Question

which transformation will always map a parallelogram onto itself?
a. a 90° rotation about its center
b. a reflection across one of its diagonals
c. a 180° rotation about its center
d. a reflection across a line joining the midpoints of opposite sides

Explanation:

Step1: Recall properties of parallelograms

A parallelogram has opposite sides equal and parallel, and opposite angles equal. The center of a parallelogram is the midpoint of its diagonals.

Step2: Analyze each transformation

• Option A: A \(90^\circ\) rotation about the center. Most parallelograms (e.g., non - square rectangles, non - rhombus parallelograms) will not map onto themselves with a \(90^\circ\) rotation. For a figure to be mapped onto itself by a \(90^\circ\) rotation, it should have rotational symmetry of order 4 (like a square), which is not true for all parallelograms.
• Option B: A reflection across one of its diagonals. Only rhombuses (a special type of parallelogram) have this property. In a general parallelogram (e.g., a rectangle that is not a square), reflecting across a diagonal will not map the figure onto itself.
• Option C: A \(180^\circ\) rotation about its center. Since the center is the midpoint of the diagonals, rotating a parallelogram \(180^\circ\) about its center will map each vertex to the vertex opposite to it. Because opposite sides are equal and parallel, and opposite angles are equal, the image of the parallelogram after a \(180^\circ\) rotation about the center will coincide with the original parallelogram.
• Option D: A reflection across a line joining the midpoints of opposite sides. Only rectangles (a special type of parallelogram) have this property. In a general parallelogram (e.g., a non - rectangular parallelogram), reflecting across such a line will not map the figure onto itself.

Answer:

C. a \(180^\circ\) rotation about its center