Overview
Introduction: Symmetry is the idea that a shape or object can be divided or transformed so that two parts match exactly. In Class 7 (Mathematics – VII, Chapter: Symmetry) students explore mirror (line) symmetry and introduce rotational symmetry through visual activities, paper folding, and simple constructions. Importance: Symmetry develops spatial reasoning and geometric understanding that are foundational for later topics (congruence, transformations, geometry of polygons). It connects mathematics to art, nature, architecture and everyday observation, sharpening students' ability to recognise patterns and reason about shapes. Key themes: line (mirror) symmetry and axes of symmetry; identifying and drawing lines of symmetry for letters, shapes and plane figures; using paper folding and mirrors as tests for symmetry; introduction to rotational symmetry and order of symmetry; symmetry of regular polygons and circles; constructing symmetric figures and symmetric designs. What the student will learn: Students will learn to define symmetry in simple terms, identify whether a figure is symmetric, and draw its line(s) of symmetry. They will practise paper folding and mirror tests to…
Learning Objectives
- Define line symmetry and mirror image for plane figures.
- Explain the concept of line of symmetry using paper folding and mirror methods.
- Identify and mark all lines of symmetry in given 2-D shapes (triangles, quadrilaterals, regular polygons, circle).
- Draw lines of symmetry for irregular figures and alphabetic letters.
- Construct mirror images of simple figures and letters across a given line of symmetry.
- Determine the number of lines of symmetry for regular polygons and common shapes.
- Describe rotational symmetry and state the order of rotational symmetry for regular polygons.
- Apply symmetry to complete half-drawn figures and repeating patterns accurately.
Topics in this chapter
7 topics · tap a topic title to jump straight to it.
Introduction to Symmetry
Introduction to Symmetry
Key Point: Lines of symmetry of a regular n-sided polygon = n
What is symmetry? Symmetry means exact matching in size, shape and position of parts on opposite sides of a dividing line or around a centre. If a figure can be folded or reflected so that one part exactly covers the other, the figure is called symmetric.
Line (mirror) symmetry: A figure has line symmetry if there is a straight line (called the line of symmetry or axis of symmetry) that divides it into two mirror-image halves. Each point on one side has a corresponding point on the other side at the same distance from the line.
Rotational symmetry: A figure has rotational symmetry if it can be rotated (about a point) by a certain angle less than 360° and still look the same. The number of distinct orientations in which the figure maps onto itself is called the order of rotational symmetry. For a regular n-sided polygon, the order is n.
Symmetry in the coordinate plane: In graphs, symmetry is often described with respect to the y-axis, x-axis or origin. If replacing x by −x leaves the rule unchanged (f(−x)=f(x)), the graph is symmetric about the y-axis (even function). If f(−x)=−f(x), the graph is symmetric about the origin (odd function).
How to test for symmetry (simple methods):
- Folding: Fold the paper along a candidate line. If the halves match exactly, that line is a line of symmetry.
- Reflection: Imagine a mirror placed along the candidate line; if the reflected half matches the other, the line is a symmetry axis.
- Coordinate test: For graphs use algebraic substitution (f(−x)=f(x) or f(−x)=−f(x)).
Why symmetry matters: It helps in recognizing shapes quickly, solving geometry problems, understanding patterns in nature and design, and simplifying algebraic work on graphs.
- Butterfly: usually has a vertical line of symmetry along its body (mirror image wings).
- Human face: approximate vertical symmetry (left and right halves similar).
- Rectangle: 2 lines of symmetry (two perpendicular lines through midpoints of opposite sides).
- Square: 4 lines of symmetry (two diagonals and two midlines) and rotational symmetry of order 4 (angles 90°, 180°, 270°, 360°).
- Equilateral triangle: 3 lines of symmetry (each through a vertex and opposite side midpoint) and rotational order 3 (120°).
- Circle: infinite lines of symmetry (any diameter is an axis) and infinite rotational symmetry.
- \[Lines of symmetry of a regular n-sided polygon = n\]
- \[Order of rotational symmetry of a regular n-sided polygon = n\]
- \[Angle of smallest rotation that maps a regular n-gon onto itself = 360° / n\]
- \[Even function (y-axis symmetry): f(−x) = f(x)\]
- \[Odd function (origin symmetry): f(−x) = −f(x)\]
Line (Mirror) Symmetry
Line (Mirror) Symmetry
Key Point: Number of lines of symmetry for a regular n-gon = n.
Definition: A figure has line (or mirror) symmetry if there exists a straight line (called the line of symmetry) such that the figure is exactly the same on both sides of that line. The line of symmetry acts like a mirror: one half is the mirror image (reflection) of the other half.
Key ideas and properties:
- Line of symmetry (also called axis of symmetry or mirror line) divides a figure into two congruent mirror-image parts.
- If two corresponding points on the figure are mirror images, the line of symmetry is the perpendicular bisector of the segment joining those points.
- A figure can have zero, one, or many lines of symmetry. For example: some letters (A, H, M, T, U, V, W, X, Y) have one or more lines; a circle has infinitely many lines of symmetry; a regular n-gon has n lines.
- To test symmetry physically, fold the paper along a candidate line: if the two halves match exactly, that line is a line of symmetry.
- In coordinate geometry, reflecting a point across a line gives its mirror-image coordinates (see formulas below).
Why it matters: Line symmetry helps in identifying shapes, solving geometry problems (congruence, construction), and appears often in art, architecture and nature.
- Butterfly wings: almost exact mirror images on either side of the body (one line of symmetry along the body axis).
- Human face (approximate symmetry): the vertical midline of the face is a mirror line (approximate symmetry).
- Square: 4 lines of symmetry (two diagonals and two medians).
- Rectangle (non-square): 2 lines of symmetry (two medians — vertical and horizontal).
- Equilateral triangle: 3 lines of symmetry (each median is also an axis of symmetry).
- Isosceles triangle: 1 line of symmetry (the perpendicular from the apex to the base).
- \[Number of lines of symmetry for a regular n-gon = n.\]
- \[Circle: infinite lines of symmetry (any line through the center).\]
- \[Reflection across x-axis: (x\]\[y) → (x, −y).\]
- \[Reflection across y-axis: (x\]\[y) → (−x\]\[y).\]
- \[Reflection across the line y = x: (x\]\[y) → (y\]\[x).\]
- \[General reflection of point (x0\]\[y0) across the line ax + by + c = 0 gives x' = x0 − 2a(ax0 + by0 + c)/(a^2 + b^2)\]\[y' = y0 − 2b(ax0 + by0 + c)/(a^2 + b^2).\]
Lines of Symmetry of Common Plane Figures
Lines of Symmetry of Common Plane Figures
Key Point: Lines of symmetry of a regular n-gon = n.
Line (or mirror) symmetry means a figure can be folded along a line so that the two halves match exactly. That line is called a line of symmetry. To identify a line of symmetry, imagine reflecting the figure across a line; if the figure maps onto itself, the line is a symmetry axis.
Common plane figures and their lines of symmetry (with short reasons):
- Circle: infinitely many. Any line through the center is an axis.
- Regular n-gon (all sides and angles equal): n lines. Each axis goes through a vertex and the opposite side midpoint (or two opposite vertices) depending on n.
- Equilateral triangle: 3 lines. Each axis from a vertex to midpoint of opposite side.
- Isosceles triangle: 1 line. Axis passes through the apex and midpoint of base.
- Scalene triangle: 0 lines. No equal sides/angles, so no mirror axis.
- Square: 4 lines. Two medians (vertical & horizontal) and two diagonals.
- Rectangle (non-square): 2 lines. Vertical and horizontal lines through midpoints of opposite sides (diagonals are not symmetry axes).
- Rhombus (non-square): 2 lines. The diagonals are symmetry axes (they bisect opposite angles).
- Parallelogram (non-rectangular, non-rhombic): 0 lines. Opposite sides equal but no mirror symmetry.
- Kite (general): 1 line. The axis runs through the pair of equal adjacent sides' common vertex and the midpoint of the opposite side.
- Semicircle: 1 line. The perpendicular bisector of the diameter (or the diameter if semicircle is symmetric about it, depending on orientation) — typically the diameter's perpendicular through center is the axis if the flat side is horizontal the axis is vertical through center.
Key idea: Regularity or equal opposite parts give symmetry. To test for a line of symmetry, fold or reflect the figure digitally or trace and compare halves.
- Circle: dinner plate, clock face — infinite axes (any diameter).
- Square: tiled floor tile, chessboard square — 4 symmetry lines (2 medians + 2 diagonals).
- Rectangle: door, book cover — 2 symmetry lines (vertical and horizontal through midpoints).
- Equilateral triangle: triangular road sign or triangular logo — 3 symmetry lines (each vertex to opposite midpoint).
- Isosceles triangle: tent or roof cross-section — 1 symmetry line through apex and base midpoint.
- Rhombus: diamond-shaped window pane — 2 symmetry lines (diagonals).
- \[Lines of symmetry of a regular n-gon = n.\]
- \[Circle: infinitely many symmetry lines (every diameter).\]
- \[Equilateral triangle: 3\]\[Isosceles triangle: 1\]\[Scalene triangle: 0.\]
- \[Square: 4\]\[Rectangle (non-square): 2\]\[Rhombus (non-square): 2\]\[Parallelogram (non-special): 0\]\[Kite: typically 1.\]
- \[General rule: if a reflection across a line maps every vertex to another vertex and edges to edges\]\[that line is a symmetry axis.\]
Symmetry in Alphabets, Numbers and Patterns
Symmetry in Alphabets, Numbers and Patterns
Key Point: Order of rotational symmetry (n): the number of times a figure maps onto itself during a full 360° rotation.
What is symmetry? Symmetry means exact correspondence in size, form and arrangement of parts on opposite sides of a line or around a centre. Two main types used in Class 7 are line (reflection) symmetry and rotational symmetry.
Line (reflection) symmetry: An object has line symmetry if it can be folded along a line (called the line/axis of symmetry) so that the two halves match exactly. The line may be vertical, horizontal or slanted.
Rotational symmetry: An object has rotational symmetry if it can be rotated (less than a full turn) about a fixed point and still look the same. The number of times it matches itself in one full 360° rotation is called the order of rotational symmetry. The smallest angle through which it can be rotated to map onto itself is 360°/order.
How to test symmetry: For line symmetry use a mirror placed along a suspected axis or imagine folding along that line. For rotational symmetry rotate the figure about its centre and see if it coincides with the original.
Symmetry in Alphabets (uppercase): Examples of common reflectional symmetry depend on font style, but typical lists are:
- Vertical axis symmetry: A, H, I, M, O, T, U, V, W, X, Y
- Horizontal axis symmetry: B, C, D, E, H, I, O, X
- Both vertical and horizontal (two axes): H, I, O, X
- Some letters have rotational symmetry of order 2 (180°): H, I, O, S, X, Z (font-dependent)
Note: Symmetry of a letter can vary with typeface; always check with the actual printed shape.
Symmetry in Numbers (digits 0–9): Common symmetry properties (standard digital/printed style):
- Vertical axis symmetry: 0, 1, 8 (often)
- Horizontal axis symmetry: 0, 8
- Rotational symmetry (order 2, 180°): 0, 8
- Digits 6 and 9 are rotational images of each other (180° rotation swaps them)
Symmetry in patterns: Patterns (like rangoli, tiles, mandalas, rosettes) show one or more types of symmetry. A regular polygon (n sides) has n lines of symmetry and rotational symmetry of order n. A square has 4 lines and order 4; a rectangle has 2 lines and order 2; a circle has infinite lines and infinite rotational symmetry.
Why it matters: Recognising symmetry helps in geometry problems, design, art, and real-life tasks such as folding, cutting, logo design and identifying shapes.
- Alphabet: Fold paper through the vertical middle of letter 'A' — both halves match, so 'A' has vertical symmetry.
- Number: Rotate the digit '8' by 180° and you still get '8' — '8' has rotational symmetry of order 2 and both axes of reflection.
- Shape: A regular hexagon can be rotated by 60° increments (360°/6) and match itself — rotational order = 6; it also has 6 lines of symmetry.
- Real life: A butterfly's wings are mirror images (line symmetry); a bicycle wheel rotated about its centre shows rotational symmetry.
- Design: Rangoli patterns often use repeated reflection and rotation to create symmetric designs.
- \[Order of rotational symmetry (n): the number of times a figure maps onto itself during a full 360° rotation.\]
- \[Angle of rotation for one match = 360° / order\]\[Example: if order = 4\]\[smallest rotation = 360°/4 = 90°.\]
- \[Regular n-gon: number of lines of symmetry = n\]\[rotational order = n.\]
- \[Common shapes: square → 4 lines of symmetry\]\[rotational order 4\]\[rectangle → 2 lines\]\[order 2\]\[circle → infinite lines and rotational symmetry.\]
Drawing and Constructing Symmetric Figures
Drawing and Constructing Symmetric Figures
Key Point: Reflection in coordinate geometry: Across x-axis: (x, y) → (x, −y).
What is symmetry? Symmetry means balance or exact correspondence between parts of a figure. A figure is symmetric if one part is a mirror image of the other. In Class 7 we mainly study line (mirror) symmetry and basic constructions to draw symmetric figures.
Types relevant here: (a) Line (reflection) symmetry — a line (axis of symmetry) divides the figure into two mirror halves. (b) Point (central) symmetry — every point has a corresponding point directly opposite with respect to a center (rare in basic Class 7 constructions). We also briefly note rotational symmetry (figure looks same after rotation by certain angle).
Principles used when drawing/constructing symmetric figures:
- Corresponding points are at equal perpendicular distances from the axis of symmetry.
- The midpoint of a pair of symmetric points lies on the axis of symmetry.
- When reflecting using coordinates, specific formulas give reflected points (see formulas section).
How to draw or construct a symmetric figure (basic methods):
- Folding method (paper): Fold the paper along the proposed symmetry axis and trace the image of the original figure to get the mirror half. This is the simplest for visual checks.
- Ruler & compass — reflecting a point across a line:
- Given point P and axis line l. Draw a perpendicular from P to l (construct using compass or by drawing two equal arcs on l and joining their intersections with an arc centered at P to get the foot).
- Let H be the foot of the perpendicular on l. Measure distance PH with the compass and mark point P' on the other side of l such that HP' = HP and H lies between P and P'. Then P' is the reflection of P across l.
- Reflect a polygon: Reflect each vertex as above and join corresponding reflected vertices in the same order to obtain the reflected polygon.
- To find the axis of symmetry from two symmetric points A and A': Construct the perpendicular bisector of segment AA' — this will be the axis (midpoint lies on it and it is perpendicular to AA').
Constructing symmetric patterns: Use repeated reflections of a motif around one or several axes. For rotationally symmetric designs, regularly space reflections or rotate the motif about a centre by equal angles (for example, 60° for a 6-fold snowflake).
Tools and tips: Use graph paper or coordinate grid for accuracy. GeoGebra or simple graphing software helps show reflections visually by entering coordinates and applying reflection transformations. For hand construction practice, a compass, ruler and protractor are enough.
- Example 1 (Simple reflection): Reflect triangle ABC with A(2,1), B(4,1), C(3,3) about the y-axis. Reflected vertices: A'(-2,1), B'(-4,1), C'(-3,3). Join A'B'C' to get the symmetric triangle.
- Example 2 (Using ruler and compass): Given point P and line l, construct P' the reflection of P across l by dropping a perpendicular from P to l to find the foot H and marking P' such that HP' = HP on the other side.
- Example 3 (Finding axis): Given two corresponding points A and A' of a symmetric shape, construct the perpendicular bisector of AA'. The bisector is the axis of symmetry for those points.
- Real-life example 1: Human face (approximate vertical line symmetry).
- Real-life example 2: Butterflies and many leaves show bilateral (line) symmetry.
- Real-life example 3: Starfish and some flowers show rotational symmetry (central symmetry for certain rotations).
- \[Reflection in coordinate geometry: Across x-axis: (x\]\[y) → (x, −y).\]
- \[Across y-axis: (x\]\[y) → (−x\]\[y).\]
- \[Across origin (central symmetry): (x\]\[y) → (−x, −y).\]
- \[Across line y = x: (x\]\[y) → (y\]\[x).\]
- \[Midpoint formula (useful to check axis): midpoint of (x1,y1) and (x2,y2) = ((x1+x2)/2\]\[(y1+y2)/2)\]\[The midpoint of symmetric points lies on the axis.\]
- \[Distance formula (used to check equal distances): distance between (x1,y1) and (x2,y2) = sqrt((x2−x1)^2 + (y2−y1)^2).\]
Multiple and Special Cases of Symmetry
Multiple and Special Cases of Symmetry
Key Point: Number of lines of symmetry for a regular n-gon = n.
What is meant by 'multiple and special cases of symmetry'?
In symmetry, we check if a figure can be folded or reflected so that the two parts match exactly. A figure may have more than one line (axis) of symmetry. Special cases are shapes that have many or unusual symmetries — for example, shapes with infinitely many symmetry axes or with rotational symmetry of various orders.
Key ideas
- Line (mirror) symmetry: An axis is a line such that reflecting the figure across it maps the figure onto itself. A figure can have 0, 1, 2, 3, 4, ... such lines.
- Multiple lines of symmetry: Some figures have more than one axis. Example: a square has 4 axes of symmetry (2 medians + 2 diagonals); an equilateral triangle has 3 axes.
- Special cases: A circle has infinitely many lines of symmetry (every diameter is an axis). Regular n-gons (regular polygons with n equal sides) have exactly n axes of symmetry.
- Rotational symmetry (order): A figure has rotational symmetry of order k if it can be rotated by 360°/k (or multiples) about its center and map onto itself. A regular n-gon has rotational symmetry of order n.
- Point symmetry (180° rotation): Sometimes a figure looks the same after a rotation of 180° — this is also called central symmetry or half-turn symmetry.
Relationships to remember
- For a regular n-gon: number of axes of line symmetry = n, and rotational symmetry order = n.
- Circle: infinite axes, infinite rotational symmetry (any angle).
How to test
- Fold-method: fold along a candidate axis — if parts match exactly, it is an axis.
- Reflection-method: draw the perpendicular mirror line and check whether every point has a corresponding mirror point.
- Rotation-method: rotate the figure about its center by candidate angles and check for coincidence.
- Square: 4 lines of symmetry (two diagonals and two medians) and rotational symmetry of order 4 (rotations by 90°, 180°, 270°, 360°).
- Rectangle (not a square): 2 lines of symmetry (two medians) and rotational symmetry of order 2 (180°).
- Equilateral triangle: 3 lines of symmetry (each median) and rotational symmetry of order 3 (rotations by 120°).
- Circle: infinitely many lines of symmetry (every diameter) and infinite rotational symmetry (any angle).
- Regular pentagon: 5 lines of symmetry and rotational symmetry of order 5 (rotations by 72°).
- Butterfly or human face (approx.): typically 1 vertical axis of symmetry (bilateral symmetry).
- \[Number of lines of symmetry for a regular n-gon = n.\]
- \[Order of rotational symmetry for a regular n-gon = n.\]
- \[Smallest rotation angle that maps a figure onto itself = 360° / (order of rotational symmetry).\]
- \[If a figure has k-fold rotational symmetry\]\[it will coincide with itself after rotations of multiples of 360°/k.\]
Applications and Problem Solving
Applications and Problem Solving
Key Point: Order of rotational symmetry for a regular n-gon = n.
Applications and Problem Solving in the chapter Symmetry teaches how to use the ideas of mirror (line) symmetry and rotational symmetry to solve geometric problems and to recognise symmetry in real life. You learn to identify axes of symmetry, calculate order of rotational symmetry, use transformations (reflection and rotation) to find images of points and figures, and apply simple coordinate rules for reflections and rotations.
Key problem-solving approaches:
- Visual method: Fold or use tracing paper to check if two halves match — this identifies an axis of symmetry.
- Geometric method: Check equal distances from a candidate axis and perpendicularity (axis should be a perpendicular bisector of corresponding segments).
- Transformations and coordinates: Use algebraic rules for reflection across axes or lines and rotations about the origin to find images quickly.
- Count symmetries: For regular polygons, use known results (see formulas); compare shapes (for example, rectangle vs square) to decide number of axes.
Why this matters in real life: symmetry is used in design (architecture, logos, textiles), engineering (balanced parts), nature (flowers, leaves, animals), and art (patterns, tiling). Understanding symmetry helps simplify problems by reducing what must be checked and by using transformations to map a complex figure to a simpler one.
- Find axes of symmetry: A rectangle has 2 lines of symmetry (two perpendicular lines through midpoints of opposite sides). A square has 4 lines of symmetry (two diagonals and two midlines).
- Regular polygon: A regular hexagon has 6 lines of symmetry and rotational symmetry order 6.
- Reflection in coordinate plane: Reflect point (3, 4) across the y-axis. Image is (-3, 4).
- Reflection in a vertical line: Reflect (3, 4) across the line x = 2. Image is (1, 4) because x' = 2*2 - 3 = 1.
- Rotation about origin: Rotate point (2, 0) by 90 degrees counterclockwise about the origin. Image is (0, 2).
- Composition idea: Two reflections in intersecting lines with angle θ between them is equivalent to a rotation by 2θ about the intersection point.
- \[Order of rotational symmetry for a regular n-gon = n.\]
- \[Angle of rotation for one step in rotational symmetry = 360° / n (for order n).\]
- \[Reflection across x-axis: (x\]\[y) → (x, -y).\]
- \[Reflection across y-axis: (x\]\[y) → (-x\]\[y).\]
- \[Reflection across line y = x: (x\]\[y) → (y\]\[x).\]
- \[Reflection across vertical line x = a: (x\]\[y) → (2a - x\]\[y).\]
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 line that divides a figure into two mirror-image halves.
- Reflective symmetry
- Also called mirror symmetry; when one half of a figure is the mirror image of the other across a line.
- Rotational symmetry
- When a figure can be rotated about a point and still look the same at certain angles.
- Order of rotational symmetry
- The number of times a figure matches itself during a full 360° rotation.
- Angle of rotation
- The smallest angle through which a figure can be rotated to map onto itself.
- Centre of rotation
- The fixed point about which a figure is rotated.
- Mirror image
- The image of a figure produced by reflection across a line (appears reversed).
- Symmetric figure
- A figure that has at least one line or order of symmetry.
- Asymmetric figure
- A figure that has no line or rotational symmetry (does not match itself under these transformations).
- Point symmetry
- A kind of symmetry where every part has a matching part directly opposite through a central point (180° rotation).
- Axis of symmetry
- Another name for a line of symmetry; an axis where reflection produces identical halves.
- Regular polygon
- A polygon with all sides equal and all angles equal; it has many symmetries.
- Fold
- A physical method to test symmetry by folding paper along a line to see if halves match.
- Reflection (transformation)
- A transformation that flips a figure over a line producing a mirror image.
- Rotation (transformation)
- A transformation that turns a figure about a fixed point through a specified angle.
- Half-turn
- A rotation of 180°; special case of rotational symmetry often equal to point symmetry.
- Quarter-turn
- A rotation of 90°; commonly the smallest rotation for squares and some patterns.
- Congruent
- Two figures that are identical in shape and size; one can be obtained from the other by rigid motions (translation, rotation, reflection).
- Bilateral symmetry
- Symmetry with exactly one line (or plane in 3D) dividing a figure into two mirrored halves.
Practice Questions
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How many lines of symmetry does a square have? / एक वर्ग में कितनी सममिति रेखाएं होती हैं? (a) 1 (b) 2 (c) 4 (d) 6
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(c) 4 / (c) 4 — A square has 4 lines of symmetry: 2 through midpoints of opposite sides and 2 diagonal lines. / वर्ग में 4 सममिति रेखाएं होती हैं: 2 विपरीत भुजाओं के मध्यबिंदुओं से और 2 विकर्ण रेखाएं।
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The order of rotational symmetry of an equilateral triangle is: / समबाहु त्रिभुज की घूर्णी सममिति की कोटि है: (a) 1 (b) 2 (c) 3 (d) 6
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(c) 3 / (c) 3 — An equilateral triangle maps onto itself after rotations of 120°, 240° and 360°, so the order is 3. / समबाहु त्रिभुज 120°, 240° और 360° के घुमाव के बाद खुद पर आ जाता है, इसलिए कोटि 3 है।
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Which capital letter has both a vertical and a horizontal line of symmetry? / किस बड़े अक्षर में ऊर्ध्वाधर और क्षैतिज दोनों सममिति रेखाएं होती हैं? (a) A (b) B (c) H (d) S
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(c) H / (c) H — The letter H has a vertical line through its middle and a horizontal line through the crossbar, giving 2 lines of symmetry. / H में मध्य में ऊर्ध्वाधर रेखा और क्षैतिज रेखा दोनों होती हैं, इसलिए 2 सममिति रेखाएं हैं।
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A regular hexagon has ______ lines of symmetry and rotational symmetry of order ______. / एक समषट्भुज में ______ सममिति रेखाएं और ______ कोटि की घूर्णी सममिति होती है।
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6 lines; order 6 / 6 रेखाएं; कोटि 6 — A regular n-gon has n lines of symmetry and rotational symmetry of order n; for n = 6 both values are 6. / एक सम n-भुज में n सममिति रेखाएं और n कोटि की घूर्णी सममिति होती है; n = 6 के लिए दोनों 6 हैं।
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When a point P(3, 2) is reflected across the y-axis, the image P' is ______. / जब बिंदु P(3, 2) को y-अक्ष के पार परावर्तित किया जाता है, तो प्रतिबिंब P' ______ है।
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(−3, 2) / (−3, 2) — Reflection across the y-axis changes the sign of the x-coordinate: (x, y) → (−x, y). / y-अक्ष के पार परावर्तन में x-निर्देशांक का चिह्न बदलता है: (x, y) → (−x, y)।
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True or False: A circle has exactly 4 lines of symmetry. / सत्य या असत्य: एक वृत्त में ठीक 4 सममिति रेखाएं होती हैं।
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False / असत्य — A circle has infinitely many lines of symmetry because every diameter is a line of symmetry. / वृत्त में अनंत सममिति रेखाएं होती हैं क्योंकि प्रत्येक व्यास एक सममिति रेखा है।
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Describe the line(s) of symmetry of an isosceles triangle and explain using paper folding. / एक समद्विबाहु त्रिभुज की सममिति रेखा का वर्णन करें और कागज मोड़कर समझाएं।
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An isosceles triangle has exactly 1 line of symmetry that passes through the apex (top vertex) and the midpoint of the base. / समद्विबाहु त्रिभुज में ठीक 1 सममिति रेखा होती है जो शीर्ष (ऊपरी कोने) से आधार के मध्यबिंदु तक जाती है। Folding along this line makes the two equal sides coincide exactly. / इस रेखा पर मोड़ने पर दोनों समान भुजाएं एक-दूसरे पर बिल्कुल आ जाती हैं।
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A figure has rotational symmetry of order 4. What is the smallest angle of rotation at which it maps onto itself? / एक आकृति में 4 कोटि की घूर्णी सममिति है। न्यूनतम घूर्णन कोण क्या है जिस पर यह खुद पर आ जाती है?
Show answer
90° / 90° — Smallest angle = 360° ÷ order = 360° ÷ 4 = 90°. / न्यूनतम कोण = 360° ÷ कोटि = 360° ÷ 4 = 90°।
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