Rhombus PQRS is shown on the coordinate plane. Points M and N are midpoints of their respective sides.
Select ALL of the transformations that map the rhombus onto itself
a 90 degrees clockwise rotation around the center of the rhombus
A 180 degrees clockwise rotation around the center of the rhombus
A reflection across PR
A reflection across NM
a reflection across QS
The transformations that map the rhombus onto itself are:
- A 180 degrees clockwise rotation around the center of the rhombus
- A reflection across PR
To determine which transformations map the rhombus onto itself, let's analyze the properties of the rhombus PQRS:
1. Rhombus PQRS:
- PQRS is a parallelogram with opposite sides parallel.
- Opposite angles of PQRS are congruent.
- The diagonals of PQRS intersect at right angles and bisect each other.
Now, let's consider each transformation option:
1. A 90 degrees clockwise rotation around the center of the rhombus:
- This transformation does not preserve the properties of the rhombus: it changes the angles and the parallelism of the sides. Therefore, this transformation does not map the rhombus onto itself.
2. A 180 degrees clockwise rotation around the center of the rhombus:
- This transformation does preserve the properties of the rhombus: it keeps the angles, the parallelism of sides, and the intersection of diagonals. Therefore, this transformation maps the rhombus onto itself.
3. A reflection across PR:
- This reflection preserves the properties of the rhombus since it maintains the parallelism of sides, congruent angles, and the intersection of diagonals. Therefore, this transformation maps the rhombus onto itself.
4. A reflection across NM:
- This reflection does not preserve the properties of the rhombus: it changes the parallelism of the sides. Therefore, this transformation does not map the rhombus onto itself.
5. A reflection across QS:
- This reflection preserves the properties of the rhombus since it maintains the parallelism of sides, congruent angles, and the intersection of diagonals. Therefore, this transformation maps the rhombus onto itself.
Based on the analysis above, the transformations that map the rhombus PQRS onto itself are:
- A 180 degrees clockwise rotation around the center of the rhombus
- A reflection across PR
- A reflection across QS
To determine the transformations that map the rhombus onto itself, we need to visualize how each transformation affects the shape. Let's break it down step by step:
1. 90 degrees clockwise rotation around the center of the rhombus:
For this transformation, each point of the rhombus will be rotated by 90 degrees clockwise around the center. This transformation will retain the shape of the rhombus and map it onto itself.
2. 180 degrees clockwise rotation around the center of the rhombus:
Similar to the previous transformation, this rotation will map the rhombus onto itself by rotating each point by 180 degrees clockwise around the center.
3. Reflection across PR:
This reflection will flip the rhombus over a line passing through the midpoint of PR, which is also the center of the rhombus. It will preserve the shape, mapping it onto itself.
4. Reflection across NM:
Similarly, this reflection will flip the rhombus over a line passing through the midpoint of NM, again the center of the rhombus. It will also retain the shape of the rhombus.
5. Reflection across QS:
This reflection will not map the rhombus onto itself because it does not preserve the shape. It will change the orientation of the rhombus and, therefore, is not a valid transformation that maps the rhombus onto itself.
Based on the explanations above, the transformations that map the rhombus onto itself are:
- A 90 degrees clockwise rotation around the center of the rhombus
- A 180 degrees clockwise rotation around the center of the rhombus
- A reflection across PR
- A reflection across NM
Therefore, the correct selections are the first four options.