Rotations and Reflections of Rectangles, Parallelograms, Trapezoids, and Regular Polygons: CCSS.Math.Content.HSG-CO.A.3 - Geometry

To determine whether the statement is true or false, identify any example that would make the statement false.

"Reflections and rotations never result in the same image."

Imagine a square that exists in quadrant two. When this image is reflected across the y-axis it is still a square that is now in quadrant one. When the square from quadrant two is rotated around the origin, it results in the same image in quadrant one. Therefore, it is possible for a reflection and rotation to result in the same image.

Thus, the statement "Reflections and rotations never result in the same image." is false.

Recall that a hexagon is a six sided figure. Now recall that a line of symmetry creates two mirrored, identical figures which is also known as a reflection.

Knowing these two pieces of information draw the image of the hexagon and the lines of symmetry

Counting up these lines results 6 lines of symmetry.

Therefore, the answer is 6.

Since a cylinder is composed of two, identical circular bases that are separated by the height of the cylinder, the transformation that must occur to have one carried onto the other is a translation. Recall that a translation is the sliding of an object without changing its size or shape. Therefore, for the one base to be carried onto the other, translation is the geometric transformation that must occur.

Looking at the statement and the image given, it is seen that the object is a trapezoid.

A trapezoid has two bases of differing lengths and two side pieces of equal length in this particular case.

For an object to be divided into images that can be reflected onto one another, the images must be identically mirrored.

To identify the possible solutions, draw the lines of symmetry.

Only one line of symmetry could be drawn to allow for two images of the trapezoid to be reflected over that line. The trapezoid cannot be divided in any other way that will result in a reflected image.

Therefore, the original statement is false.

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 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 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 degrees to get to the 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 where the x and y axis intersect.

Looking at the original image and making one 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 degrees to get to the degree rotated image, results in the following

From here one final rotation must occur to reach 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 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 degrees to get to the degree rotated image, results in the following

From here another rotation must occur to reach degrees.

Lastly, complete one more degree rotation to land at the 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 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 degrees:

The reflection of an isosceles triangle.

Therefore, the statement is true.

Since a trapezoidal based prism is composed of two, identical trapezoidal bases that are separated by the height of the prism, the transformation that must occur to have one base carried onto the other is a translation. Recall that a translation is the sliding of an object without changing its size or shape. Therefore, for the one base to be carried onto the other, translation is the geometric transformation that must occur.

To determine whether this statement is true or false first recall what a 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 degree rotation and the reflection of the original image to verify the truth of the statement.

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

The original equilateral diamond with rotation of degrees on its right most corner is as follows:

The rotated image results in the reflected original image.

Therefore, the statement is true.

In order to create the resulting image of a reflection first recall what a reflection is.

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

In this specific situation the line of reflection is the x-axis.

The original image is,

When the image is reflected across the x-axis result in,

This is the correct answer.

To determine whether the statement is true or false, identify any example that would make the statement false.

"Reflections and rotations never result in the same image."

Imagine a square that exists in quadrant two. When this image is reflected across the y-axis it is still a square that is now in quadrant one. When the square from quadrant two is rotated around the origin, it results in the same image in quadrant one. Therefore, it is possible for a reflection and rotation to result in the same image.

Thus, the statement "Reflections and rotations never result in the same image." is false.

Recall that a hexagon is a six sided figure. Now recall that a line of symmetry creates two mirrored, identical figures which is also known as a reflection.

Knowing these two pieces of information draw the image of the hexagon and the lines of symmetry

Counting up these lines results 6 lines of symmetry.

Therefore, the answer is 6.

Since a cylinder is composed of two, identical circular bases that are separated by the height of the cylinder, the transformation that must occur to have one carried onto the other is a translation. Recall that a translation is the sliding of an object without changing its size or shape. Therefore, for the one base to be carried onto the other, translation is the geometric transformation that must occur.

Looking at the statement and the image given, it is seen that the object is a trapezoid.

A trapezoid has two bases of differing lengths and two side pieces of equal length in this particular case.

For an object to be divided into images that can be reflected onto one another, the images must be identically mirrored.

To identify the possible solutions, draw the lines of symmetry.

Only one line of symmetry could be drawn to allow for two images of the trapezoid to be reflected over that line. The trapezoid cannot be divided in any other way that will result in a reflected image.

Therefore, the original statement is false.

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 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 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 degrees to get to the 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 where the x and y axis intersect.

Looking at the original image and making one 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 degrees to get to the degree rotated image, results in the following

From here one final rotation must occur to reach 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 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 degrees to get to the degree rotated image, results in the following

From here another rotation must occur to reach degrees.

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