Symmetries of Polygons: Rotations and Reflections - Geometry
Card 1 of 30
Identify the four reflection axes of a square.
Identify the four reflection axes of a square.
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Two diagonals and two midlines through the center. Diagonals connect opposite vertices, midlines bisect opposite sides.
Two diagonals and two midlines through the center. Diagonals connect opposite vertices, midlines bisect opposite sides.
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What is the smallest positive rotation that carries any parallelogram onto itself?
What is the smallest positive rotation that carries any parallelogram onto itself?
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$180^\circ$. All parallelograms have 2-fold rotational symmetry about their center.
$180^\circ$. All parallelograms have 2-fold rotational symmetry about their center.
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Which rotation angles carry any parallelogram onto itself (include $0^\circ$ and $360^\circ$)?
Which rotation angles carry any parallelogram onto itself (include $0^\circ$ and $360^\circ$)?
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$0^\circ,\ 180^\circ,\ 360^\circ$. These are multiples of the fundamental $180^\circ$ symmetry rotation.
$0^\circ,\ 180^\circ,\ 360^\circ$. These are multiples of the fundamental $180^\circ$ symmetry rotation.
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How many lines of reflection symmetry does a general parallelogram have?
How many lines of reflection symmetry does a general parallelogram have?
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$0$. General parallelograms have no reflection symmetry lines.
$0$. General parallelograms have no reflection symmetry lines.
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What is the smallest positive rotation that carries a rhombus (non-square) onto itself?
What is the smallest positive rotation that carries a rhombus (non-square) onto itself?
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$180^\circ$. Non-square rhombus has 2-fold rotational symmetry like other parallelograms.
$180^\circ$. Non-square rhombus has 2-fold rotational symmetry like other parallelograms.
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What is the smallest positive rotation that carries a general trapezoid onto itself?
What is the smallest positive rotation that carries a general trapezoid onto itself?
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$360^\circ$. General trapezoid has no rotational symmetry except the identity.
$360^\circ$. General trapezoid has no rotational symmetry except the identity.
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Identify the order of rotational symmetry of a general trapezoid.
Identify the order of rotational symmetry of a general trapezoid.
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$1$. Only the identity rotation maps general trapezoid to itself.
$1$. Only the identity rotation maps general trapezoid to itself.
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What is the smallest positive rotation that carries a non-square rectangle onto itself?
What is the smallest positive rotation that carries a non-square rectangle onto itself?
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$180^\circ$. Rectangle maps to itself by rotation about its center every half-turn.
$180^\circ$. Rectangle maps to itself by rotation about its center every half-turn.
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Which rotation angles carry any rectangle onto itself (include $0^\circ$ and $360^\circ$)?
Which rotation angles carry any rectangle onto itself (include $0^\circ$ and $360^\circ$)?
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$0^\circ,\ 180^\circ,\ 360^\circ$. These are multiples of the fundamental $180^\circ$ symmetry rotation.
$0^\circ,\ 180^\circ,\ 360^\circ$. These are multiples of the fundamental $180^\circ$ symmetry rotation.
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How many lines of reflection symmetry does a non-square rectangle have?
How many lines of reflection symmetry does a non-square rectangle have?
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$2$. Rectangle has reflection symmetry across horizontal and vertical midlines only.
$2$. Rectangle has reflection symmetry across horizontal and vertical midlines only.
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Identify the reflection axes of a non-square rectangle in relation to its sides.
Identify the reflection axes of a non-square rectangle in relation to its sides.
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The two midlines through the center, parallel to the sides. These lines bisect opposite sides and pass through the center.
The two midlines through the center, parallel to the sides. These lines bisect opposite sides and pass through the center.
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What is the smallest positive rotation that carries a square onto itself?
What is the smallest positive rotation that carries a square onto itself?
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$90^\circ$. Square has 4-fold rotational symmetry, rotating by quarter-turns.
$90^\circ$. Square has 4-fold rotational symmetry, rotating by quarter-turns.
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Which rotation angles carry a square onto itself (include $0^\circ$ and $360^\circ$)?
Which rotation angles carry a square onto itself (include $0^\circ$ and $360^\circ$)?
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$0^\circ,\ 90^\circ,\ 180^\circ,\ 270^\circ,\ 360^\circ$. These are multiples of the fundamental $90^\circ$ symmetry rotation.
$0^\circ,\ 90^\circ,\ 180^\circ,\ 270^\circ,\ 360^\circ$. These are multiples of the fundamental $90^\circ$ symmetry rotation.
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How many lines of reflection symmetry does a square have?
How many lines of reflection symmetry does a square have?
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$4$. Square has reflections across two diagonals and two midlines.
$4$. Square has reflections across two diagonals and two midlines.
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How many lines of reflection symmetry does a rhombus (non-square) have?
How many lines of reflection symmetry does a rhombus (non-square) have?
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$2$. Rhombus has reflection symmetry across both diagonal lines only.
$2$. Rhombus has reflection symmetry across both diagonal lines only.
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Identify the reflection axes of a rhombus (non-square).
Identify the reflection axes of a rhombus (non-square).
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The two diagonals. The diagonals of a rhombus are perpendicular bisectors of each other.
The two diagonals. The diagonals of a rhombus are perpendicular bisectors of each other.
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What is the smallest positive rotation that carries a kite (non-rhombus) onto itself?
What is the smallest positive rotation that carries a kite (non-rhombus) onto itself?
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$360^\circ$. Non-rhombus kite has only one reflection symmetry, no rotations.
$360^\circ$. Non-rhombus kite has only one reflection symmetry, no rotations.
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How many lines of reflection symmetry does a kite (non-rhombus) have?
How many lines of reflection symmetry does a kite (non-rhombus) have?
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$1$. Kite has reflection symmetry across one diagonal only.
$1$. Kite has reflection symmetry across one diagonal only.
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For an isosceles trapezoid, how many lines of reflection symmetry are there?
For an isosceles trapezoid, how many lines of reflection symmetry are there?
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$1$. Isosceles trapezoid has reflection symmetry across its perpendicular bisector.
$1$. Isosceles trapezoid has reflection symmetry across its perpendicular bisector.
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Identify the reflection axis of an isosceles trapezoid.
Identify the reflection axis of an isosceles trapezoid.
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The line through midpoints of the bases, perpendicular to the bases. This line is equidistant from corresponding points on the legs.
The line through midpoints of the bases, perpendicular to the bases. This line is equidistant from corresponding points on the legs.
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What is the smallest positive rotation that carries an isosceles trapezoid onto itself?
What is the smallest positive rotation that carries an isosceles trapezoid onto itself?
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$360^\circ$. Isosceles trapezoid has only reflection symmetry, no rotational symmetry.
$360^\circ$. Isosceles trapezoid has only reflection symmetry, no rotational symmetry.
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For a general trapezoid (not isosceles), how many reflection symmetries are there?
For a general trapezoid (not isosceles), how many reflection symmetries are there?
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$0$. General trapezoid has no symmetries except the identity.
$0$. General trapezoid has no symmetries except the identity.
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What is the smallest positive rotation that carries a general trapezoid onto itself?
What is the smallest positive rotation that carries a general trapezoid onto itself?
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$360^\circ$. General trapezoid has no rotational symmetry except the identity.
$360^\circ$. General trapezoid has no rotational symmetry except the identity.
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What is the smallest positive rotation that carries a regular $n$-gon onto itself?
What is the smallest positive rotation that carries a regular $n$-gon onto itself?
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$\frac{360^\circ}{n}$. Regular polygon divides full rotation equally among $n$ symmetries.
$\frac{360^\circ}{n}$. Regular polygon divides full rotation equally among $n$ symmetries.
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How many rotational symmetries does a regular $n$-gon have (including $0^\circ$)?
How many rotational symmetries does a regular $n$-gon have (including $0^\circ$)?
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$n$. Includes identity rotation plus $n-1$ non-trivial rotations.
$n$. Includes identity rotation plus $n-1$ non-trivial rotations.
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Which rotation angles carry a regular $n$-gon onto itself?
Which rotation angles carry a regular $n$-gon onto itself?
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$k\cdot\frac{360^\circ}{n}$ for $k=0,1,\dots,n-1$. These are all multiples of the fundamental rotation $\frac{360^\circ}{n}$.
$k\cdot\frac{360^\circ}{n}$ for $k=0,1,\dots,n-1$. These are all multiples of the fundamental rotation $\frac{360^\circ}{n}$.
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How many lines of reflection symmetry does a regular $n$-gon have?
How many lines of reflection symmetry does a regular $n$-gon have?
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$n$. Regular $n$-gon has one reflection axis through each vertex or edge.
$n$. Regular $n$-gon has one reflection axis through each vertex or edge.
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In a regular polygon with odd $n$, what does each reflection axis pass through?
In a regular polygon with odd $n$, what does each reflection axis pass through?
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A vertex and the midpoint of the opposite side. For odd $n$, reflection axes pass through vertices and opposite edge midpoints.
A vertex and the midpoint of the opposite side. For odd $n$, reflection axes pass through vertices and opposite edge midpoints.
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In a regular polygon with even $n$, how many reflection axes go through opposite vertices?
In a regular polygon with even $n$, how many reflection axes go through opposite vertices?
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$\frac{n}{2}$. Half the reflection axes connect pairs of opposite vertices.
$\frac{n}{2}$. Half the reflection axes connect pairs of opposite vertices.
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In a regular polygon with even $n$, how many reflection axes go through midpoints of opposite sides?
In a regular polygon with even $n$, how many reflection axes go through midpoints of opposite sides?
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$\frac{n}{2}$. Half the reflection axes connect midpoints of opposite edges.
$\frac{n}{2}$. Half the reflection axes connect midpoints of opposite edges.
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