Question

what transformation is always the same as rotating a figure 180 degrees about the origin?
rotating 90 degrees and then 180 degrees
reflecting over the line y = x
reflecting over the origin
translating left 4 units and down 4 units

Explanation:

A 180-degree rotation about the origin maps a point $(x,y)$ to $(-x,-y)$. Reflecting a point over the origin also maps $(x,y)$ to $(-x,-y)$, so this transformation is equivalent to a 180-degree rotation. Rotating 90 degrees then 180 degrees results in a 270-degree rotation, which is different. Reflecting over $y=x$ swaps $x$ and $y$, which is not a 180 rotation. Translation is a shift, not a rotation.

Answer:

reflecting over the origin