Rotation, Reflection, and Transformation Definitions: CCSS.Math.Content.HSG-CO.A.4 - Geometry

Drawing the described circle is the first step in answering this given question.

To calculate the number of symmetrical rotations, take one piece and rotate it clockwise until it results in the exact image as originally started with. Since the circle is divided into six equal pieces that means that rotating one of the pieces can be done six different times and still keep rotational symmetry, which also means keeping the angles of each piece the same and returning to the original image.

Therefore the correct answer is six.

Before solving this problem recall the definition of a reflection.

Reflection: The transformation of each and every point on a line or figure across some line where is some constant or linear function.

Applying this definition to the situation at hand result in the following depiction. The pencils are represented by the blue and black lines and the reflection line is the red dashed line. After the reflection occurs, the pencils essentially swap positions. Since the pencils were parallel to start with, they remain parallel after the reflection occurred.

Therefore, the correct answer is,

"The two pencils are still parallel."

For this particular question recall what rotational symmetry means. For an object to achieve rotational symmetry it must return to its original form. This can be accomplished by rotating the figure or equal parts of the figure in either the clockwise or non clockwise direction.

For this particular problem, it is stated that the polygon is "regular" meaning all sides and angles are equal to one another. It is also said that the polygon has sides. This means that the figure can be rotated times to return to its original form.

Therefore, to obtain rotational symmetry of a regular polygon, it must have an order equal to the number of sides.

Thus, the answer is .

For this particular question recall what rotational symmetry means. For an object to achieve rotational symmetry it must return to its original form. This can be accomplished by rotating the figure or equal parts of the figure in either the clockwise or non clockwise direction.

For this particular problem, look at the following image.

The polygon has six sides. This means that the figure can be rotated times to return to its original form.

Therefore, to obtain rotational symmetry of a regular polygon, it must have an order equal to the number of sides.

Thus, the answer is .

Recall that the circle is degrees. It is also important to recall that an equilateral triangle is a triangle whose side lengths are equal and the angles are equal. When the triangle is inscribed in the circle three lines can be drawn from the vertex on each point to the midpoint of the opposite side. This results in the following image.

From here, calculate the central angle between two points of the triangle. Since the circle is degrees and the triangle is divided into six equal sectors then each sector is degrees. Now, there are two sectors between two points of the triangle. Therefore, the minimum angle rotation that can be done to result in the same triangle is degrees.

To calculate the number of symmetrical rotations, take one piece and rotate it clockwise until, it results in the exact image as originally started with. Since the circle is divided into equal pieces that means that rotating one of the pieces can be done two different times and still keep rotational symmetry, which also means keeping the angles of each piece the same and returning to the original image.

Therefore the correct answer is .

To calculate the number of symmetrical rotations, take one piece and rotate it clockwise until, it results in the exact image as originally started with. Since the circle is divided into equal pieces that means that rotating one of the pieces can be done 8 different times and still keep rotational symmetry, which also means keeping the angles of each piece the same and returning to the original image.

Therefore the correct answer is .

To calculate the number of symmetrical rotations, take one piece and rotate it clockwise until, it results in the exact image as originally started with. Since the circle is divided into 16 equal pieces that means that rotating one of the pieces can be done different times and still keep rotational symmetry, which also means keeping the angles of each piece the same and returning to the original image.

Therefore the correct answer is .

To calculate the number of symmetrical rotations, take one piece and rotate it clockwise until, it results in the exact image as originally started with. Since the circle is divided into equal pieces that means that rotating one of the pieces can be done different times and still keep rotational symmetry, which also means keeping the angles of each piece the same and returning to the original image.

Therefore the correct answer is .

To calculate the number of symmetrical rotations, take one piece and rotate it clockwise until, it results in the exact image as originally started with. Since the circle is divided into equal pieces that means that rotating one of the pieces can be done different times and still keep rotational symmetry, which also means keeping the angles of each piece the same and returning to the original image.

Therefore the correct answer is .

To calculate the number of symmetrical rotations, take one piece and rotate it clockwise until, it results in the exact image as originally started with. Since the circle is divided into equal pieces that means that rotating one of the pieces can be done different times and still keep rotational symmetry, which also means keeping the angles of each piece the same and returning to the original image.

Therefore the correct answer is .

To calculate the number of symmetrical rotations, take one piece and rotate it clockwise until, it results in the exact image as originally started with. Since the circle is divided into equal pieces that means that rotating one of the pieces can be done different times and still keep rotational symmetry, which also means keeping the angles of each piece the same and returning to the original image.

Therefore the correct answer is .

Drawing the described circle is the first step in answering this given question.

To calculate the number of symmetrical rotations, take one piece and rotate it clockwise until it results in the exact image as originally started with. Since the circle is divided into six equal pieces that means that rotating one of the pieces can be done six different times and still keep rotational symmetry, which also means keeping the angles of each piece the same and returning to the original image.

Therefore the correct answer is six.

Before solving this problem recall the definition of a reflection.

Reflection: The transformation of each and every point on a line or figure across some line where is some constant or linear function.

Applying this definition to the situation at hand result in the following depiction. The pencils are represented by the blue and black lines and the reflection line is the red dashed line. After the reflection occurs, the pencils essentially swap positions. Since the pencils were parallel to start with, they remain parallel after the reflection occurred.

Therefore, the correct answer is,

"The two pencils are still parallel."

For this particular question recall what rotational symmetry means. For an object to achieve rotational symmetry it must return to its original form. This can be accomplished by rotating the figure or equal parts of the figure in either the clockwise or non clockwise direction.

For this particular problem, it is stated that the polygon is "regular" meaning all sides and angles are equal to one another. It is also said that the polygon has sides. This means that the figure can be rotated times to return to its original form.

Therefore, to obtain rotational symmetry of a regular polygon, it must have an order equal to the number of sides.

Thus, the answer is .

For this particular question recall what rotational symmetry means. For an object to achieve rotational symmetry it must return to its original form. This can be accomplished by rotating the figure or equal parts of the figure in either the clockwise or non clockwise direction.

For this particular problem, look at the following image.

The polygon has six sides. This means that the figure can be rotated times to return to its original form.

Therefore, to obtain rotational symmetry of a regular polygon, it must have an order equal to the number of sides.

Thus, the answer is .

Recall that the circle is degrees. It is also important to recall that an equilateral triangle is a triangle whose side lengths are equal and the angles are equal. When the triangle is inscribed in the circle three lines can be drawn from the vertex on each point to the midpoint of the opposite side. This results in the following image.

From here, calculate the central angle between two points of the triangle. Since the circle is degrees and the triangle is divided into six equal sectors then each sector is degrees. Now, there are two sectors between two points of the triangle. Therefore, the minimum angle rotation that can be done to result in the same triangle is degrees.

To calculate the number of symmetrical rotations, take one piece and rotate it clockwise until, it results in the exact image as originally started with. Since the circle is divided into equal pieces that means that rotating one of the pieces can be done two different times and still keep rotational symmetry, which also means keeping the angles of each piece the same and returning to the original image.

Therefore the correct answer is .

To calculate the number of symmetrical rotations, take one piece and rotate it clockwise until, it results in the exact image as originally started with. Since the circle is divided into equal pieces that means that rotating one of the pieces can be done 8 different times and still keep rotational symmetry, which also means keeping the angles of each piece the same and returning to the original image.

Therefore the correct answer is .

To calculate the number of symmetrical rotations, take one piece and rotate it clockwise until, it results in the exact image as originally started with. Since the circle is divided into 16 equal pieces that means that rotating one of the pieces can be done different times and still keep rotational symmetry, which also means keeping the angles of each piece the same and returning to the original image.

Therefore the correct answer is .