

- Reflecting the original triangle across the horizontal line creates a second image.

- Reflecting the first image (which was the reflection across the vertical line) across the horizontal line creates a third image.

In the final design, we have:

- The original triangle.

- The image from the first reflection (vertical).

- The image from reflecting the original triangle across the horizontal line.

- The image from reflecting the first image (vertical reflection) across the horizontal line.

Assuming the reflections result in distinct copies (which is typical unless the original triangle has specific placement and symmetry relative to the lines), the total number of triangles in the final design is $1 + 1 + 1 + 1 = 4$.

Therefore, there will be 4 triangles in the final design, including the original.

The correct option is (D) 4.

We need to find the letter among the options that cannot be folded along any line (horizontal, vertical, or diagonal) such that the two halves coincide exactly.

Let's examine the symmetry of each letter (assuming standard block capital letters):

(A) E: Has a horizontal line of symmetry.

(B) S: Does not have horizontal or vertical line symmetry. However, it does have point symmetry (rotational symmetry of order 2 around its center).

(C) V: Has a vertical line of symmetry.

(D) W: Has a vertical line of symmetry.

The letter that has no line symmetry is 'S'.

Therefore, the correct option is (B) S.

When an object is reflected in a plane mirror, the image formed is laterally inverted.

Lateral inversion means that the image is flipped along the horizontal axis. Specifically, the left side of the object appears as the right side in the image, and the right side of the object appears as the left side in the image.

The inversion occurs across the plane of the mirror, which typically results in a left-right reversal while maintaining the up-down orientation.

Therefore, the reflection of an image in a mirror is laterally inverted, meaning the left appears as Right and vice versa.

The correct option is (C) Right.

A semi-circle is half of a circle, bounded by a diameter and the arc of the circle.

Consider a semi-circle. The only line along which it can be folded such that the two halves coincide is the line that passes through the center of the original circle (and the midpoint of the diameter) and is perpendicular to the diameter.

This single line bisects both the diameter (the straight edge) and the circular arc, making the two resulting parts identical.

Any other line would not divide the semi-circle into two congruent halves.

Therefore, a semi-circle has exactly 1 line of symmetry.

The correct option is (B) 1.

Let's evaluate each statement:

(A) A figure can have only one line of symmetry.

This statement is False. Many figures have more than one line of symmetry (e.g., a square has 4, a rectangle has 2, a circle has infinite).

(B) If a figure has line symmetry, the line of symmetry is a mirror line.

This statement is True. The line of symmetry acts like a mirror; one side of the figure is the reflection (mirror image) of the other side across this line.

(C) Reflection changes the size and shape of an object.

This statement is False. Reflection is an isometric transformation, meaning it preserves the size and shape of the object. Only the orientation changes.

(D) All triangles have at least one line of symmetry.

This statement is False. Scalene triangles (triangles with no equal sides) have no line of symmetry.

Based on the evaluation, only statement (B) is true.

Therefore, the correct option is (B) If a figure has line symmetry, the line of symmetry is a mirror line.

A regular polygon has all sides equal in length and all interior angles equal in measure.

The lines of symmetry in a regular polygon pass through its center.

If 'n' is an odd number (e.g., equilateral triangle, regular pentagon), the lines of symmetry pass through each vertex and the midpoint of the opposite side. There are 'n' such vertices (and opposite sides), so there are 'n' lines of symmetry.

If 'n' is an even number (e.g., square, regular hexagon), the lines of symmetry pass either through opposite vertices or through the midpoints of opposite sides. There are $n/2$ pairs of opposite vertices and $n/2$ pairs of opposite sides. Each pair contributes one line of symmetry, for a total of $n/2 + n/2 = n$ lines of symmetry.

In both cases (n is odd or n is even), a regular polygon with 'n' sides has exactly n lines of symmetry.

Therefore, the number of lines of symmetry in a regular polygon with 'n' sides is $n$.

The correct option is (B) $n$.

Symmetry is commonly observed in various natural objects and phenomena.

Let's consider each option:

(A) A snowflake: Snowflakes typically exhibit beautiful hexagonal symmetry (rotational symmetry of order 6 and multiple lines of symmetry).

(B) A sunflower: Sunflowers display radial symmetry in the arrangement of their seeds and petals. The pattern often follows Fibonacci sequences and golden ratio properties, leading to approximate rotational symmetry.

(C) A rock: Rocks are typically irregular in shape and do not generally exhibit symmetry, although some crystals or specific rock formations might show symmetry.

(D) A human hand: A human hand exhibits bilateral symmetry, but the left and right hands are mirror images of each other (chirality). Each individual hand itself does not have line symmetry in a strict sense (a line drawn through the middle would not produce identical halves).

Among the given options, snowflakes and sunflowers are well-known examples of natural objects exhibiting clear forms of symmetry (line and rotational symmetry in snowflakes, and approximate radial/rotational symmetry in sunflowers).

Therefore, the objects that exhibit symmetry in nature among the options are a snowflake and a sunflower.

The correct options are (A) and (B).

The definition provided describes the property of a figure or design that can be divided into two congruent halves by a specific line such that one half is the mirror image of the other across that line.

When a figure is folded along this line, the two halves fit exactly on top of each other, or "exactly covers" each other.

This property is the defining characteristic of Line symmetry.

Rotational symmetry involves rotating a figure around a central point. Point symmetry is a specific type of rotational symmetry ($180^\circ$). Translation involves sliding a figure without rotation or reflection.

Therefore, the definition given is the definition of Line symmetry.

The correct option is (C) Line.

Let's examine the symmetry of the uppercase letter 'Z' (assuming standard representation).

Can 'Z' be folded along a line (horizontal, vertical, or diagonal) such that the two halves coincide?

- A horizontal line through the middle does not result in coinciding halves.

- A vertical line through the middle does not result in coinciding halves.

- Diagonal lines do not result in coinciding halves.

The letter 'Z' does not possess any line symmetry. However, it does have rotational symmetry of order 2 about its center (it looks the same after a $180^\circ$ rotation).

Since 'Z' has no line symmetry, the number of lines of symmetry is 0.

The correct option is (A) 0.

Reflection is a geometric transformation where a point is mapped to another point such that the line of reflection is the perpendicular bisector of the line segment connecting the original point and its image.

This property means that the distance from the original point to the line of reflection is exactly the same as the distance from the image point to the line of reflection.

The line of reflection lies exactly in the middle of the original point and its image.

Therefore, the distance of any point on the figure from the line of reflection is equal to the distance of its corresponding image point from the line.

The correct option is (C) Equal to.

Let's check for line symmetry:

(A) Swastika (standard form): Has rotational symmetry but no line symmetry.

(B) Ashok Chakra: Has multiple lines of symmetry.

(C) Lotus flower: Has multiple lines of symmetry.

Based on strict geometric definition, only (B) and (C) have line symmetry. However, given the option (D) "All of the above", it suggests the question might assume a representation or context where all three are considered to have line symmetry, or there is an error in the question/options. Assuming the intended answer is among the choices, and acknowledging that (B) and (C) definitely have line symmetry, option (D) is the most likely intended answer if it is assumed that the Swastika is also considered to have line symmetry in this context.

Therefore, assuming the intended scope where all three might be considered as exhibiting line symmetry (potentially in idealized forms), the correct option is (D) All of the above (considering ideal forms).

In geometry, symmetry refers to a property where a figure or shape can be divided into parts that are the same in some way. It means that one half is a mirror image of the other half, or that the figure looks the same after it has been rotated by a certain angle around a central point.

There are several types of symmetry, the most common ones being Line Symmetry and Rotational Symmetry.

Line Symmetry (or Reflectional Symmetry)

A figure has line symmetry if it can be divided into two identical halves by a line. This line is called the axis of symmetry. If you fold the figure along the axis of symmetry, the two halves will perfectly match each other.

For example, a butterfly has line symmetry along a vertical line down its body. A square has four axes of symmetry (two diagonals and two lines joining the midpoints of opposite sides).

Rotational Symmetry

A figure has rotational symmetry if it looks exactly the same after being rotated by a certain angle (less than $360^\circ$) about a fixed central point. The fixed point is called the center of rotation.

The number of times a figure fits onto itself during a full rotation of $360^\circ$ is called the order of rotational symmetry. The angle of rotation is $360^\circ$ divided by the order of rotational symmetry.

For example, a square has rotational symmetry of order 4 because it looks the same after rotations of $90^\circ$, $180^\circ$, $270^\circ$, and $360^\circ$ (which brings it back to the original position). The order is 4, and the angle of rotation is $360^\circ / 4 = 90^\circ$. A circle has infinite rotational symmetry as it looks the same after rotation by any angle about its center.

A line of symmetry (also known as an axis of symmetry or a mirror line) of a figure is a line that divides the figure into two identical, mirror-image halves. If the figure is folded along this line, the two halves will coincide perfectly. This type of symmetry is called line symmetry or reflectional symmetry.

You can check if a figure has a line of symmetry using the folding method by following these steps:

Step 1: Prepare the Figure

If the figure is on a piece of paper, you can directly work with the paper. If the figure is drawn on a surface, you might need to trace it onto paper first.

Step 2: Identify a Potential Line

Mentally (or by drawing a light line), identify a straight line that you suspect might be a line of symmetry for the figure.

Step 3: Fold the Paper along the Line

Carefully fold the paper exactly along the suspected line of symmetry. Make sure the fold is sharp and precise.

Step 4: Check if the Halves Coincide

Observe the two parts of the figure formed by the fold. If the figure has line symmetry along the folded line, the two halves of the figure on either side of the fold will match up perfectly, lying exactly on top of each other when the paper is folded.

Step 5: Conclude

If the two halves coincide perfectly, the line you folded along is indeed a line of symmetry. If the halves do not match, that particular line is not a line of symmetry.

You can repeat this process by trying different potential lines to find all possible lines of symmetry for the figure.

A circle has an infinite number of lines of symmetry.

This is because any straight line that passes through the center of the circle divides it into two identical semicircles. Since there are infinitely many such lines that can pass through the center, a circle has infinitely many lines of symmetry.

First, draw a scalene triangle. A scalene triangle is a triangle in which all three sides have different lengths, and consequently, all three angles have different measures.

To check for lines of symmetry in a scalene triangle, imagine folding it along any possible line. Because all sides and angles are unequal, there is no line you can fold the triangle along such that the two halves perfectly coincide.

Therefore, a scalene triangle has no lines of symmetry.

First, draw an isosceles triangle. An isosceles triangle is a triangle that has at least two sides of equal length. The angles opposite these equal sides are also equal.

An isosceles triangle has one line of symmetry.

This line of symmetry is the line segment drawn from the vertex angle (the angle between the two equal sides) to the midpoint of the opposite side (the base). This line is also the altitude to the base and the angle bisector of the vertex angle.

If you fold the isosceles triangle along this line, the two equal sides and the two equal angles will perfectly match, causing the two halves of the triangle to coincide.

First, draw an equilateral triangle. An equilateral triangle is a triangle in which all three sides are of equal length, and consequently, all three angles are also equal (each measuring $60^\circ$).

An equilateral triangle has three lines of symmetry.

These three lines of symmetry are:

1. The line segment drawn from each vertex to the midpoint of the opposite side (this is also the median, altitude, and angle bisector for that vertex).

Since there are three vertices, there are three such lines. Each of these lines divides the equilateral triangle into two congruent (identical) right-angled triangles. Folding the triangle along any of these lines will make the two halves coincide perfectly.

A square has four lines of symmetry.

These four lines are:

1. Two lines joining the midpoints of opposite sides (one vertical and one horizontal).

2. Two lines that are its diagonals.

If you fold a square along any of these four lines, the two resulting halves will match up perfectly, demonstrating line symmetry.

A rectangle has two lines of symmetry.

These two lines of symmetry are:

1. The line segment joining the midpoints of the two longer sides.

2. The line segment joining the midpoints of the two shorter sides.

Folding the rectangle along either of these lines will make the two halves coincide perfectly. The diagonals of a rectangle are not lines of symmetry.

First, draw a parallelogram. A parallelogram is a quadrilateral with two pairs of parallel sides.

A typical parallelogram (one that is not a rectangle, rhombus, or square) has no lines of symmetry.

In a general parallelogram, there is no line along which you can fold the figure so that the two halves match perfectly.

It is worth noting that some special types of parallelograms do have lines of symmetry:

- A rectangle (a parallelogram with four right angles) has two lines of symmetry.

- A rhombus (a parallelogram with four equal sides) has two lines of symmetry.

- A square (a parallelogram with four equal sides and four right angles) has four lines of symmetry.

However, for a parallelogram that is not one of these special cases, the number of lines of symmetry is zero.

First, draw a rhombus. A rhombus is a quadrilateral with all four sides of equal length. It is a special type of parallelogram where the adjacent sides are equal.

A rhombus has two lines of symmetry.

These two lines of symmetry are its diagonals.

Each diagonal of a rhombus divides it into two congruent triangles. Folding the rhombus along either diagonal will make the two halves coincide perfectly, demonstrating line symmetry.

Note that a square is a special type of rhombus where all angles are $90^\circ$. A square has 4 lines of symmetry (its two diagonals and the two lines joining the midpoints of opposite sides).

A capital English letter has a vertical line of symmetry if it looks the same when reflected across a vertical line drawn through its center.

The capital English letters that have a vertical line of symmetry are:

A, H, I, M, O, T, U, V, W, X, Y.

A capital English letter has a horizontal line of symmetry if it looks the same when reflected across a horizontal line drawn through its center.

The capital English letters that have a horizontal line of symmetry are:

B, C, D, E, H, I, K, O, X.

A capital English letter has both a vertical and a horizontal line of symmetry if it remains unchanged when reflected across a vertical line and also when reflected across a horizontal line.

We need to find the letters that are present in the list of letters with vertical symmetry (from Question 12) and the list of letters with horizontal symmetry (from Question 13).

Letters with vertical symmetry: A, H, I, M, O, T, U, V, W, X, Y

Letters with horizontal symmetry: B, C, D, E, H, I, K, O, X

The letters that appear in both lists are the ones with both vertical and horizontal lines of symmetry.

These letters are:

H, I, O, X.

A digit has a horizontal line of symmetry if it looks the same when reflected across a horizontal line drawn through its center.

Let's examine the digits from 0 to 9:

- Digit 0: Has a horizontal line of symmetry.

- Digit 1: Does not have a horizontal line of symmetry.

- Digit 2: Does not have a horizontal line of symmetry.

- Digit 3: Has a horizontal line of symmetry.

- Digit 4: Does not have a horizontal line of symmetry.

- Digit 5: Does not have a horizontal line of symmetry.

- Digit 6: Does not have a horizontal line of symmetry.

- Digit 7: Does not have a horizontal line of symmetry.

- Digit 8: Has a horizontal line of symmetry.

- Digit 9: Does not have a horizontal line of symmetry.

The digits that have a horizontal line of symmetry are 0, 3, and 8.

In the context of symmetry, reflection is a type of transformation that flips a figure across a line, creating a mirror image of the original figure.

The line across which the reflection occurs is called the line of reflection or the axis of reflection. Every point in the original figure has a corresponding point in the reflected figure (its image) on the opposite side of the line of reflection. The distance from any point to the line of reflection is equal to the distance from its image to the line of reflection. The line segment connecting a point and its image is perpendicular to the line of reflection.

Line symmetry is based on reflection. A figure has line symmetry if, when reflected across a specific line (its line of symmetry), the figure maps onto itself, meaning the reflected image is identical to the original figure and occupies the same position.

When an object is reflected in a mirror, the surface of the mirror itself acts as the line of symmetry (in a 2D representation) or the plane of symmetry (in a 3D representation).

In the context of reflectional symmetry, the line (or plane) of symmetry is the line (or plane) across which the reflection occurs. The mirror's surface is where the image appears to be "flipped" from the object.

If an object were placed directly on the mirror's surface and had line symmetry along that surface line, its reflection would perfectly coincide with the part of the object on the other side of the line.

One example of a symmetrical object found in nature is a butterfly.

A butterfly typically exhibits bilateral symmetry (a form of line symmetry). If an imaginary vertical line is drawn down the center of its body, the two wings and sides of the body are nearly identical mirror images of each other.

One common example of a symmetrical object found in architecture is the facade of a classical building.

Many historical and modern buildings are designed with a sense of balance and proportion. The main entrance, windows, and decorative elements on the left side of the central point or line are often mirrored on the right side.

This design typically demonstrates line symmetry, with the line of symmetry running vertically down the center of the building's front face. If you were to draw a vertical line through the middle, the two halves of the facade would appear as mirror images of each other, creating a visually harmonious and balanced appearance.

Yes, a kite shape (with two distinct pairs of equal adjacent sides) always has a line of symmetry.

A kite has exactly one line of symmetry.

This line of symmetry is the diagonal connecting the vertices where the two pairs of equal sides meet. For example, if the kite has vertices A, B, C, and D such that sides AB = BC and AD = CD, the line of symmetry is the diagonal BD.

If you fold the kite along this diagonal (BD), the side AB will perfectly match the side AD (since AB = AD) and the side CB will perfectly match the side CD (since CB = CD). The vertices A and C will coincide.

The other diagonal (AC in the example) is generally not a line of symmetry for a typical kite, unless the kite is also a rhombus or a square.

Yes, a figure can definitely have more than one line of symmetry.

Some figures have multiple lines along which they can be folded to produce two identical, coinciding halves.

Here are some examples of figures with more than one line of symmetry:

- A square has 4 lines of symmetry.

- A rectangle has 2 lines of symmetry.

- An equilateral triangle has 3 lines of symmetry.

- A rhombus has 2 lines of symmetry.

- A circle has an infinite number of lines of symmetry.

- A regular hexagon has 6 lines of symmetry.

First, draw a crescent moon shape. A typical crescent shape is formed by the intersection of two circles or arcs.

A crescent moon shape typically does not have a line of symmetry.

While it might appear balanced, there is usually no single line along which you can fold the standard crescent shape such that the two halves perfectly coincide. The outer curve and the inner curve have different radii and positions, preventing a mirror image match across any straight line.

Note: A highly stylized or specific crescent shape might be designed with symmetry (e.g., a perfect semi-circle with a smaller semi-circle removed from the same center), but the standard astronomical crescent shape lacks line symmetry.

Here is an example of a figure with no line of symmetry:

Consider the capital English letter 'F' (assuming standard block letter font).

If you try to draw a vertical line through the center of the 'F' and fold it, the two halves will not match. The top bar on the left has no corresponding part on the right side below the horizontal arm.

If you try to draw a horizontal line through the center and fold it, the top part (with the top bar and horizontal arm) will not match the bottom part (just the vertical stroke).

There is no line along which you can fold the letter 'F' so that the two halves are identical mirror images.

Therefore, the letter 'F' is an example of a figure with no line of symmetry.

A regular pentagon is a polygon with 5 equal sides and 5 equal interior angles.

For any regular polygon with $n$ sides, the number of lines of symmetry is equal to $n$.

Since a regular pentagon has $n=5$ sides, it has 5 lines of symmetry.

These lines of symmetry pass through each vertex and the midpoint of the opposite side. There are 5 vertices, so there are 5 such lines.

Mirror symmetry is also known as Line Symmetry.

It is also frequently referred to as Reflectional Symmetry.

These terms are used because the symmetry is based on the concept of reflection across a line (the line of symmetry), much like an image is reflected in a mirror.

Concept of Line Symmetry

Line symmetry, also known as reflectional symmetry or mirror symmetry, is a property of a figure where it can be divided into two identical halves by a straight line. This dividing line is called the line of symmetry or the axis of symmetry.

If a figure has line symmetry, it means that if you were to reflect the figure across the line of symmetry, the figure would map onto itself, appearing exactly the same as the original figure. Alternatively, if the figure is on paper, folding the paper along the line of symmetry would cause the two halves of the figure to coincide perfectly.

Every point on one side of the line of symmetry has a corresponding point on the other side, such that the line of symmetry is the perpendicular bisector of the line segment connecting these two points.

Lines of Symmetry of a Square

A square is a quadrilateral with four equal sides and four right ($90^\circ$) angles. It is a highly symmetrical shape.

A square has a total of four lines of symmetry.

Let's consider a square ABCD.

Types of Lines of Symmetry in a Square

1. Lines Joining the Midpoints of Opposite Sides: There are two such lines.

- One horizontal line joining the midpoints of the top and bottom sides (say, AB and CD). If you fold the square along this line, the top half aligns perfectly with the bottom half (vertex A maps to D, B maps to C).

- One vertical line joining the midpoints of the left and right sides (say, AD and BC). If you fold the square along this line, the left half aligns perfectly with the right half (vertex A maps to B, D maps to C).

These lines pass through the center of the square and are perpendicular to its sides.

2. The Diagonals: There are two diagonals in a square.

- The diagonal connecting opposite vertices (say, A and C). If you fold the square along this diagonal, the two triangular halves (ABC and ADC) are congruent and one fits exactly over the other (vertex B maps to D).

- The diagonal connecting the other pair of opposite vertices (say, B and D). If you fold the square along this diagonal, the two triangular halves (BAD and BCD) are congruent and one fits exactly over the other (vertex A maps to C).

These lines also pass through the center of the square.

Explanation of Why These are Lines of Symmetry

Each of these four lines divides the square into two congruent parts such that one part is the mirror image of the other across the line. Whether you physically fold a paper square or mathematically reflect it across any of these four lines, the resulting figure covers the original figure perfectly.

For example, consider the vertical line joining the midpoints of the left and right sides. Every point on the left side of the square has a corresponding point on the right side at the same distance from the line. For instance, the vertices A and B are equidistant from the vertical midline and are reflections of each other across it. The entire left half of the square is a reflection of the entire right half across this line.

Similarly, for a diagonal, say AC. The vertex B is the reflection of the vertex D across the line AC. The side AB is reflected onto AD, and CB is reflected onto CD. The triangle ABC is congruent to triangle ADC, and one perfectly overlaps the other when folded along the diagonal AC.

Since these four lines (two mid-segment lines and two diagonals) possess the property of dividing the square into perfect mirror images, they are the lines of symmetry for a square.

First, draw a rectangle. A rectangle is a quadrilateral with four right ($90^\circ$) angles. Opposite sides are equal in length.

A rectangle has two lines of symmetry.

These two lines of symmetry are:

1. The line segment joining the midpoints of the two longer sides. This is a horizontal line of symmetry (if the longer sides are horizontal).

2. The line segment joining the midpoints of the two shorter sides. This is a vertical line of symmetry (if the shorter sides are vertical).

Explanation of Why These are Lines of Symmetry

Consider the horizontal line joining the midpoints of the top and bottom sides. If you fold the rectangle along this line, the top half of the rectangle will perfectly overlap the bottom half. Every point in the top half has a corresponding mirror image point in the bottom half, equidistant from the fold line.

Similarly, consider the vertical line joining the midpoints of the left and right sides. Folding the rectangle along this line will cause the left half to perfectly overlap the right half. Every point in the left half has a corresponding mirror image point in the right half across this line.

The diagonals of a rectangle are generally not lines of symmetry. If you fold a rectangle along a diagonal, the two halves (triangles) are congruent, but they are not mirror images that coincide when folded along the diagonal (unless the rectangle is a square). The vertices on one side of the diagonal would not map onto the vertices on the other side (except the endpoints of the diagonal itself).

Comparison with a Square

A square is a special type of rectangle where all four sides are equal. This additional property gives the square more symmetry than a non-square rectangle.

Key Differences:

1. Number of Lines of Symmetry: A rectangle has 2 lines of symmetry, while a square has 4 lines of symmetry.

2. Position of Lines of Symmetry: Both figures have lines of symmetry joining the midpoints of opposite sides. A rectangle has these two lines.

However, a square also has its two diagonals as lines of symmetry, which a general rectangle does not.

In essence, a square possesses the two lines of symmetry that a rectangle has, plus an additional two lines of symmetry corresponding to its diagonals.

Here are the lines of symmetry for the three types of triangles:

1. Equilateral Triangle

An equilateral triangle has all three sides equal in length and all three angles equal ($60^\circ$ each).

Number of lines of symmetry: 3

Description of lines of symmetry: Each line of symmetry passes through a vertex and the midpoint of the opposite side. Since there are three vertices, there are three such lines.

2. Isosceles Triangle (not equilateral)

An isosceles triangle has exactly two sides of equal length and the two angles opposite those sides are equal. It does not have all three sides equal (unlike an equilateral triangle).

Number of lines of symmetry: 1

Description of line of symmetry: The single line of symmetry passes through the vertex angle (the angle between the two equal sides) and the midpoint of the opposite side (the base). This line is perpendicular to the base and bisects the vertex angle.

3. Scalene Triangle

A scalene triangle has all three sides of different lengths and all three angles of different measures.

Number of lines of symmetry: 0

Description of lines of symmetry: A scalene triangle has no lines of symmetry. There is no line you can fold it along such that the two halves are identical mirror images.

Concept of Reflection Symmetry

Reflection symmetry (also called line symmetry or mirror symmetry) describes a property where a figure remains unchanged after it is reflected across a specific line. This line is the line of symmetry. Essentially, one half of the figure is a mirror image of the other half with respect to this line.

Mirror as an Example

A common way to understand reflection is by looking into a mirror. When you stand in front of a mirror, you see your image. This image is a reflection of yourself across the surface of the mirror. Your left side appears as the right side of your image, and vice versa. Every point on your body has a corresponding point in the image, and the line connecting these two points is perpendicular to the mirror surface, with the mirror being exactly halfway between the point and its image.

Drawing and Reflection of Letter 'F'

Imagine drawing the capital letter 'F' and a vertical line next to it, perhaps a few units away to the right of the 'F'.

(Imagine drawing a standard block letter 'F' on the left side, and a vertical line a short distance to its right.)

Now, imagine reflecting the 'F' across this vertical line. To do this, for every point on the original 'F', find its corresponding point on the other side of the vertical line, at the same distance from the line. The shape formed by all these reflected points is the reflection of the original 'F'.

(Imagine drawing the letter 'F' as seen in a mirror, reflected across the vertical line. It will appear as if flipped horizontally.)

Relationship Between Original Shape and Reflection

The reflection of the letter 'F' across the vertical line is a mirror image of the original 'F'.

- The size and shape of the reflected 'F' are identical to the original 'F'. It is a congruent figure.

- The orientation is reversed horizontally. What was on the left side of the original 'F' (like the vertical stroke) is now on the right side of the reflected 'F', and what was on the right (like the horizontal arms) is now on the left.

- The distance of any point on the reflected 'F' from the vertical line is the same as the distance of the corresponding point on the original 'F' from that line.

- The line segment connecting any point on the original 'F' to its reflected image on the other side is perpendicular to the vertical line.

In the case of the letter 'F' reflected across a vertical line *next to it*, the original shape and its reflection are separate but related figures. If the original 'F' itself had a vertical line of symmetry (which it doesn't), then its reflection *across a line running through its center* would make the two halves of the 'F' coincide. Since 'F' does not have this symmetry, its reflection across an external line creates a distinct, flipped image.

Here are the lines of symmetry for the given capital English letters:

Letter A

Imagine drawing the capital letter 'A'.

It has one line of symmetry.

This line is a vertical line passing through the center of the letter.

(Imagine a vertical line bisecting the letter 'A'.)

Letter B

Imagine drawing the capital letter 'B' (assuming a symmetrical shape).

It has one line of symmetry.

This line is a horizontal line passing through the center of the letter.

(Imagine a horizontal line passing through the middle of the letter 'B'.)

Letter H

Imagine drawing the capital letter 'H'.

It has two lines of symmetry.

These lines are:

1. A vertical line passing through the center of the letter.

2. A horizontal line passing through the center (along the crossbar).

(Imagine a vertical line and a horizontal line crossing at the center of the 'H'.)

Letter M

Imagine drawing the capital letter 'M'.

It has one line of symmetry.

This line is a vertical line passing through the center of the letter.

(Imagine a vertical line bisecting the letter 'M'.)

Letter O

Imagine drawing the capital letter 'O' (like a circle or an oval).

It has an infinite number of lines of symmetry.

Any straight line passing through the center of the letter 'O' is a line of symmetry.

(Imagine many lines passing through the center of the 'O' in all directions.)

Letter X

Imagine drawing the capital letter 'X'.

It has four lines of symmetry.

These lines are:

1. A vertical line passing through the center.

2. A horizontal line passing through the center.

3. The two diagonal lines of the 'X'.

(Imagine a vertical line, a horizontal line, and two diagonal lines crossing at the center of the 'X'.)

Difference Between Line Symmetry and No Symmetry

A figure is said to have line symmetry if there exists a straight line (the line of symmetry) such that if the figure were folded along this line, the two halves would coincide perfectly, or if reflected across this line, the figure would map onto itself.

A figure is said to have no symmetry (specifically, no line symmetry in this context) if there is no such line. No matter which straight line you choose to fold or reflect the figure across, the two resulting halves or the reflected image will not perfectly coincide with the original figure.

In essence, line symmetry implies a specific type of balance or mirroring within the figure along a particular line, whereas having no line symmetry means the figure lacks this specific reflective balance.

Examples of Figures with Different Numbers of Lines of Symmetry

Figure with Exactly One Line of Symmetry:

An isosceles triangle that is not equilateral has exactly one line of symmetry.

Number of lines of symmetry: 1

(Imagine drawing an isosceles triangle with two equal sides and one base side of a different length. Then, draw a single vertical line from the vertex angle down to the midpoint of the base.)

Figure with More Than One Line of Symmetry:

A square has more than one line of symmetry (specifically, four lines of symmetry).

Number of lines of symmetry: 4

(Imagine drawing a square. Draw two lines connecting the midpoints of opposite sides (one horizontal, one vertical) and two lines along its diagonals. These are the four lines of symmetry.)

Figure with No Line of Symmetry:

A scalene triangle has no lines of symmetry.

Number of lines of symmetry: 0

(Imagine drawing a triangle where all three sides have different lengths and all three angles have different measures. There is no line you can draw that divides it into two mirror images.)

Here are three examples of objects or structures that exhibit line symmetry, commonly found in our surroundings:

Object 1: A Human Face

Most human faces have approximate line symmetry. While not perfectly symmetrical, there is a clear sense of balance.

Line of Symmetry: A single vertical line running down the center of the face, from the forehead, between the eyes, down the nose, and through the middle of the mouth and chin. The left side of the face is roughly a mirror image of the right side across this line.

Object 2: A Standard Rectangular Door

A plain rectangular door often exhibits line symmetry, especially when viewed from the front, excluding hinges and handles.

Lines of Symmetry: There are typically two lines of symmetry.

1. A vertical line passing through the center, dividing the door into a left and right half.

2. A horizontal line passing through the center, dividing the door into a top and bottom half (assuming symmetrical panels or design).

Object 3: A Common Leaf (like a Maple or Oak leaf)

Many types of leaves in nature display line symmetry.

Line of Symmetry: A single vertical line running along the main central vein of the leaf. The shape and veins on one side of the central vein are typically mirrored on the other side.

First, draw a regular hexagon. A regular hexagon is a polygon with 6 equal sides and 6 equal interior angles ($120^\circ$ each).

A regular hexagon has 6 lines of symmetry.

These lines of symmetry can be found in two ways:

1. Lines connecting opposite vertices: There are 3 pairs of opposite vertices in a regular hexagon. The lines segment connecting each pair of opposite vertices passes through the center of the hexagon and is a line of symmetry.

(Imagine drawing a regular hexagon and then drawing the three main diagonals connecting opposite corners.)

2. Lines connecting the midpoints of opposite sides: There are 3 pairs of opposite sides in a regular hexagon. The lines segment connecting the midpoint of each side to the midpoint of its opposite side passes through the center of the hexagon and is a line of symmetry. These lines are perpendicular to the pairs of sides they connect.

(Imagine drawing a regular hexagon and then drawing three lines, each starting from the midpoint of one side, going through the center, and ending at the midpoint of the opposite side.)

Explanation of Why These are Lines of Symmetry

Each of these 6 lines divides the regular hexagon into two congruent halves. If you fold the hexagon along any of these lines, the two halves will perfectly coincide. This is because the properties of a regular hexagon (equal sides, equal angles) are balanced across each of these lines. For example, the distance from any point on the hexagon to one of these lines is equal to the distance from its mirror image point on the other side, and the line segment connecting the point and its image is perpendicular to the line of symmetry.

Thus, a regular hexagon has 3 lines of symmetry through opposite vertices and 3 lines of symmetry through the midpoints of opposite sides, totaling $3 + 3 = 6$ lines of symmetry.

A trapezoid (or trapezium) is a quadrilateral with at least one pair of parallel sides.

Yes, a trapezoid can have a line of symmetry, but only a specific type of trapezoid.

The type of trapezoid that has line symmetry is an isosceles trapezoid.

An isosceles trapezoid has:

- One pair of parallel sides (the bases).

- The non-parallel sides are equal in length.

- The base angles are equal (angles on the same base are equal).

Example of an Isosceles Trapezoid with Line Symmetry

Imagine drawing a trapezoid where the top and bottom sides are parallel, and the two non-parallel sides are equal in length.

(Imagine drawing an isosceles trapezoid.)

Lines of Symmetry

An isosceles trapezoid has exactly one line of symmetry.

This line of symmetry is the straight line segment that connects the midpoint of the longer base to the midpoint of the shorter base.

(Draw the isosceles trapezoid and then draw a vertical line segment connecting the midpoint of the top base to the midpoint of the bottom base. This line should be perpendicular to both bases.)

Explanation

If you fold an isosceles trapezoid along the line connecting the midpoints of its parallel sides, the two non-parallel sides will coincide because they are equal in length. The corresponding angles at the bases will also coincide. The two halves of the trapezoid formed by this line are mirror images of each other.

A general trapezoid that is not isosceles (where the non-parallel sides have different lengths or the base angles are unequal) does not have line symmetry, as there is no line along which it can be folded to create perfect mirror images.

Among the digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$, the digits that have line symmetry are 0, 3, and 8.

Digit 0

The digit 0 (shaped like a circle or an oval) has line symmetry.

Number of lines of symmetry: 2

Lines of symmetry:

1. A horizontal line passing through the center.

2. A vertical line passing through the center.

(Imagine drawing the digit '0' and drawing a horizontal line and a vertical line crossing at its center.)

Digit 3

The digit 3 has line symmetry.

Number of lines of symmetry: 1

Line of symmetry:

1. A horizontal line passing through the center.

(Imagine drawing the digit '3' and drawing a horizontal line passing through its middle.)

Digit 8

The digit 8 has line symmetry.

Number of lines of symmetry: 2

Lines of symmetry:

1. A horizontal line passing through the center.

2. A vertical line passing through the center.

(Imagine drawing the digit '8' and drawing a horizontal line and a vertical line crossing at its center.)

The digits $1, 2, 4, 5, 6, 7, 9$ do not have any line of symmetry (in standard written forms).

Relation between Reflection and Symmetry

In geometry, reflection is a transformation that creates a mirror image of a figure across a line (called the line of reflection). Every point in the original figure is mapped to a corresponding point in the reflected figure, such that the line of reflection is the perpendicular bisector of the segment connecting the original point and its image.

Line symmetry (or reflectional symmetry) is a specific type of symmetry. A figure possesses line symmetry if, and only if, the figure is identical to its reflection across a particular line within or associated with the figure. This line is called the line of symmetry.

The relation is that line symmetry is the property a figure has when it is invariant under reflection across a specific line. If a figure has line symmetry, it means there is a line such that reflecting the figure across that line leaves the figure unchanged in its position and appearance.

What Happens When a Symmetrical Figure is Reflected Across Its Line of Symmetry

If a figure is symmetrical about a line (i.e., it has that line as a line of symmetry), then when you reflect the figure across that specific line of symmetry, the figure maps onto itself.

This means the reflected figure is exactly the same as the original figure and occupies the exact same space. The first half of the figure is reflected onto the second half, and the second half is reflected onto the first half, resulting in the entire figure coinciding with its original position.

Illustration using a Diagram (Isosceles Triangle)

Consider an isosceles triangle ABC, where AB = AC. This triangle has one line of symmetry, which is the altitude from vertex A to the midpoint of the base BC. Let's call the midpoint M.

(Imagine drawing an isosceles triangle ABC with A at the top, BC as the base, and M as the midpoint of BC. Draw a vertical line segment AM, which is the line of symmetry.)

Here, the line AM is the line of symmetry for triangle ABC.

Now, consider reflecting the triangle ABC across the line AM:

- Vertex A lies on the line AM, so its reflection is A itself.

- Vertex B is reflected across AM to its corresponding point on the other side, which is vertex C (because BM = CM and angle AMB = angle AMC = $90^\circ$).

- Vertex C is reflected across AM to its corresponding point on the other side, which is vertex B.

- Any point on the side AB is reflected onto a corresponding point on the side AC.

- Any point on the side AC is reflected onto a corresponding point on the side AB.

- Any point on the base BC is reflected onto a corresponding point on BC (points on BM are reflected onto CM, and vice versa).

When all points of triangle ABC are reflected across the line AM, the resulting reflected figure is triangle ACB, which is the exact same triangle as ABC, just with vertices named in a different order (reflecting B to C and C to B). The reflected triangle perfectly occupies the space of the original triangle.

This illustrates that when a figure is symmetrical about a line, reflecting it across that line leaves the figure unchanged.

If you fold a piece of paper with a design along a line, and the design on one side of the fold exactly matches the design on the other side, it means that one side of the design is a perfect mirror image of the other side with respect to the fold line.

Based on this result from the folding test, we can say the following:

1. About the design: The design possesses line symmetry (or reflectional symmetry). This means the design has a property of balance where one half is the mirror image of the other half.

2. About the fold line: The fold line is the line of symmetry (or axis of symmetry) for the design. It is the specific line that divides the figure into two congruent mirror-image halves.

The folding method is a practical way to identify whether a figure has line symmetry and to locate its line(s) of symmetry.

Illustration: Simple Design (Heart Shape)

Consider a standard heart shape drawn on a piece of paper.

(Imagine drawing a symmetrical heart shape.)

If you find the exact vertical center of the heart shape and fold the paper along this vertical line, the left half of the heart will precisely overlap the right half.

(Imagine the heart shape with a vertical line drawn exactly through its middle, top to bottom. Label this line "Line of Symmetry".)

In this illustration, the heart shape has line symmetry, and the vertical fold line represents its single line of symmetry because folding along this line results in the left and right sides matching exactly.