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a quadrilateral with a rotational symmetry of order more than 1 but not a line symmetry.

Question

A quadrilateral with a rotational symmetry of order more than 1 but not a line symmetry.

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Solution

Understanding the Problem

We are looking for a quadrilateral that has rotational symmetry of order greater than 1, meaning it can be rotated by a certain angle and still appear unchanged, but does not exhibit line symmetry, meaning there are no lines of reflection that divide it into two identical halves.

Identifying Suitable Quadrilaterals

  1. Rotational Symmetry of Order Greater Than 1:

    • A shape has rotational symmetry of order 2 if it can be rotated 180 degrees and looks the same. For order greater than 2, it must also appear identical at other angles (like 120 degrees for order 3).
  2. Line Symmetry:

    • A shape has line symmetry if it can be divided into two identical parts by a line. We need to avoid this characteristic in our quadrilateral.

Example of a Suitable Quadrilateral

A kite without being a rhombus serves as a good example of a quadrilateral meeting these criteria. A kite has:

  • Rotational Symmetry of Order 2: It looks the same when rotated by 180 degrees.
  • Lacks Line Symmetry: In general, a kite does not have a line of symmetry unless special cases are considered (like a rhombus, which does not suit our needs).

Summary

A kite fulfills the criteria of having rotational symmetry (specifically of order 2) while not possessing line symmetry when it is not a rhombus. Other non-standard shapes that may exhibit such properties could be crafted, but they may lack regularity. Thus, a kite stands as a classic example.

This problem has been solved

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