a regular hexagon is shown below.
line w bisects each side it passes through.
line u passes through a vertex and bisects a side.
point y is the center of the hexagon.

image of a regular hexagon with center y, dashed line w (vertical, bisecting top and bottom sides) and dashed line u (passing through a vertex and bisecting an opposite side)

which transformation(s) must map the hexagon exactly onto itself? choose all that apply.
reflection across line u
clockwise rotation about y by 180°
counterclockwise rotation about y by 144°
reflection across line w
none of the above

Question

a regular hexagon is shown below.
line w bisects each side it passes through.
line u passes through a vertex and bisects a side.
point y is the center of the hexagon.

image of a regular hexagon with center y, dashed line w (vertical, bisecting top and bottom sides) and dashed line u (passing through a vertex and bisecting an opposite side)

which transformation(s) must map the hexagon exactly onto itself? choose all that apply.
  reflection across line u
  clockwise rotation about y by 180°
  counterclockwise rotation about y by 144°
  reflection across line w
  none of the above

Accepted Answer

# Brief Explanations:
1. **Reflection across line $ u $**: Line $ u $ passes through a vertex and bisects a side. In a regular hexagon, a line through a vertex and the midpoint of the opposite side (or bisecting a side) is a line of symmetry? Wait, no—wait, a regular hexagon has lines of symmetry: 6 lines, 3 through opposite vertices, 3 through midpoints of opposite sides. Wait, line $ w $ bisects each side it passes through—so line $ w $ is through midpoints of opposite sides (since it bisects two sides, maybe? Wait, the hexagon: line $ w $ is vertical, bisecting the top and bottom sides. So line $ w $ is a line of symmetry (through midpoints of opposite sides). Line $ u $: passes through a vertex and bisects a side. Let's check reflection over line $ u $: would that map the hexagon to itself? Let's see the hexagon's vertices: a regular hexagon has vertices at 0°, 60°, 120°, 180°, 240°, 300°. Line $ u $: let's say it's at 30° (since it passes through a vertex (60°? Wait, no, the vertex is at, say, 60°, and it bisects a side. Wait, maybe line $ u $ is not a line of symmetry. Wait, let's check rotation: 180° rotation about center $ Y $: in a regular hexagon, rotating 180° will map each vertex to the opposite vertex (since 60°*3=180°), so that's a symmetry. Counterclockwise rotation by 144°: the regular hexagon has rotational symmetry of order 6, so the angle is 360/6=60°. So 60°, 120°, 180°, 240°, 300°, 360°. 144° is not a multiple of 60°, so that's not a symmetry. Reflection across line $ w $: line $ w $ is through midpoints of opposite sides (top and bottom), so that's a line of symmetry (since reflecting over it will map top to bottom, left to right appropriately). Wait, let's re-examine:

- **Reflection across line $ w $**: Line $ w $ bisects the top and bottom sides (since it "bisects each side it passes through"—so it passes through two sides, bisecting them. So that's a line through midpoints of opposite sides, which is a line of symmetry for the regular hexagon. So reflection over $ w $ maps the hexagon to itself.

- **Clockwise rotation about $ Y $ by 180°**: A regular hexagon has rotational symmetry of 180° (since 180° is 3*60°, and 60° is the basic rotation). Rotating 180° will map each vertex to the one opposite (e.g., 0°→180°, 60°→240°, 120°→300°), so that works.

- **Reflection across line $ u $**: Let's see, line $ u $ passes through a vertex and bisects a side. Let's take coordinates: center at (0,0), vertices at (1,0), (0.5, √3/2), (-0.5, √3/2), (-1,0), (-0.5, -√3/2), (0.5, -√3/2). Line $ u $: passes through (0.5, √3/2) (a vertex) and bisects a side. The side between (-0.5, -√3/2) and (0.5, -√3/2) is the bottom side? No, wait, the bottom side is between (-0.5, -√3/2) and (0.5, -√3/2)? No, wait, regular hexagon with side length 1, center at (0,0): vertices at (1,0), (0.5, √3/2), (-0.5, √3/2), (-1,0), (-0.5, -√3/2), (0.5, -√3/2). So the top side is between (0.5, √3/2) and (-0.5, √3/2), bottom between (0.5, -√3/2) and (-0.5, -√3/2). Line $ w $ is vertical, through (0, √3/2) and (0, -√3/2)? No, wait, the problem says "Line $ w $ bisects each side it passes through"—so line $ w $ passes through two sides, bisecting each. So top side: midpoint is (0, √3/2), bottom side: midpoint is (0, -√3/2). So line $ w $ is the vertical line through (0,0), passing through midpoints of top and bottom sides. So that's a line of symmetry (reflecting over it will map left to right, top to bottom appropriately).

- **Reflection across line $ u $**: Line $ u $ passes through a vertex (e.g., (0.5, √3/2)) and bisects a side. Let's say the side it bisects is the left side? Wait, no, the hexagon has sides: between (1,0) and (0.5, √3/2) (side 1), (0.5, √3/2) and (-0.5, √3/2) (side 2), (-0.5, √3/2) and (-1,0) (side 3), (-1,0) and (-0.5, -√3/2) (side 4), (-0.5, -√3/2) and (0.5, -√3/2) (side 5), (0.5, -√3/2) and (1,0) (side 6). Line $ u $ passes through the vertex (0.5, √3/2) (end of side 1, start of side 2) and bisects a side—maybe side 4? Wait, side 4 is between (-1,0) and (-0.5, -√3/2), midpoint at (-0.75, -√3/4). No, line $ u $ is the dashed line from (0.5, √3/2) down to, say, (-0.5, -√3/2)? Wait, that's a line through a vertex (0.5, √3/2) and the midpoint of side 4? Wait, no, the midpoint of side 4 is (-0.75, -√3/4), not (-0.5, -√3/2). Wait, maybe line $ u $ is a line through a vertex and the midpoint of the opposite side? No, the opposite side of (0.5, √3/2) would be side 4? No, the opposite vertex of (0.5, √3/2) (60°) is (0.5, -√3/2) (300°)? No, 60° and 240° are opposite (60+180=240). Wait, 60° vertex: (0.5, √3/2), opposite vertex: (0.5, -√3/2)? No, that's 300°, which is 60°+240°, not 180°. Wait, no, in a regular hexagon, opposite vertices are 180° apart: 0° (1,0) and 180° (-1,0); 60° (0.5, √3/2) and 240° (-0.5, -√3/2); 120° (-0.5, √3/2) and 300° (0.5, -√3/2). So line $ u $ passes through (0.5, √3/2) (60° vertex) and bisects a side—maybe side 5? Side 5 is between (-0.5, -√3/2) and (0.5, -√3/2), midpoint at (0, -√3/2). Wait, line $ u $ goes from (0.5, √3/2) to (0, -√3/2)? No, the dashed line in the diagram: from the top-right vertex (0.5, √3/2) down to the bottom-left side? Wait, maybe line $ u $ is not a line of symmetry. Let's check reflection over line $ u $: would that map the hexagon to itself? Probably not, because line $ u $ is not a line of symmetry (since it's through a vertex and a side midpoint, not opposite vertex or opposite side midpoint).

- **Clockwise rotation about $ Y $ by 180°**: As mentioned, rotating 180° in a regular hexagon: each vertex (60°) maps to 60°+180°=240° (opposite vertex), so that works. So that's a symmetry.

- **Counterclockwise rotation about $ Y $ by 144°**: 144° is not a multiple of 60° (360/6=60), so 144° is 2.4*60°, not an integer multiple, so no.

- **Reflection across line $ w $**: Line $ w $ is through midpoints of opposite sides (top and bottom), so that's a line of symmetry (since reflecting over it will flip the hexagon vertically, mapping top to bottom, and the left and right parts appropriately).

So the correct transformations are: Clockwise rotation about $ Y $ by 180°, Reflection across line $ w $. Wait, wait, earlier I thought reflection across line $ u $ was no, but let's recheck the options. Wait, the options are:

- Reflection across line $ u $: No (line $ u $ is not a line of symmetry)

- Clockwise rotation about $ Y $ by 180°: Yes (rotational symmetry)

- Counterclockwise rotation about $ Y $ by 144°: No (not multiple of 60°)

- Reflection across line $ w $: Yes (line of symmetry)

Wait, but let's confirm the regular hexagon's symmetries:

- Lines of symmetry: 6 lines, 3 through opposite vertices (0°, 120°, 240°), 3 through midpoints of opposite sides (60°, 180°, 300°? Wait, no: 0° line through (1,0) and (-1,0) (opposite vertices), 60° line through (0.5, √3/2) and (-0.5, -√3/2) (opposite vertices), 120° line through (-0.5, √3/2) and (0.5, -√3/2) (opposite vertices); and 30° line through midpoint of (1,0)-(0.5, √3/2) and midpoint of (-1,0)-(-0.5, -√3/2), 90°? No, wait, no—wait, a regular hexagon has 6 lines of symmetry: 3 through opposite vertices, 3 through midpoints of opposite sides. So the midpoints of opposite sides: top side midpoint (0, √3/2), bottom side midpoint (0, -√3/2) (line $ w $ is this, vertical line), and two others: midpoint of (0.5, √3/2)-(1,0) and midpoint of (-0.5, -√3/2)-(-1,0), and midpoint of (-0.5, √3/2)-(-1,0) and midpoint of (0.5, -√3/2)-(1,0). Wait, maybe line $ w $ is the vertical line through midpoints of top and bottom sides (so that's a line of symmetry, 300°? No, 90°? Wait, no, the vertical line through (0,0) is 90°? No, 0° is horizontal, 90° is vertical. Wait, in standard position, 0° is right (1,0), 90° is up (0,1), but in the hexagon, the top side is horizontal? Wait, maybe the hexagon is oriented with a horizontal top and bottom side. So the top side is horizontal, midpoint at (0, h), bottom side horizontal, midpoint at (0, -h), so line $ w $ is the vertical line (90°) through (0,0), which is a line of symmetry (through midpoints of opposite horizontal sides). So reflection over line $ w $ (vertical) will map left to right, top to bottom, which works.

So the correct options are:

- Clockwise rotation about $ Y $ by 180°: Yes

- Reflection across line $ w $: Yes

Wait, but the original problem's line $ u $: let's see the diagram. Line $ u $ is a dashed line from the top-right vertex to the bottom-left side, bisecting a side. So line $ u $ is not a line of symmetry. So the correct transformations are Clockwise rotation about $ Y $ by 180°, Reflection across line $ w $. Wait, but the options include "Reflection across line $ u $"—is that correct? Let's think again. Maybe line $ u $ is through a vertex and the midpoint of the opposite side? No, the problem says "Line $ u $ passes through a vertex and bisects a side". In a regular hexagon, a line through a vertex and the midpoint of the adjacent side? No, that's not a symmetry. Wait, maybe I made a mistake. Let's check the regular hexagon's symmetries again.

Regular hexagon:

- Rotational symmetry: order 6, angles 60°, 120°, 180°, 240°, 300°, 360°.

- Reflection symmetry: 6 lines, 3 through opposite vertices, 3 through midpoints of opposite sides.

So line $ w $: bisects each side it passes through—so it passes through two sides, bisecting them. So that's a line through midpoints of opposite sides (so one of the 3 reflection lines through midpoints of opposite sides). So reflection over line $ w $ is a symmetry.

Line $ u $: passes through a vertex and bisects a side. So it's a line through a vertex and the midpoint of a side (not opposite side). Is that a line of symmetry? Let's take coordinates: vertex at (1,0) (0°), midpoint of adjacent side (between (1,0) and (0.5, √3/2)) is (0.75, √3/4). A line through (1,0) and (0.75, √3/4) is not a line of symmetry, because reflecting over it would not map the hexagon to itself. So reflection over line $ u $ is not a symmetry.

Clockwise rotation 180°: yes, as 180° is a multiple of 60° (3*60°), so that's a rotational symmetry.

Counterclockwise rotation 144°: 144/60=2.4, not integer, so no.

So the correct options are:

- Clockwise rotation about $ Y $ by 180°

- Reflection across line $ w $

Wait, but the options given are:

Options:

- Reflection across line $ u $: No

- Clockwise rotation about $ Y $ by 180°: Yes

- Counterclockwise rotation about $ Y $ by 144°: No

- Reflection across line $ w $: Yes

- None of the above: No

So the correct choices are "Clockwise rotation about $ Y $ by 180°" and "Reflection across line $ w $". Wait, but the problem says "Choose all that apply"—so we need to check which of the options are correct.

Wait, let's re-express:

1. Reflection across line $ u $: No (line $ u $ is not a line of symmetry)

2. Clockwise rotation about $ Y $ by 180°: Yes (rotational symmetry)

3. Counterclockwise rotation about $ Y $ by 144°: No (144 is not multiple of 60)

4. Reflection across line $ w $: Yes (line of symmetry through midpoints of opposite sides)

5. None of the above: No

So the correct options are the second and fourth: "Clockwise rotation about $ Y $ by 180°" and "Reflection across line $ w $".
# Answer:
- Clockwise rotation about $ Y $ by $ 180^\circ $
- Reflection across line $ w $

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