Answer :
To determine which polygon always has 4-fold reflectional symmetry and order 4 rotational symmetry, we should review the properties of each given option.
1. Rectangle:
- Reflectional symmetry: A rectangle has 2-fold reflectional symmetry. It can be reflected over its two axes—the midlines passing through the centers of opposite sides.
- Rotational symmetry: A rectangle has order 2 rotational symmetry; it looks the same after rotating by 180 degrees.
2. Trapezoid:
- Reflectional symmetry: In general, a trapezoid does not have reflectional symmetry, though an isosceles trapezoid has one line of symmetry along its vertical axis.
- Rotational symmetry: A trapezoid can only have order 1 rotational symmetry; it does not look the same after any rotation.
3. Right Triangle:
- Reflectional symmetry: A right triangle generally has no lines of symmetry unless it's isosceles, in which case it has one line of symmetry along its vertical axis.
- Rotational symmetry: A right triangle has order 1 rotational symmetry; it does not look the same after any rotation unless it’s equilateral, but that's not specified here.
4. Square:
- Reflectional symmetry: A square has 4-fold reflectional symmetry. It can be reflected over four axes: the two diagonals and the vertical and horizontal midlines passing through its center.
- Rotational symmetry: A square has order 4 rotational symmetry. It looks identical after rotations of 90 degrees, 180 degrees, 270 degrees, and 360 degrees.
Considering the properties, a square is the polygon that satisfies both conditions: 4-fold reflectional symmetry and order 4 rotational symmetry.
Thus, the correct answer is:
- Square (Option 4)
1. Rectangle:
- Reflectional symmetry: A rectangle has 2-fold reflectional symmetry. It can be reflected over its two axes—the midlines passing through the centers of opposite sides.
- Rotational symmetry: A rectangle has order 2 rotational symmetry; it looks the same after rotating by 180 degrees.
2. Trapezoid:
- Reflectional symmetry: In general, a trapezoid does not have reflectional symmetry, though an isosceles trapezoid has one line of symmetry along its vertical axis.
- Rotational symmetry: A trapezoid can only have order 1 rotational symmetry; it does not look the same after any rotation.
3. Right Triangle:
- Reflectional symmetry: A right triangle generally has no lines of symmetry unless it's isosceles, in which case it has one line of symmetry along its vertical axis.
- Rotational symmetry: A right triangle has order 1 rotational symmetry; it does not look the same after any rotation unless it’s equilateral, but that's not specified here.
4. Square:
- Reflectional symmetry: A square has 4-fold reflectional symmetry. It can be reflected over four axes: the two diagonals and the vertical and horizontal midlines passing through its center.
- Rotational symmetry: A square has order 4 rotational symmetry. It looks identical after rotations of 90 degrees, 180 degrees, 270 degrees, and 360 degrees.
Considering the properties, a square is the polygon that satisfies both conditions: 4-fold reflectional symmetry and order 4 rotational symmetry.
Thus, the correct answer is:
- Square (Option 4)
