Overview
Introduction: This chapter introduces symmetry — a basic geometric idea about balanced shapes and mirror images. Students learn what it means for a figure to be symmetric about a line (line or mirror symmetry) and how to locate and draw the line(s) of symmetry. Importance: Symmetry builds visual reasoning and geometric intuition important for higher mathematics, art, design and observing patterns in nature and architecture. It strengthens skills like reflection, measurement and spatial visualization. Key themes: mirror images and reflection; identifying and drawing lines (axes) of symmetry; symmetry in letters, numbers and common shapes; completing half-drawings by reflection; counting lines of symmetry for simple figures (triangle types, quadrilaterals, regular polygons, circle). Activities: paper-folding, tracing, and drawing to test symmetry and create symmetric designs. What the student will learn: recognise symmetric and non-symmetric figures; use paper folding or drawing to find mirror lines; draw and label axes of symmetry; determine the number of symmetry lines for standard shapes (e.g., equilateral triangle — 3, isosceles triangle — 1, scalene triangle — 0; square — 4,…
Learning Objectives
- Define line symmetry and related terms such as mirror line and mirror image
- Explain the difference between symmetrical and asymmetrical figures
- Identify the line(s) of symmetry in given plane figures and shapes
- Determine the number of lines of symmetry in regular polygons and common 2-D shapes
- Draw line(s) of symmetry on figures presented on square and triangular grids
- Complete a given half-figure to make it symmetric about a specified axis
- Construct mirror images of simple figures across vertical, horizontal, and diagonal axes using grid or tracing methods
- Classify letters, numerals, and everyday objects according to their lines of symmetry
Topics in this chapter
8 topics · tap a topic title to jump straight to it.
Introduction to Symmetry
Introduction to Symmetry
Key Point: Number of lines of symmetry for a regular n-sided polygon = n
What is symmetry?
A figure is said to be symmetric if one part is exactly like the other part when a line (called the line or axis of symmetry) is drawn so that the two parts match exactly. In other words, if you fold the figure along that line, the two halves will coincide.
Key ideas:
- Line (reflection) symmetry: A line that divides a shape into two mirror-image halves is called a line (or axis) of symmetry.
- Mirror image: Each point on one side of the axis has a corresponding point at the same distance on the other side.
- Point symmetry (rotational by 180°): A shape has point symmetry if rotating it 180° about a point (center) gives the same shape. (Introduced briefly for contrast.)
- Regular polygons: Many regular polygons have several lines of symmetry. A regular n-sided polygon has n lines of symmetry.
How to test symmetry: Fold test: fold the paper along a suspected line of symmetry — if the halves match exactly, it is a line of symmetry. Mirror test: place a mirror on the suspected axis — the reflected half should complete the shape.
Common counts of lines of symmetry (useful examples):
- Square: 4 lines of symmetry
- Rectangle (non-square): 2 lines of symmetry
- Equilateral triangle: 3 lines of symmetry
- Isosceles triangle: 1 line of symmetry
- Scalene triangle: 0 lines of symmetry
- Regular n-gon: n lines of symmetry
- Circle: infinitely many lines of symmetry (every diameter is an axis)
Why symmetry matters: Symmetry appears in nature, art, architecture and helps in design, pattern recognition and simplification of geometry problems.
- Butterfly wings — vertical line of symmetry down the middle of the body.
- Human face — approximately vertical symmetry (eyes, ears, cheeks) though faces are not perfectly symmetric.
- Leaves — many leaves are symmetric along the midrib (central vein).
- Geometric shapes: square (4 axes), equilateral triangle (3 axes), rectangle (2 axes).
- A circular plate — any diameter is an axis (infinite lines of symmetry).
- Letters and digits: A, H, I, M, O, T, U, V, W, X, Y have vertical symmetry (some also have horizontal).
- \[Number of lines of symmetry for a regular n-sided polygon = n\]
- \[A circle has infinitely many lines of symmetry (every diameter is an axis).\]
- \[Reflection (mirror) across the vertical line x = a: a point (x\]\[y) maps to (2a - x\]\[y).\]
- \[Reflection across the horizontal line y = b: a point (x\]\[y) maps to (x, 2b - y).\]
- \[Reflection across the line y = x: (x\]\[y) maps to (y\]\[x).\]
- \[Point (180° rotational) symmetry about the origin: (x\]\[y) maps to (−x, −y).\]
Line (Reflection) Symmetry
Line (Reflection) Symmetry
Key Point: Reflection across the y-axis: (x, y) -> (-x, y).
What is line (reflection) symmetry?
Line symmetry (also called reflection symmetry) means a figure can be folded or reflected across a line so that the two halves match exactly. The dividing line is called the line of symmetry or axis of symmetry. Each point on one half has a corresponding point on the other half at the same distance from the axis, forming mirror images.
Key properties
- The axis of symmetry splits the figure into two congruent mirror-image parts.
- Each point and its reflected image are equidistant from the axis.
- The segment joining a point and its image is perpendicular to the axis and its midpoint lies on the axis.
Types of symmetry lines
- Vertical line of symmetry (example: letter A if symmetric)
- Horizontal line of symmetry (example: a simple smiling face drawn symmetrically)
- Oblique (slanted) line of symmetry
How to check for line symmetry
- Folding test: fold along a candidate line — if halves match, it is a symmetry line.
- Mirror test: place a mirror on the line — if the reflected image fills the other half exactly, it is a symmetry line.
- Geometric test: join corresponding points; the perpendicular bisector of the segment through a point and its image is the axis.
Number of symmetry lines for common shapes
- Circle: infinitely many axes (any line through center)
- Regular n-gon: n axes
- Square: 4 axes (2 diagonals + 2 medians)
- Rectangle (not a square): 2 axes (two medians)
- Equilateral triangle: 3 axes; isosceles triangle: 1 axis; scalene triangle: 0 axes
- Butterfly wings — usually a vertical line of symmetry down the middle.
- Human face (approximate) — vertical symmetry through the nose.
- Leaf shapes — many leaves show a vertical axis of symmetry.
- Taj Mahal and many buildings — vertical symmetry in architecture.
- Square — four lines of symmetry (two diagonals and two medians).
- Rectangle — two lines of symmetry (vertical and horizontal medians).
- \[Reflection across the y-axis: (x\]\[y) -> (-x\]\[y).\]
- \[Reflection across the x-axis: (x\]\[y) -> (x, -y).\]
- \[Reflection across the vertical line x = a: (x\]\[y) -> (2a - x\]\[y).\]
- \[Reflection across the horizontal line y = b: (x\]\[y) -> (x, 2b - y).\]
- \[Reflection across the line y = x: (x\]\[y) -> (y\]\[x).\]
- \[Midpoint property: If A(x\]\[y) maps to A'(x'\]\[y')\]\[then midpoint M = ((x + x')/2\]\[(y + y')/2) lies on the axis of symmetry.\]
Lines of Symmetry in Common Plane Figures
Lines of Symmetry in Common Plane Figures
Key Point: Regular n-gon: number of lines of symmetry = n.
Definition: A line of symmetry (axis of symmetry) of a plane figure is a line such that if the figure is folded along it, the two parts match exactly. Equivalently, reflection of the figure across that line maps the figure onto itself.
How to recognize a line of symmetry:
- Fold test: If folding along the line makes the two halves coincide, it is a line of symmetry.
- Mirror test: Place a mirror on the line; the reflection of one half should show the other half.
- Analytic test: In coordinate geometry, a symmetry line is a line whose reflection maps every vertex to another vertex of the shape.
Common plane figures and their lines of symmetry (short):
- Circle: infinitely many (every line through the centre).
- Regular n-gon (regular polygon with n sides): n lines (each through a vertex and centre or through midpoints depending on n).
- Square: 4 lines (2 medians through midpoints of opposite sides and 2 diagonals).
- Rectangle (non-square): 2 lines (perpendicular bisectors through centre — one vertical, one horizontal if axis-aligned).
- Equilateral triangle: 3 lines (each median is an axis).
- Isosceles triangle (not equilateral): 1 line (the median from apex to base midpoint).
- Scalene triangle: 0 lines (no sides or angles equal).
- Rhombus (not square): 2 lines (its diagonals).
- Kite (with reflection symmetry): 1 line (the symmetry axis through unequal vertex pair); some special kites may have 2.
- Parallelogram (non-rectangular, non-rhombic): 0 lines of symmetry.
Why lines of symmetry matter: They help classify shapes (regular vs. irregular), solve construction and congruence problems, and appear in nature and design (balancing, aesthetics, engineering).
How to draw a symmetry line: For many figures draw the line joining a vertex to the midpoint of the opposite side (for triangles), or draw diagonals and perpendicular bisectors (for quadrilaterals). For regular polygons draw lines from centre to vertices or midpoints as required.
- Circle — a round plate or wheel: infinite symmetry lines (every diameter).
- Square — a chessboard square or a photo frame: 4 symmetry lines (two diagonals and two medians).
- Rectangle — a door or book cover: 2 symmetry lines (vertical and horizontal through centre).
- Equilateral triangle — a triangular road sign (if equilateral): 3 symmetry lines (medians).
- Isosceles triangle — a triangle roof cross-section: 1 symmetry line (through apex and midpoint of base).
- Regular hexagon — a honeycomb cell or some nuts (bolt head): 6 symmetry lines (through opposite vertices or midpoints).
- \[Regular n-gon: number of lines of symmetry = n.\]
- \[Circle: number of lines of symmetry = infinite (every diameter is an axis).\]
- \[Square: 4\]\[Rectangle (non-square): 2\]\[Equilateral triangle: 3\]\[Isosceles triangle (not equilateral): 1\]\[Scalene triangle: 0.\]
- \[Rhombus (not square): 2 (its diagonals)\]\[Kite (with bilateral symmetry): 1.\]
- \[Parallelogram (general): 0 lines of symmetry unless it is a rectangle or rhombus (special cases).\]
Symmetry by Paper Folding and Cutting
Symmetry by Paper Folding and Cutting
Key Point: Reflection across y-axis: (x, y) → (−x, y)
What is symmetry? Symmetry is a property of a figure when one half is a mirror image of the other half. In plane figures, this mirror line is called a line (or axis) of symmetry.
Symmetry by paper folding: Paper folding is a simple way to find lines of symmetry. When you fold a paper so that one edge or part exactly matches another, the fold line is a line of symmetry for the shape on the paper. If after folding, all points and edges match exactly, the shape is symmetric about that fold.
Steps to test or make symmetry by folding:
- Fold the paper so that two matching parts coincide (e.g., fold a rectangle along its midline). The crease is a candidate axis of symmetry.
- Open the paper. If the crease divides the shape into two equal mirror-image halves, it is a line of symmetry.
- For cutting activities (like making snowflakes), fold the paper several times, cut a pattern on the folded paper, and then unfold. The resulting pattern has repeated symmetric sections (rotational and reflection symmetry depending on how you fold).
Why folding works: Folding moves each point to its mirror point across the fold line. If every point on one side maps to a corresponding point on the other side, the fold line is an axis of symmetry.
Types and quick examples of results:
- Line symmetry (reflectional): shapes like isosceles triangle, rectangle, circle (infinite axes) — visible by folding along appropriate lines.
- Rotational symmetry from cutting folded sectors: if you fold a paper into equal sectors and cut a design, the unfolded figure may show rotational symmetry (for example, fold into 6 equal sectors to get 6-fold symmetry).
Classroom activity idea: Fold a square twice (once along vertical midline and once along horizontal midline). Cut a small shape through all layers near the centre. Unfold to see four identical holes arranged symmetrically. Change folds (diagonal, multiple sectors) to explore other symmetries.
How to check symmetry on a coordinate grid (visual idea): Plot a point (x, y). Its mirror across the y-axis is (−x, y); across the x-axis is (x, −y); across the line y = x is (y, x). Draw the axis and confirm distances to the axis are equal for each pair of mirrored points.
Key tips:
- Every fold-line you make is a candidate axis; test by checking whether halves match exactly.
- Regular polygons: folding from center to vertices can show all axes of symmetry (for a regular n-gon there are n axes).
- Careful, some shapes look symmetric but are only approximately so (e.g., many faces); folding reveals exact symmetry.
- Fold a rectangle along its vertical midline — the crease is an axis of symmetry; the two halves match exactly.
- Fold a square along its diagonals and midlines. A square has 4 lines of symmetry (two diagonals and two midlines).
- Fold a circle (approximate) by folding paper so edges match — a circle has infinitely many axes of symmetry (any diameter).
- Make a paper snowflake: fold a paper into several equal sectors, cut a small pattern, then unfold to get a design with rotational and reflection symmetry.
- Butterfly wings: folding an image of a butterfly along its body line shows near-perfect mirror symmetry.
- \[Reflection across y-axis: (x\]\[y) → (−x\]\[y)\]
- \[Reflection across x-axis: (x\]\[y) → (x, −y)\]
- \[Reflection across line y = x: (x\]\[y) → (y\]\[x)\]
- \[Rotation by 180° about origin (point symmetry): (x\]\[y) → (−x, −y)\]
- \[Number of lines of symmetry of a regular n-gon = n (e.g.\]\[equilateral triangle: 3\]\[square: 4\]\[regular pentagon: 5)\]
- \[Rectangle: 2 lines of symmetry\]\[Circle: infinitely many lines of symmetry\]\[Scalene triangle: 0 lines of symmetry\]\[Isosceles triangle: 1 line of symmetry.\]
Symmetry in Alphabets, Numbers and Designs
Symmetry in Alphabets, Numbers and Designs
Key Point: Definition (line symmetry): A figure has line symmetry if there exists a line such that reflection across that line maps the figure onto itself.
What is symmetry? Symmetry (line or reflection symmetry) means a figure can be folded or reflected about a line (called the line of symmetry) so that the two halves match exactly. A mirror placed along that line shows identical halves.
Types of symmetry covered in Class 6:
- Line (reflection) symmetry: The figure is identical on both sides of a straight line (axis). The axis may be vertical, horizontal or diagonal.
- Rotational symmetry (basic idea): A figure has rotational symmetry if it looks the same after rotation by some angle less than 360°. The number of times it maps onto itself in one full turn is the order of rotational symmetry.
Symmetry in alphabets and numbers: Whether a letter or number is symmetric depends on its style (font). In a simple block (sans-serif) style:
- Vertical line symmetry (mirror left-right): A, H, I, M, O, T, U, V, W, X, Y
- Horizontal line symmetry (mirror top-bottom): H, I, O, X (these letters are symmetric about a horizontal midline in block form)
- Both vertical and horizontal symmetry: H, I, O, X (they have two perpendicular lines of symmetry)
- Numbers (block style): 0 and 8 usually have both vertical and horizontal symmetry; other digits depend on font.
Symmetry in designs and shapes: Regular polygons and many decorative patterns have symmetry. For example, a regular equilateral triangle has 3 lines of symmetry, a square has 4, and a circle has infinitely many. Simple designs like butterflies, leaves and mandalas often show vertical symmetry.
How to test for symmetry (classroom activities):
- Fold-and-check: Fold a printed figure along a suspected axis — if halves match, it's symmetric.
- Mirror test: Place a small mirror on the suspected axis; if the reflected half completes the figure, that axis is a line of symmetry.
Why this matters (real life): Symmetry appears in nature (flowers, leaves, animals), engineering (bridges, wheels), architecture (windows, facades), signs and logos. Recognising symmetry helps in design, art, and geometry.
- Letters: Vertical symmetry — A, H, I, M, O, T, U, V, W, X, Y (block letters). Horizontal symmetry — H, I, O, X (in many block fonts).
- Numbers: 0 and 8 usually have both vertical and horizontal symmetry (in standard digital-style forms).
- Shapes: A regular equilateral triangle has 3 lines of symmetry; a square has 4; a regular pentagon has 5; a circle has infinitely many lines of symmetry.
- Nature: A butterfly's wings (approximate vertical symmetry), many flowers (radial symmetry), leaves (often one line of symmetry).
- Man-made: Road signs (some are symmetric), building facades, car grills and logos (BMW, Toyota have symmetric elements).
- \[Definition (line symmetry): A figure has line symmetry if there exists a line such that reflection across that line maps the figure onto itself.\]
- \[For a regular n-sided polygon: number of lines of symmetry = n.\]
- \[Order of rotational symmetry (k): k = number of times the figure matches itself during a full 360° turn\]\[The smallest angle of rotation that maps the figure onto itself = 360° / k.\]
- \[Relationship (regular polygon): order of rotational symmetry = n and smallest rotation angle = 360° / n.\]
Drawing and Completing Symmetric Figures
Drawing and Completing Symmetric Figures
Key Point: Midpoint formula (useful when the axis is the perpendicular bisector of a segment joining corresponding points): midpoint of (x1,y1) and (x2,y2) = ((x1+x2)/2, (y1+y2)/2).
What is symmetry? Symmetry in a figure means one part of the figure is a mirror image of the other part. When a figure can be folded along a line so that the two halves match exactly, that line is called the axis (or line) of symmetry.
Line (mirror) symmetry: A figure has line symmetry if there exists a line such that every point on one side of the line has a matching point on the other side at the same distance from the line. The matching halves are mirror images.
How to draw or complete a symmetric figure (step-by-step):strong>
- Identify the axis of symmetry (a vertical, horizontal, or slanted line) shown on the paper.
- If you have a set of points or one half of a shape, take each key point and draw a perpendicular from the point to the axis. Measure its distance to the axis and mark a point the same distance on the opposite side. (On paper, this can be done by folding along the axis or using a ruler and right angle.)
- Repeat for all important points, then join the reflected points in the same order to complete the figure.
- Check that each point and its reflected point are at equal distances from the axis, and corresponding line segments preserve length and angles.
Using fold or mirror method: Place a mirror on the axis or fold the paper along the axis. The visible shape in the mirror (or the folded half) shows how the other half should look. This is the simplest method for classroom use.
Compass-and-ruler construction method (classroom geometric method): For a point P not on the axis, draw any radius using compass centered at P that meets the axis at two places; alternatively construct a perpendicular from P to the axis, find the foot of the perpendicular, measure distance and mark the reflected point P' the same distance on the other side. Repeat for other points and join.
Things to remember: Reflection preserves lengths and angles — corresponding sides and angles remain equal. The axis of symmetry is the perpendicular bisector of the segment joining corresponding points.
- Butterfly wings: draw one half of a wing and complete the other half by reflecting across the central line of the body.
- Human face sketch: eyes, nose and mouth positions can be completed by reflecting features across the vertical midline.
- Leaves and flowers: many leaves and some petals are symmetric about their midrib or central line.
- Road signs and logos: e.g., the plus sign (+) has vertical and horizontal symmetry; reflect one quadrant to complete others.
- \[Midpoint formula (useful when the axis is the perpendicular bisector of a segment joining corresponding points): midpoint of (x1,y1) and (x2,y2) = ((x1+x2)/2\]\[(y1+y2)/2).\]
- \[Reflection across the y-axis: a point (x,y) reflects to (-x,y).\]
- \[Reflection across the x-axis: a point (x,y) reflects to (x,-y).\]
- \[Reflection across the line y = x: a point (x,y) reflects to (y,x).\]
- \[Perpendicular bisector property: If A and A' are corresponding points\]\[the axis of symmetry is the perpendicular bisector of segment AA' (so it meets AA' at its midpoint and at right angle).\]
Properties and Problem Solving
Properties and Problem Solving
Key Point: Number of lines of symmetry of a regular n-gon = n
What is symmetry? Symmetry means a balanced arrangement. An object is symmetric if it can be divided by a line (called a line of symmetry) into two parts that are mirror images of each other. In Class 6 we mainly study line (reflection) symmetry and basic rotational symmetry ideas.
Key properties of line symmetry
- The line of symmetry (mirror line) divides a figure into two congruent halves that are mirror images.
- Each point on one half has a corresponding point on the other half such that the mirror line is the perpendicular bisector of the segment joining the two points.
- Distance to the mirror line is equal for corresponding points; the perpendicular from each point to the mirror line meets the line at the same midpoint for the pair.
- A figure can have 0, 1, 2, 3, 4, or infinitely many lines of symmetry (e.g., circle = infinite; regular n-gon = n lines).
Rotational symmetry (basic idea)
- A figure has rotational symmetry of order k if it looks the same after rotating by 360/k degrees about its center. Regular polygons are good examples (equilateral triangle: order 3; square: order 4).
Problem-solving strategies
- Visual test (folding): Fold the paper along a guessed line — if the parts match, it is a line of symmetry.
- Perpendicular-bisector method: To check if a line is a mirror line, pick a point A on one side and its image A' on the other. The mirror line must be the perpendicular bisector of AA'.
- Use tracing/overlay: Trace a figure, fold or flip the trace to see if it matches.
- Use coordinates for precise checks: Reflect a point (x,y) across axes: across x-axis -> (x, -y); across y-axis -> (-x, y). For a vertical line x = c, the reflection of (x,y) is (2c-x, y).
- Count symmetry lines for regular polygons: regular n-gon has n lines of symmetry passing through vertices and/or midpoints depending on n.
Tips for solving questions
- Always draw clearly and label points. Use a ruler and compass when constructing perpendicular bisectors.
- Start by checking easy candidates: vertical, horizontal, and diagonal lines for common shapes (rectangle, square, triangle).
- Remember special cases: a square has 4 lines; a rectangle (non-square) has 2; circle has infinite; regular pentagon has 5, etc.
Short worked idea: To find the line of symmetry of an isosceles triangle, draw the perpendicular from the apex to the base — it will bisect the base and be the axis of symmetry.
- Butterfly: The line down the middle of the body is a line of symmetry—wing patterns on one side mirror the other.
- Human face (approximate): Nose and chin form a central vertical mirror line—left and right sides are similar.
- Square: Has 4 lines of symmetry (two diagonals and two medians through opposite sides).
- Rectangle (non-square): Has 2 lines of symmetry (vertical and horizontal medians).
- Equilateral triangle: Has 3 lines of symmetry, each passing through a vertex and the midpoint of the opposite side.
- \[Number of lines of symmetry of a regular n-gon = n\]
- \[Angle of smallest rotation that maps a regular n-gon onto itself = 360° / n (rotational symmetry of order n)\]
- \[Reflection across x-axis: (x\]\[y) → (x, -y)\]
- \[Reflection across y-axis: (x\]\[y) → (-x\]\[y)\]
- \[Reflection across vertical line x = c: (x\]\[y) → (2c - x\]\[y)\]
- \[Perpendicular-bisector property: If L is a mirror line and A\]\[A' are corresponding points\]\[then L ⟂ AA' and L passes through midpoint of AA'.\]
Symmetry in Nature and Real Life
Symmetry in Nature and Real Life
Key Point: Reflection across the x-axis: (x, y) → (x, -y).
What is symmetry? Symmetry means a balanced and regular arrangement. An object is symmetric if one part is a mirror image of another part or if it repeats after a rotation. In Class 6, we mainly study line (mirror) symmetry and basic rotational and radial ideas.
Types seen in nature and life:
- Line (bilateral) symmetry: One line divides a figure into two identical mirror halves. Many animals and humans show bilateral symmetry (left and right sides).
- Radial symmetry: Parts repeat around a central point (e.g., flowers, starfish, many fruits). You can draw many lines through the centre that make similar parts.
- Rotational symmetry: A figure looks the same after rotating by a certain angle around a center (e.g., some wheels, regular polygons).
How to test symmetry: Fold or use a mirror along a suspected line of symmetry. If both halves match exactly, the line is an axis of symmetry. For rotation, rotate the figure about its centre and see if it matches itself before completing a full turn.
Why it matters: Symmetry helps in biology (balance and movement), art and design (beauty and pattern), architecture (strength and proportion), and technology (aerodynamics and manufacturing).
- Butterfly: bilateral (mirror) symmetry — left wing mirrors the right wing.
- Leaf: many leaves have a central vein which acts as a line of symmetry.
- Flowers (e.g., daisy): radial symmetry — petals arranged around the centre.
- Starfish: radial symmetry — body parts repeated around a central point.
- Human face and body: approximate bilateral symmetry (left and right sides).
- Snowflakes: often show six-fold radial symmetry.
- \[Reflection across the x-axis: (x\]\[y) → (x, -y).\]
- \[Reflection across the y-axis: (x\]\[y) → (-x\]\[y).\]
- \[Reflection through the origin: (x\]\[y) → (-x, -y).\]
- \[Reflection across the line y = x: (x\]\[y) → (y\]\[x).\]
- \[Rotation of a point (x\]\[y) about the origin by angle θ: (x'\]\[y') = (x cosθ - y sinθ\]\[x sinθ + y cosθ).\]
- \[Order of rotational symmetry = number of times a figure maps onto itself during a full 360° turn\]\[If the smallest rotation that maps the figure to itself is θ°\]\[then order = 360° / θ° (e.g.\]\[square: θ = 90°\]\[order = 4\]\[rectangle: θ = 180°\]\[order = 2).\]
Key Concepts
- Symmetry
- A property where a shape or object looks the same after a specific transformation (like reflection or rotation).
- Line of symmetry
- A straight line that divides a figure into two identical mirror-image halves.
- Axis of symmetry
- Another name for a line of symmetry; the line about which a figure is symmetric.
- Reflection
- A transformation that flips a figure across a line (mirror) to produce its mirror image.
- Mirror image
- The result of reflecting a figure across a line; it looks like what you see in a mirror.
- Reflectional symmetry
- Symmetry that occurs when a figure is identical to its mirror image across a line.
- Rotational symmetry
- When a figure can be rotated (less than a full turn) about a point and still look the same.
- Order of rotational symmetry
- The number of distinct positions in one full turn where the figure matches itself.
- Angle of rotation
- The smallest angle through which a figure is rotated to map onto itself.
- Centre of rotation
- The fixed point about which a figure is rotated to test for rotational symmetry.
- Point symmetry
- A type of symmetry where every part has a matching part directly opposite through a central point (also called centre of symmetry).
- Bilateral symmetry
- Symmetry in which a single line (midline) divides a body or figure into two mirror halves, common in living organisms.
- Radial symmetry
- Symmetry around a central point so that multiple lines through the centre give mirror images.
- Asymmetry
- When a figure or object has no symmetry; it does not match itself under reflection or rotation (other than full turn).
- Regular polygon
- A polygon with all sides equal and all interior angles equal; it has multiple symmetries.
- Congruent
- Two shapes that have the same size and shape; one can be moved to match the other using translations, rotations or reflections.
- Perpendicular bisector
- A line that cuts a segment into two equal parts at a right angle; often a line of symmetry for simple shapes.
- Folding (fold test)
- A method to find lines of symmetry by folding a paper figure to see if the halves match exactly.
- Symmetry operation
- A movement (reflection, rotation, or combination) that maps a figure onto itself.
- Symmetrical figure
- A figure that has one or more symmetry elements (lines or centres) so that part(s) match under symmetry operations.
Practice Questions
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How many lines of symmetry does an equilateral triangle have? / एक समबाहु त्रिभुज में कितनी सममिति रेखाएँ होती हैं? (a) 0 / 0 (b) 1 / 1 (c) 2 / 2 (d) 3 / 3
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(d) 3 / 3 — An equilateral triangle has all three sides equal, so it has 3 lines of symmetry — one from each vertex to the midpoint of the opposite side. A regular n-gon has n lines of symmetry; n=3 gives 3. / समबाहु त्रिभुज की सभी तीन भुजाएँ बराबर होती हैं, इसलिए इसमें 3 सममिति रेखाएँ होती हैं — प्रत्येक शीर्ष से विपरीत भुजा के मध्य बिंदु तक।
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Which of the following letters has both a vertical and a horizontal line of symmetry? / निम्न में से किस अक्षर में लंबवत् और क्षैतिज दोनों सममिति रेखाएँ होती हैं? (a) A / A (b) T / T (c) H / H (d) V / V
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(c) H / H — In block (sans-serif) font, the letter H is symmetric about both a vertical midline and a horizontal midline, giving it two perpendicular lines of symmetry. / ब्लॉक फ़ॉन्ट में, अक्षर H लंबवत् और क्षैतिज दोनों मध्य रेखाओं के बारे में सममित है, जिससे इसमें दो लंबवत् सममिति रेखाएँ होती हैं।
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A circle has ______ lines of symmetry. / एक वृत्त में ______ सममिति रेखाएँ होती हैं। (a) 2 / 2 (b) 4 / 4 (c) 8 / 8 (d) Infinitely many / अनगिनत
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(d) Infinitely many / अनगिनत — Every line passing through the centre (every diameter) of a circle is a line of symmetry. Since there are infinitely many diameters, a circle has infinitely many lines of symmetry. / वृत्त के केंद्र से गुजरने वाली प्रत्येक रेखा (प्रत्येक व्यास) एक सममिति रेखा है। अनगिनत व्यास होने से वृत्त में अनगिनत सममिति रेखाएँ होती हैं।
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If a point P(3, 4) is reflected across the y-axis, the image is P'(______, ______). / यदि बिंदु P(3, 4) को y-अक्ष के पार परावर्तित किया जाए, तो प्रतिबिंब P'(______, ______) होगा।
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P'(−3, 4) / P'(−3, 4) — Reflection across the y-axis changes the sign of the x-coordinate and keeps the y-coordinate unchanged: (x, y) → (−x, y). So (3, 4) → (−3, 4). / y-अक्ष के पार परावर्तन में x-निर्देशांक का चिह्न बदल जाता है और y-निर्देशांक अपरिवर्तित रहता है: (x, y) → (−x, y)। अतः (3, 4) → (−3, 4)।
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A scalene triangle has ______ line(s) of symmetry. / एक विषमबाहु त्रिभुज में ______ सममिति रेखा(एँ) होती है/हैं।
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0 / 0 — A scalene triangle has all three sides different in length and all three angles different. No fold line can make the two halves match exactly, so it has no line of symmetry. / विषमबाहु त्रिभुज की तीनों भुजाएँ और तीनों कोण भिन्न होते हैं। कोई भी मोड़-रेखा दो हिस्सों को पूरी तरह मिला नहीं सकती, अतः इसमें कोई सममिति रेखा नहीं होती।
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True or False: A rectangle (non-square) has 4 lines of symmetry. / सत्य या असत्य: एक आयत (जो वर्ग नहीं है) में 4 सममिति रेखाएँ होती हैं।
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False / असत्य — A rectangle (non-square) has only 2 lines of symmetry: one vertical median and one horizontal median through its centre. Its diagonals are NOT lines of symmetry (they do not produce matching halves). / एक आयत (जो वर्ग नहीं है) में केवल 2 सममिति रेखाएँ होती हैं: एक लंबवत् और एक क्षैतिज माध्यिका। इसके विकर्ण सममिति रेखाएँ नहीं हैं।
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In nature, which of the following shows radial symmetry? / प्रकृति में निम्न में से कौन-सा अरीय सममिति (radial symmetry) दर्शाता है?
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A flower / एक फूल — Many flowers have petals arranged equally around a central point, showing radial (multi-axis) symmetry. Multiple lines through the centre divide the flower into identical parts. Examples: sunflower, daisy. / कई फूलों की पंखुड़ियाँ केंद्रीय बिंदु के चारों ओर समान रूप से व्यवस्थित होती हैं, जो अरीय सममिति दर्शाती हैं। उदाहरण: सूरजमुखी, डेज़ी।
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Draw a horizontal axis of symmetry on the letter 'B'. If you fold 'B' along this axis, what do you observe? / अक्षर 'B' पर एक क्षैतिज सममिति रेखा खींचें। इस रेखा के साथ 'B' को मोड़ने पर आप क्या देखते हैं?
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The upper and lower halves of 'B' match exactly / 'B' का ऊपरी और निचला आधा भाग बिल्कुल मेल खाते हैं — The letter B (in standard block form) has a horizontal line of symmetry through its middle. When folded along this line, the two bumps (upper and lower) coincide perfectly, showing mirror symmetry. / अक्षर B (मानक ब्लॉक रूप में) अपने मध्य में एक क्षैतिज सममिति रेखा रखता है। इस रेखा के साथ मोड़ने पर ऊपरी और निचले उभार पूरी तरह मेल खाते हैं।
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