Math / Geometry

QuestionA regular pentagon is shown below. Line $c$ bisects each side it passes through. Line $d$ passes through a vertex and bisects a side. Point $X$ is the center of the pentagon.
Which transformation(s) must map the pentagon exactly onto itself? Choose all that apply. Clockwise rotation about $X$ by $120^{\circ}$ Reflection across line $d$ Reflection across line $c$ Counterclockwise rotation about $X$ by $180^{\circ}$ None of the above Explanation Check

Studdy Solution

STEP 1

1. The pentagon is regular, meaning all sides and angles are equal.
2. Line $c$ bisects two sides of the pentagon.
3. Line $d$ passes through a vertex and bisects the opposite side.
4. Point $X$ is the center of the pentagon.

STEP 2

1. Analyze the properties of a regular pentagon.
2. Determine the effect of each transformation.
3. Identify transformations that map the pentagon onto itself.

STEP 3

Analyze the properties of a regular pentagon: - A regular pentagon has rotational symmetry and reflectional symmetry. - It can be rotated by multiples of $72^\circ$ about its center to map onto itself. - It has lines of symmetry that pass through vertices and midpoints of opposite sides.

STEP 4

Determine the effect of each transformation:
- Clockwise rotation about $X$ by $120^\circ$ : - A regular pentagon has rotational symmetry of $72^\circ$ , $144^\circ$ , $216^\circ$ , and $288^\circ$ . - $120^\circ$ is not a multiple of $72^\circ$ , so this does not map the pentagon onto itself.
- Reflection across line $d$ : - Line $d$ is a line of symmetry as it passes through a vertex and bisects the opposite side. - Reflection across line $d$ maps the pentagon onto itself.
- Reflection across line $c$ : - Line $c$ bisects two sides but does not pass through a vertex. - It is not a line of symmetry for a regular pentagon, so this does not map the pentagon onto itself.
- Counterclockwise rotation about $X$ by $180^\circ$ : - $180^\circ$ is not a multiple of $72^\circ$ , so this does not map the pentagon onto itself.
- None of the above: - Based on the analysis, reflection across line $d$ is a valid transformation.

STEP 5

Identify transformations that map the pentagon onto itself: - Reflection across line $d$ .
The transformation that maps the pentagon onto itself is:
- Reflection across line $d$ .

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