A rectangle in a coordinate plane has vertices (−1,9), (5,−1), (−1,−1) and (5,9). Identify a reflection that carries the rectangle onto itself.(1 point)

Question

Responses

rx=3
r subscript x equals 3 baseline

ry=5
r subscript y equals 5 baseline

rx=−1
r subscript x equals negative 1 baseline

ry=4

Answer 1

To analyze the given rectangle with vertices at (−1,9), (5,−1), (−1,−1), and (5,9), we want to find which reflection will carry the rectangle onto itself.

1. Identify the center of the rectangle: The center can be found by averaging the x-coordinates and y-coordinates of the rectangle's vertices.

   • The average x-coordinate: \[ \frac{-1 + 5}{2} = \frac{4}{2} = 2 \]

   • The average y-coordinate: \[ \frac{9 + (-1)}{2} = \frac{8}{2} = 4 \]

   Thus, the center of the rectangle is (2, 4).

2. Consider the possible reflections:

   • Reflection over the line \(x = 3\) (vertical line):
     • Reflecting a point to the left of the line will not give the same vertices on the right.
   • Reflection over the line \(y = 5\) (horizontal line):
     • Similar reasoning shows that this will not match the vertices.
   • Reflection over the line \(x = -1\) (vertical line):
     • This line would cause points to swap positions symmetrically but does not align with the center (2, 4).
   • Reflection over the line \(y = 4\) (horizontal line):
     • This line goes through the center of the rectangle (2, 4), and thus reflects the top part of the rectangle down onto the bottom part and vice versa, maintaining the vertices' positions.

3. Conclusion: The reflection that carries the rectangle onto itself is over the line \(y = 4\). Hence, the answer is:

ry=4

Answer 2

Which of the following summarizes the reflections that will carry a regular pentagon onto itself?(1 point)
Responses

A line drawn through each vertex and through the midpoint of the opposite side will create five lines of reflection that will carry a regular pentagon onto itself.
A line drawn through each vertex and through the midpoint of the opposite side will create five lines of reflection that will carry a regular pentagon onto itself.

A line through each vertex and through the opposite vertex and then through each side’s midpoint and through the opposite side’s midpoint will create five lines of reflection that will carry a regular pentagon onto itself.
A line through each vertex and through the opposite vertex and then through each side’s midpoint and through the opposite side’s midpoint will create five lines of reflection that will carry a regular pentagon onto itself.

A line through the midpoint of each side and through the midpoint of each opposite side will create five lines of reflection that carry a regular pentagon onto itself.
A line through the midpoint of each side and through the midpoint of each opposite side will create five lines of reflection that carry a regular pentagon onto itself.

A regular pentagon does not have any lines of reflection because it has an odd number of sides.

Answer 3

To summarize the reflections that will carry a regular pentagon onto itself, we need to understand the properties of the symmetries of a regular pentagon.

A regular pentagon has the following characteristics:

1. Lines of symmetry: A regular pentagon has 5 lines of symmetry.
2. Reflection lines: Each line of symmetry can be drawn through a vertex and the midpoint of the opposite side.

Based on this understanding of the regular pentagon, let's evaluate the given responses:

1. A line drawn through each vertex and through the midpoint of the opposite side will create five lines of reflection that will carry a regular pentagon onto itself.

   • This is correct because each vertex can be connected to the midpoint of the opposite side to form lines of symmetry.

2. A line through each vertex and through the opposite vertex and then through each side’s midpoint and through the opposite side’s midpoint will create five lines of reflection that will carry a regular pentagon onto itself.

   • This is incorrect because it describes lines that do not properly correspond to the lines of symmetry of a regular pentagon.

3. A line through the midpoint of each side and through the midpoint of each opposite side will create five lines of reflection that carry a regular pentagon onto itself.

   • This is incorrect because this description does not accurately represent the lines of symmetry of a regular pentagon.

4. A regular pentagon does not have any lines of reflection because it has an odd number of sides.

   • This is incorrect; a regular pentagon does have lines of reflection.

Therefore, the most accurate summary is:

A line drawn through each vertex and through the midpoint of the opposite side will create five lines of reflection that will carry a regular pentagon onto itself.

Answer 4

△ABC is reflected over the line x=2 to create △A′B′C′ , which is then reflected over the line y=−2 . Which of the following could describe this composition transformation using a single transformation?(1 point)
Responses

a translation in the same direction as the reflections
a translation in the same direction as the reflections

a reflection over the line
a reflection over the line

a rotation about the intersection of the two perpendicular lines
a rotation about the intersection of the two perpendicular lines

a dilation by a certain factor

Answer 5

To determine which option describes the composition of the transformations involving the two reflections, we need to analyze the effect of reflecting a triangle over the line \( x = 2 \) and then over the line \( y = -2 \).

1. First Reflection (over \( x = 2 \)):

   • A reflection over a vertical line \( x = c \) means each point of the triangle moves to the opposite side of the line. For example, if \( A(x, y) \) is a point, after reflection, it becomes \( A'(4 - x, y) \).

2. Second Reflection (over \( y = -2 \)):

   • Similarly, reflecting a point over \( y = c \) means moving it to the opposite side of that horizontal line. Thus, reflecting \( A'(x', y') \) from the first step over \( y = -2 \) results in \( A''(x', -4 - y') \).

3. Composition of Reflections:

   • The result of reflecting over two perpendicular lines (like \( x = 2 \) and \( y = -2 \)) is equivalent to performing a rotation. Specifically, reflecting across two intersecting lines results in a rotation about the point where these two lines intersect (which in this case is the point \( (2, -2) \)).
   • The angle of rotation will be double the angle between the two lines (which is 90 degrees here), resulting in a 180-degree rotation.

Therefore, the composition of the two reflections can be summarized as:

a rotation about the intersection of the two perpendicular lines.

This is the correct response.

Answer 6

Rectangle ABCD , with four 90° angles, undergoes a translation followed by a dilation. Which of the following properties are not preserved after this composition transformation?(1 point)
Responses

distance
distance

angle measure
angle measure

orientation
orientation

shape

Answer 7

To determine the properties that are not preserved after a translation followed by a dilation of rectangle ABCD, let's analyze the transformations:

1. Translation:

   • A translation moves every point of a shape the same distance in a given direction. Under translation, the following properties are preserved:
     • Distance: The distance between any two points remains the same.
     • Angle Measure: The angle measures remain the same.
     • Orientation: The orientation of the shape (the order of the vertices) remains the same.
     • Shape: The overall shape remains the same.

2. Dilation:

   • A dilation changes the size of a shape but keeps the shape similar. In this case, it retains the following properties:
     • Angle Measure: The angles remain the same after dilation.
     • Shape: The shape is preserved in terms of similarity (the rectangle remains similar to itself but changes in size).
     • Orientation: The orientation remains the same during dilation.
   • However, the distance between points changes because dilation scales the shape up or down.

From this analysis, the properties affected by the compositions of these transformations are:

• Distance is not preserved because the lengths of the sides change during dilation.

In conclusion, the correct answer to which property is not preserved after the translation followed by dilation is:

distance.