To rotate the rectangular object around the origin, first recall the definition for a rotation and origin.

Rotation: To rotate an object either clockwise or counter clockwise around a center point. In this particular case the center point is the origin or the point \(\displaystyle (0,0)\) where the x and y axis intersect.

Looking at the original image and making one 90 degree rotation around the origin results in the following.

When rotating, the bottom right point will become the bottom left point, the top right point becomes the bottom right point, the left bottom point becomes the top left point, and the left top point becomes the top right point. 

Now, rotating it another \(\displaystyle 90\) degrees to get to the \(\displaystyle 180\) degree rotated image, results in the following

To rotate the rectangular object around the origin, first recall the definition for a rotation and origin.

Rotation: To rotate an object either clockwise or counter clockwise around a center point. In this particular case the center point is the origin or the point \(\displaystyle (0,0)\) where the x and y axis intersect.

Looking at the original image and making one rotation around the origin results in the following.

When rotating, the bottom right point will become the bottom left point, the top right point becomes the bottom right point, the left bottom point becomes the top left point, and the left top point becomes the top right point. The visual representation for this rotation is as follows.

To rotate the rectangular object around the origin, first recall the definition for a rotation and origin.

Rotation: To rotate an object either clockwise or counter clockwise around a center point. In this particular case the center point is the origin or the point \(\displaystyle (0,0)\) where the x and y axis intersect.

Looking at the original image and making one \(\displaystyle 90\) degree rotation around the origin results in the following.

When rotating, the bottom right point will become the bottom left point, the top right point becomes the bottom right point, the left bottom point becomes the top left point, and the left top point becomes the top right point. 

Now, rotating it another \(\displaystyle 90\) degrees to get to the \(\displaystyle 180\) degree rotated image, results in the following

From here one final rotation must occur to reach \(\displaystyle 270\) degrees.

To rotate the rectangular object around the origin, first recall the definition for a rotation and origin.

Rotation: To rotate an object either clockwise or counter clockwise around a center point. In this particular case the center point is the origin or the point \(\displaystyle (0,0)\) where the x and y axis intersect.

Looking at the original image and making one 90 degree rotation around the origin results in the following.

When rotating, the bottom right point will become the bottom left point, the top right point becomes the bottom right point, the left bottom point becomes the top left point, and the left top point becomes the top right point. 

Now, rotating it another \(\displaystyle 90\) degrees to get to the \(\displaystyle 180\) degree rotated image, results in the following

From here another rotation must occur to reach \(\displaystyle 270\) degrees.

Lastly, complete one more \(\displaystyle 90\) degree rotation to land at the \(\displaystyle 360\) degree mark. Notice that rotating 360 degrees lands the image back at its original spot.

To determine whether this statement is true or false first recall what a \(\displaystyle 90\) rotation and what a reflection is.

Reflection: To flip the orientation of an object over a specific line or function.

Rotation: To rotate an object either clockwise or counter clockwise around a center point.

Now looking at the statement, draw images that depict the 90 degree rotation and the reflection of the original image to verify the truth of the statement.

"An equilateral diamond when rotated 90 degrees around its center results in the reflected image of two isosceles triangle."

The original equilateral diamond is:

The diamond rotated \(\displaystyle 90\) degrees:

The reflection of an isosceles triangle.

Therefore, the statement is true.

To determine whether this statement is true or false first recall what a \(\displaystyle 180\) rotation and what a reflection is.

Reflection: To flip the orientation of an object over a specific line or function.

Rotation: To rotate an object either clockwise or counter clockwise around a center point.

Now looking at the statement, draw images that depict the \(\displaystyle 180\) degree rotation and the reflection of the original image to verify the truth of the statement.

"An equilateral diamond when rotated \(\displaystyle 90\) degrees around its right most corner results in the reflected image of the original diamond."

The original equilateral diamond with rotation of \(\displaystyle 90\) degrees on its right most corner is as follows:

The rotated image results in the reflected original image.

Therefore, the statement is true. 

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 \(\displaystyle y=c\) where \(\displaystyle c\) 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."

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.

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 \(\displaystyle 18\) sides. This means that the figure can be rotated \(\displaystyle 18\) 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 \(\displaystyle 18\).

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 \(\displaystyle 6\) 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 \(\displaystyle 6\).