Symmetry

(Based on standard punched hole figures):

(a) A square with two holes horizontally aligned: 1 vertical line of symmetry passing exactly between the holes.

(b) A square with two holes diagonally opposed: 1 diagonal line of symmetry.

(c) A square with two holes on perpendicular bisector: 1 horizontal line or 1 vertical line depending on orientation (total 1 line).

(d) A circle with two holes horizontally aligned: 1 vertical line of symmetry passing through the center.

(e) A square with 4 holes at four corners: 4 lines of symmetry (2 along diagonals, 2 joining midpoints of opposite sides).

(f) A square with 4 holes forming a central square: 4 lines of symmetry.

(g) An equilateral triangle with a hole at one vertex and one at base center: 1 line of symmetry (the median through the top vertex).

(h) An equilateral triangle with one hole at the centroid: 3 lines of symmetry (the three medians).

(i) A right triangle with holes symmetrically about the geometric bisector: 1 line of symmetry.

(j) A circle with 3 holes forming an equilateral triangle: 3 lines of symmetry.

(k) A circle with 4 holes spaced evenly (like a square): 4 lines of symmetry.

(l) A circle with 4 holes, one central and 3 in equilateral arrangement: 3 lines of symmetry.

The holes are located by finding the mirror image of the given hole across the drawn line of symmetry (the axis of reflection).

• (a) For a square folded diagonally, the hole reflects across the diagonal.
• (b) For a square folded horizontally, the hole reflects across the horizontal midline.
• (c) For a triangle folded vertically along the altitude, the hole reflects horizontally to the symmetric opposite side.
• (d) For a circle folded vertically, reflect the hole on the left to the same position on the right.
• (e) For a circle with a diametric line, plot the hole at equal distance strictly on the opposite side.

When completed by reflection across the dotted line, the figures formed are:

(a) Half-square completes into a full Square.

(b) Half-triangle completes into a Triangle (specifically Isosceles).

(c) Half-rhombus forms a full Rhombus.

(d) A quarter-circle completes strictly into a Circle (if mirrored twice) or Semicircle (if given once).

(e) A half-pentagon completes into a regular Pentagon.

(f) A half-octagon completes into a regular Octagon.

(a) Square with inner cross: 4 lines of symmetry (2 diagonals, 2 mid-point connectors).

(b) Rectangle-like block: 2 lines of symmetry (1 horizontal, 1 vertical).

(c) Tri-part clover shape: 3 lines of symmetry.

(d) Square with corner designs: 2 lines of symmetry (the 2 diagonals).

(e) Square with edge designs: 4 lines of symmetry.

(f) Four-petal flower: 4 lines of symmetry.

(g) Six-petal flower inside hexagon: 6 lines of symmetry.

(h) Star-hexagram: 6 lines of symmetry.

Yes, there is more than one geometric way to shade additional squares so that the grid becomes symmetric about a single diagonal. The shading depends on which squares are originally dark.

If we shade the exactly corresponding squares across both the main diagonal and the anti-diagonal dynamically, then the figure will successfully become symmetric about both the diagonals.

For diagrams involving 1 or 2 intersecting mirror lines, you must trace the vertices of the given shape and plot points perpendicularly across each mirror line. If there are 2 perpendicular mirror lines (forming 4 quadrants), the final shape will span all 4 quadrants perfectly forming a symmetric kite, star, or rhombus depending on the initial shape.

(a) Vertical mirror: A, H, I, M, O, T, U, V, W, X, Y (When typed symmetrically like block capitals).

(b) Horizontal mirror: B, C, D, E, H, I, K, O, X.

(c) Both Horizontal and Vertical: H, I, O, X.

• Scalene Triangle: All 3 sides are of different lengths.
• Parallelogram: Opposites are parallel but it has rotational symmetry, not line symmetry.
• Trapezium: A general non-isosceles trapezium.

(a) Isosceles triangle: The Median or Altitude (drawn from the vertex where the two equal sides meet, to the base).

(b) Circle: Any Diameter.

(a) Circle with 4 central lines: Rotational Order = 4 (Yes, > 1).

(b) Equilateral Triangle: Rotational Order = 3 (Yes, > 1).

(c) T-shape block design (or single line): If it's a standard cross, Order = 4. If asymmetric, Order = 1.

(d) Letter H: Rotational Order = 2 (Yes, > 1). Matches after 180°.

(e) Circle with 3 distinct symmetric sectors: Rotational Order = 3 (Yes, > 1).

(f) Circular fan with 4 symmetric blades: Rotational Order = 4 (Yes, > 1).

Conclusion: Figures (a), (b), (d), (e), and (f) have rotational symmetry of order more than 1.

(a) Rhombus/Kite intersecting: Order of Rotational Symmetry = 2 (180°, 360°).

(b) 80° bounded angle (semi-hourglass): Order = 2.

(c) Equilateral Triangle: Order = 3 (Rotates every 120°).

(d) Propeller (4 blades): Order = 4 (Rotates every 90°).

(e) Cross inside a square: Order = 4.

(f) Regular Pentagon: Order = 5 (Rotates every 72°).

(g) Star (6 pointed regular): Order = 6 (Rotates every 60°).

(h) Try-star (3 intersecting legs): Order = 3.

1. Circle: Has infinite lines of symmetry and infinite order of rotational symmetry.

2. Square: Has 4 lines of symmetry and a rotational symmetry of order 4.

(i) A regular Equilateral Triangle. It has 3 lines of symmetry and rotational symmetry of order 3.

(ii) An Isosceles Triangle. It has 1 vertical line of symmetry, but its rotational order is exactly 1 (only at 360°).

(iii) A Parallelogram. It has 0 lines of symmetry, but it has a rotational symmetry of order 2 (it matches after rotating 180°).

(iv) A Kite (or an isosceles trapezium). It has 1 line of symmetry (down the middle) but order of rotational symmetry is only 1.

Yes. If a figure has two or more distinct lines of symmetry crossing at a point, they are bound to create identical symmetric parts around the center. For example, a rectangle has 2 intersecting lines of symmetry, and subsequently has rotational symmetry of order 2.

1. Square (4 Lines of symmetry, Rotational Order 4)

2. Rectangle (2 Lines of symmetry, Rotational Order 2)

3. Rhombus (2 Lines of symmetry, Rotational Order 2)

If a figure looks strictly the same after an angle of rotation of 60°, then it will continue to look the same at all subsequent multiples of 60°.

The other angles are:
60° + 60° = 120°
120° + 60° = 180°
180° + 60° = 240°
240° + 60° = 300°
300° + 60° = 360°

For a figure to have rotational symmetry of order more than 1, the angle of rotation must exactly divide 360°.

(i) 45°: Yes. Because 360 / 45 = 8. It perfectly divides 360°. It will have an order of 8 (e.g., a regular octagon).

(ii) 17°: No. Because 360 divided by 17 is approximately 21.17. Since 360 is not completely divisible by 17, it cannot act as a valid primary angle of rotational symmetry.

A regular polygon with n sides has exactly n lines of symmetry and a rotational symmetry order of n.

A decagon has 10 sides. Therefore, it has 10 lines of symmetry and a rotational symmetry order of 10.

The letters that look exactly the same when rotated by 180 degrees but cannot be folded symmetrically are: N, S, and Z.

Order of rotation = 360° / Angle of Rotation
n = 360 / 45 = 8

A regular polygon with 8 sides is a Regular Octagon.

An isosceles trapezium only overlaps completely with itself after a full 360-degree rotation. It does not have rotational symmetry before that. Thus, its order is strictly 1.

True. If a shape has 2 or more distinct lines of symmetry intersecting at a point, reflecting across them consecutively behaves like a rotation around that intersection point, forcing an order > 1.

Smallest Angle of rotation = 360° / Number of sides
Angle = 360 / 12 = 30°.

Both are quadrilaterals with equal sides. However, a rhombus only has diagonal symmetry because its internal angles are not identically equal. A square has all internal angles equal (90°), allowing lines of symmetry extending from opposite side midpoints as well as the diagonals.

The reflection reverses the order of the letters and flips each letter individually on its vertical axis.
S maps to a reversed S.
H maps to H (vertically symmetric).
T maps to T (vertically symmetric).
A maps to A (vertically symmetric).
M maps to M (vertically symmetric).

Reflected word reads exactly as: S'HTAM (where S' is the inverted S shape).

Isosceles Triangle. It has 1 line of reflectional symmetry perfectly down the middle from the odd vertex to the base, but it must be fully rotated by 360° (order 1) to look identical.

No. The degree of rotational symmetry must perfectly divide 360 into a whole integer.
360 / 50 = 7.2.
Since 7.2 is strictly not an integer, an angle of exactly 50° cannot be a standard rotational symmetry angle.

An Equilateral Triangle.

The exact Point of Intersection of its diagonals acts as the absolute center of rotation.

Every straight line universally passing strictly through the absolute center of a distinct circle structurally represents an axis of reflectional symmetry. As a result, a circle logically has an infinite count of symmetry lines.

A perfect 3D solid cube ultimately contains exactly 9 planes of reflectional symmetry (3 cutting parallel to the faces, and 6 cutting diagonally through opposite edges).

A regular polygon inherently retains rotational order absolutely equal to its sides. An icosagon strictly contains exactly 20 congruent sides. Therefore, the rotational order equals 20.

When reflecting correctly utilizing the horizontal x-axis as a mirror line, the x-coordinate rigidly remains functionally unchanged, whereas the absolute sign of the y-coordinate reverses directly.
Therefore, (3, 4) rigidly reflects purely to (3, -4).

Letters with horizontal & vertical symmetry structurally are H, I, O, X.
The letter "O" possesses curved components.
Hence, strictly evaluating the rigid straight-edge forms, the letters accurately are H, I, and X.

The Center of Rotation is distinctly defined as the fixed abstract point tightly around which a 2D geometric shape is perfectly rotated maintaining consistent radial equilibrium, generally remaining absolutely stationary strictly during the physical rotation sequence.

An Axis of Symmetry fundamentally pertains exclusively to a geometric line acting essentially as a mirror reflecting exactly two perfectly reciprocal structural halves inherently creating reflectional symmetry.
An Axis (or point) of Rotation strictly corresponds strictly to evaluating the exact pivotal rotational origin dictating specifically precisely angular rotational congruence over discrete 360° operational intervals.

Standard Angle mathematically equates sequentially matching 360 logically divided structurally by the absolute evaluated rotational Order.
Angle = exactly 360 / 12
Angle = precisely 30°.