Question

which transformation would take figure a to figure b? answer a clockwise rotation of 180° about the origin a reflection over the line y = -x a clockwise rotation of 270° about the origin a reflection over the line y = x

Explanation:

Step1: Recall rotation rules

A $180^{\circ}$ clock - wise rotation about the origin changes the coordinates $(x,y)$ of a point to $(-x,-y)$. A $270^{\circ}$ clock - wise rotation about the origin changes $(x,y)$ to $(y, - x)$. Reflection over $y = x$ changes $(x,y)$ to $(y,x)$ and reflection over $y=-x$ changes $(x,y)$ to $(-y,-x)$.

Step2: Analyze the transformation visually

By observing the orientation and position of Figure A and Figure B, we can check each transformation. For a $180^{\circ}$ clock - wise rotation about the origin, the figure would be upside - down in a different way. For a reflection over $y = x$, the orientation is not correct. For a reflection over $y=-x$, it doesn't match either.
For a $270^{\circ}$ clock - wise rotation about the origin, if we take a general point $(x,y)$ on Figure A and apply the transformation $(x,y)\to(y, - x)$, we can see that it matches the position and orientation of Figure B.

Answer:

A clockwise rotation of $270^{\circ}$ about the origin