Question

4
which transformation maps the parallelogram onto itself?
a. a 180° rotation about the origin
b. a reflection over the y - axis
c. a reflection over the x - axis
d. a 90° rotation about the origin

Explanation:

Step1: Analyze 180° rotation

A 180° rotation about the origin maps a point \((x,y)\) to \((-x,-y)\). For a parallelogram centered (or with symmetry) about the origin, this rotation will map each vertex to another vertex of the parallelogram, preserving its shape and position relative to itself.

Step2: Analyze reflection over y - axis

A reflection over the y - axis maps \((x,y)\) to \((-x,y)\). Looking at the parallelogram, the left and right sides are not symmetric with respect to the y - axis (the shape is slanted), so this reflection will not map the parallelogram onto itself.

Step3: Analyze reflection over x - axis

A reflection over the x - axis maps \((x,y)\) to \((x,-y)\). The top and bottom sides of the parallelogram are not symmetric with respect to the x - axis (the shape is slanted), so this reflection will not map the parallelogram onto itself.

Step4: Analyze 90° rotation

A 90° rotation about the origin maps \((x,y)\) to \((-y,x)\) (counter - clockwise) or \((y,-x)\) (clockwise). This rotation will change the orientation of the slanted sides of the parallelogram, so it will not map the parallelogram onto itself.

Answer:

A. a 180° rotation about the origin