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To determine which transformation would carry the rhombus onto itself, we need to examine the symmetries of the rhombus as it is graphed.
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1. 180° Rotation Counterclockwise about the Origin:
– A 180° rotation about the origin would map each point (( x, y) ) of the rhombus to (( – x, – y) ). This would carry the rhombus onto itself because the rhombus is symmetric around the origin.
2. Reflection over the Line (y = frac { 1} { 2} x + 1):
– Reflecting over the line (y = frac { 1} { 2} x + 1) would not necessarily map the rhombus onto itself as this line does not align with any of the axes of symmetry of the rhombus.
3. Reflection over the Line (x = 0):
– Reflecting over the line (x = 0) (the y-axis) would map each point (( x, y) ) to (( – x, y) ). This would carry the rhombus onto itself because the rhombus is symmetric with respect to the y-axis.
4. Reflection over the Line (y = 0):
– Reflecting over the line (y = 0) (the x-axis) would map each point (( x, y) ) to (( x, – y) ). This would carry the rhombus onto itself because the rhombus is symmetric with respect to the x-axis.
Given these options, the transformations that would carry the rhombus onto itself are:
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– A 180° rotation counterclockwise about the origin.
– Reflection over the line (x = 0).
– Reflection over the line (y = 0).
Since the question asks for which transformation would carry the rhombus onto itself, the correct answers are:
– A 180° rotation counterclockwise about the origin
– Reflection over the line (x = 0)
– Reflection over the line (y = 0)
Therefore, the correct choices are A, C, and D.
Supplemental Knowledge
Understanding the symmetries of geometric shapes, such as a rhombus, is essential in determining which transformations will map the shape onto itself. Here are some additional insights into this topic:
1. Symmetry in a Rhombus:
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– A rhombus has two lines of symmetry that pass through its diagonals.
– It also has rotational symmetry of 180 degrees around its center.
2. Types of Transformations:
– Rotation: Rotating a shape around a point by a certain angle. For a rhombus, a 180° rotation about its center will map it onto itself.
– Reflection: Reflecting a shape over a line (axis). For a rhombus, reflections over lines that coincide with its axes of symmetry (the diagonals) or the coordinate axes (if aligned properly) will map it onto itself.
3. Coordinate Transformations:
– Rotation by 180° about the origin: This transformation changes (( x, y) ) to (( – x, – y) ).
– Reflection over (x = 0): This transformation changes (( x, y) ) to (( – x, y) ).
– Reflection over (y = 0): This transformation changes (( x, y) ) to (( x, – y) ).
4. Symmetry and Invariance:
– Shapes like squares and rhombuses exhibit multiple symmetries that make them invariant under specific transformations.
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