Question

transformations
identifying transformations that map a regular polygon onto itself
a regular hexagon is shown below.
line w passes through a vertex and bisects a side.
line u passes through two vertices.
point x is the center of the hexagon.
which transformation(s) must map the hexagon exactly onto itself? choose all that apply.

• reflection across line w
• clockwise rotation about x by 288°
• reflection across line u
• counterclockwise rotation about x by 144°
• none of the above

Explanation:

Step1: Analyze Reflection across line w

A regular hexagon has reflection symmetry. Line w passes through a vertex and bisects a side. For a regular hexagon, a line through a vertex and the midpoint of the opposite side is a line of reflection. So reflection across line w will map the hexagon onto itself.

Step2: Analyze Clockwise rotation about X by 288°

The central angle between adjacent vertices in a regular hexagon is $\frac{360^{\circ}}{6} = 60^{\circ}$. A clockwise rotation of $288^{\circ}$ is equivalent to a counterclockwise rotation of $360^{\circ}- 288^{\circ}=72^{\circ}$, but also, since $288^{\circ}= 4\times60^{\circ}+ 48^{\circ}$? Wait, no, $288\div60 = 4.8$, but actually, the order of rotation symmetry is 6, so rotations by multiples of $60^{\circ}$ map it onto itself. Wait, $288^{\circ}= 60^{\circ}\times4 + 48^{\circ}$? No, $60\times4 = 240$, $60\times5=300$. Wait, no, $360 - 72=288$, and $72$ is not a multiple of $60$. Wait, no, I made a mistake. The central angle is $60^{\circ}$, so rotations by $60k^{\circ}$, where $k = 0,1,2,3,4,5$ map it onto itself. $288\div60 = 4.8$, which is not an integer. Wait, no, $288 = 360 - 72$, but $72$ is not a multiple of $60$. Wait, no, maybe I miscalculated. Wait, $60\times4 = 240$, $60\times5 = 300$. So $288$ is not a multiple of $60$. Wait, but that can't be. Wait, no, the order of the hexagon is 6, so the rotation angle should be $\frac{360}{6}=60^{\circ}$ per step. So rotations by $60^{\circ}, 120^{\circ}, 180^{\circ}, 240^{\circ}, 300^{\circ}$ (and $360^{\circ}$ which is the same as $0^{\circ}$) map it onto itself. $288^{\circ}$ is not one of these. Wait, but maybe I messed up. Wait, let's check the rotation: a clockwise rotation of $288^{\circ}$ is the same as a counterclockwise rotation of $72^{\circ}$, but $72$ is not a multiple of $60$. So that's not a symmetry? Wait, no, wait the regular hexagon has rotational symmetry of order 6, so only rotations by multiples of $60^{\circ}$ (i.e., $60k^{\circ}$, $k = 0,1,2,3,4,5$) map it onto itself. So $288^{\circ}$ is not a multiple of $60^{\circ}$? Wait, $60\times4 = 240$, $60\times5 = 300$, so $288$ is between them. So that's not a symmetry. Wait, but maybe I made a mistake here. Wait, no, let's recalculate: $360\div6 = 60$, so each rotation by $60^{\circ}$ (and multiples) works. So $288^{\circ}$: $288\div60 = 4.8$, not an integer. So that's not a symmetry. Wait, but the answer options: maybe I was wrong. Wait, let's check the other options.

Step3: Analyze Reflection across line u

Line u passes through two vertices. In a regular hexagon, a line passing through two opposite vertices is a line of reflection symmetry. So reflection across line u (which is a diameter through two vertices) will map the hexagon onto itself.

Step4: Analyze Counterclockwise rotation about X by 144°

The central angle is $60^{\circ}$, so $144^{\circ}\div60^{\circ}=2.4$, which is not an integer. So a counterclockwise rotation of $144^{\circ}$ is not a multiple of $60^{\circ}$, so it won't map the hexagon onto itself.

So let's re-express:

- Reflection across line w: Line w passes through a vertex and bisects a side. In a regular hexagon, a line through a vertex and the midpoint of the opposite side is a line of reflection symmetry. So this should work. Wait, earlier I thought line u is through two vertices, but line w is through a vertex and midpoint of a side. Let's confirm the reflection symmetries of a regular hexagon. A regular hexagon has 6 lines of reflection: 3 through opposite vertices (like line u) and 3 through the midpoints of opposite sides (like line w). So both line w and line u are lines of reflection symmetry.

Wait, so reflection across line w: yes, because line w is a line of reflection (through a vertex and midpoint of opposite side). Reflection across line u: yes, because line u is through two opposite vertices.

Now, rotation:

- Clockwise rotation by $288^{\circ}$: As $360 - 288 = 72$, but $72$ is not a multiple of $60$. Wait, no, $288 = 60\times4 + 48$? No, $60\times4 = 240$, $288 - 240 = 48$. Not a multiple. But wait, the order of rotation is 6, so rotations by $60k$ degrees. $288\div60 = 4.8$, not integer. So that's not a rotation symmetry.

- Counterclockwise rotation by $144^{\circ}$: $144\div60 = 2.4$, not integer. So not a rotation symmetry.

Wait, but let's check the rotation angles again. The regular hexagon has rotational symmetry of order 6, so the angle of rotation is $360/6 = 60$ degrees. So rotations by 60, 120, 180, 240, 300 degrees (and 360, which is 0) are symmetries. So 144 is not a multiple of 60 (144 = 60*2 + 24), so no. 288: 288 = 60*4 + 48, so no.

But reflection across line w: yes, because line w is a line of reflection (through a vertex and midpoint of opposite side). Reflection across line u: yes, because line u is through two opposite vertices.

Wait, but let's check the options again. The options are:

- Reflection across line w: yes.

- Clockwise rotation about X by 288°: no, as 288 is not a multiple of 60.

- Reflection across line u: yes.

- Counterclockwise rotation about X by 144°: no.

- None of the above: no, because reflection across w and u are valid.

Wait, but maybe I made a mistake in the rotation. Wait, 288 degrees clockwise: 360 - 288 = 72 degrees counterclockwise. But 72 is not a multiple of 60. So that's not a symmetry.

Wait, but let's confirm the reflection symmetries. A regular hexagon has 6 lines of symmetry: 3 through opposite vertices (the "vertex-vertex" lines) and 3 through the midpoints of opposite sides (the "side-side" lines, or vertex-midpoint lines). So line w is a vertex-midpoint line, line u is a vertex-vertex line. So both are lines of symmetry, so reflections across them work.

So the correct options are Reflection across line w, Reflection across line u, and wait, what about the rotation? Wait, 288 degrees clockwise: 288 = 60*4 + 48? No, 60*4 = 240, 288 - 240 = 48. Not a multiple. So that's not a symmetry.

Wait, but let's check the rotation again. The order of the hexagon is 6, so the rotation angle is 60 degrees. So rotations by 60, 120, 180, 240, 300 degrees. 288 degrees is 360 - 72, and 72 is not a multiple of 60. So that's not a symmetry.

So the correct options are Reflection across line w and Reflection across line u. Wait, but let's check the problem again. The problem says "Choose all that apply".

Wait, let's re-express:

- Reflection across line w: Line w passes through a vertex and bisects a side (i.e., midpoint of a side). In a regular hexagon, such a line is a line of reflection symmetry. So yes.

- Clockwise rotation about X by 288°: 288° is equal to 360° - 72°, but 72° is not a multiple of 60°, so this rotation does not map the hexagon onto itself. Wait, but 288° = 4*72°, but 72° is the central angle for a regular pentagon. No, hexagon is 60°. So no.

- Reflection across line u: Line u passes through two vertices (opposite vertices), so this is a line of reflection symmetry. So yes.

- Counterclockwise rotation about X by 144°: 144° is not a multiple of 60°, so no.

So the correct options are Reflection across line w and Reflection across line u. Wait, but let's check the initial analysis again.

Wait, maybe I made a mistake with the rotation. Let's calculate the rotation angle:

A regular hexagon has 6 sides, so the angle between adjacent vertices is 60 degrees. So a rotation by 60k degrees, where k is an integer between 0 and 5, will map it onto itself.

288 degrees: 288 divided by 60 is 4.8, which is not an integer, so that's not a valid rotation.

144 degrees: 144 divided by 60 is 2.4, not an integer, so not valid.

Reflections:

- Line w: through a vertex and midpoint of a side: yes, reflection symmetry.

- Line u: through two vertices: yes, reflection symmetry.

So the correct options are Reflection across line w and Reflection across line u.

Answer:

Reflection across line w, Reflection across line u (the checkboxes for these two options should be selected)