Math / Geometry

QuestionA regular pentagon is shown below. Line $g$ passes through a vertex and bisects a side. Line $h$ passes through two vertices. Point $Y$ is the center of the pentagon.
Which transformation(s) must map the pentagon exactly onto itself? Choose all that apply. Reflection across line $g$ Reflection across line $h$ Counterclockwise rotation about $Y$ by $288^{\circ}$ Clockwise rotation about $Y$ by $60^{\circ}$ None of the above Explanation Check

Studdy Solution

STEP 1

1. The pentagon is regular, meaning all sides and angles are equal.
2. Line $g$ is a line of symmetry for the pentagon.
3. Line $h$ is not a line of symmetry for the pentagon.
4. Rotations about the center $Y$ by multiples of the internal angle of rotation will map the pentagon onto itself.

STEP 2

1. Determine if reflection across line $g$ maps the pentagon onto itself.
2. Determine if reflection across line $h$ maps the pentagon onto itself.
3. Determine if counterclockwise rotation about $Y$ by $288^\circ$ maps the pentagon onto itself.
4. Determine if clockwise rotation about $Y$ by $60^\circ$ maps the pentagon onto itself.
5. Conclude which transformations map the pentagon onto itself.

STEP 3

Determine if reflection across line $g$ maps the pentagon onto itself.
- Since line $g$ passes through a vertex and bisects the opposite side, it is a line of symmetry for the regular pentagon. - Therefore, reflection across line $g$ will map the pentagon onto itself.

STEP 4

Determine if reflection across line $h$ maps the pentagon onto itself.
- Line $h$ passes through two vertices but does not bisect the pentagon symmetrically. - Therefore, reflection across line $h$ will not map the pentagon onto itself.

STEP 5

Determine if counterclockwise rotation about $Y$ by $288^\circ$ maps the pentagon onto itself.
- The internal angle of rotation for a regular pentagon is $72^\circ$ . - A counterclockwise rotation by $288^\circ$ is equivalent to rotating by $4 \times 72^\circ$ . - Therefore, this rotation will map the pentagon onto itself.

STEP 6

Determine if clockwise rotation about $Y$ by $60^\circ$ maps the pentagon onto itself.
- The internal angle of rotation for a regular pentagon is $72^\circ$ . - A clockwise rotation by $60^\circ$ does not correspond to a multiple of $72^\circ$ . - Therefore, this rotation will not map the pentagon onto itself.

STEP 7

Conclude which transformations map the pentagon onto itself.
- Reflection across line $g$ and counterclockwise rotation about $Y$ by $288^\circ$ will map the pentagon onto itself. - Reflection across line $h$ and clockwise rotation by $60^\circ$ will not map the pentagon onto itself.
The transformations that map the pentagon onto itself are: - Reflection across line $g$ - Counterclockwise rotation about $Y$ by $288^\circ$

Was this helpful?

Studdy Solution

STEP 1

1. The pentagon is regular, meaning all sides and angles are equal.
2. Line $g$ is a line of symmetry for the pentagon.
3. Line $h$ is not a line of symmetry for the pentagon.
4. Rotations about the center $Y$ by multiples of the internal angle of rotation will map the pentagon onto itself.

STEP 2

STEP 3

Determine if reflection across line $g$ maps the pentagon onto itself.
- Since line $g$ passes through a vertex and bisects the opposite side, it is a line of symmetry for the regular pentagon. - Therefore, reflection across line $g$ will map the pentagon onto itself.

STEP 4

Determine if reflection across line $h$ maps the pentagon onto itself.
- Line $h$ passes through two vertices but does not bisect the pentagon symmetrically. - Therefore, reflection across line $h$ will not map the pentagon onto itself.

STEP 5

STEP 6

STEP 7

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

1. The pentagon is regular, meaning all sides and angles are equal.2. Line g g g is a line of symmetry for the pentagon.3. Line h h h is not a line of symmetry for the pentagon.4. Rotations about the center Y Y Y by multiples of the internal angle of rotation will map the pentagon onto itself.

STEP 2

STEP 3

Determine if reflection across line g g g maps the pentagon onto itself.- Since line g g g passes through a vertex and bisects the opposite side, it is a line of symmetry for the regular pentagon. - Therefore, reflection across line g g g will map the pentagon onto itself.

STEP 4

Determine if reflection across line h h h maps the pentagon onto itself.- Line h h h passes through two vertices but does not bisect the pentagon symmetrically. - Therefore, reflection across line h h h will not map the pentagon onto itself.

STEP 5

STEP 6

STEP 7

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. The pentagon is regular, meaning all sides and angles are equal.
2. Line $g$ is a line of symmetry for the pentagon.
3. Line $h$ is not a line of symmetry for the pentagon.
4. Rotations about the center $Y$ by multiples of the internal angle of rotation will map the pentagon onto itself.

Determine if reflection across line $g$ maps the pentagon onto itself.
- Since line $g$ passes through a vertex and bisects the opposite side, it is a line of symmetry for the regular pentagon. - Therefore, reflection across line $g$ will map the pentagon onto itself.

Determine if reflection across line $h$ maps the pentagon onto itself.
- Line $h$ passes through two vertices but does not bisect the pentagon symmetrically. - Therefore, reflection across line $h$ will not map the pentagon onto itself.