Question

what is the rule for the reflection?
o (r_{y = x}(x,y)-( - y,-x))
o (r_{y=-x}(x,y)-( - y,-x))
o (r_{y = x}(x,y)-(y,x))
o (r_{y=-x}(x,y)-(y,x))

Explanation:

Step1: Recall reflection rules

When reflecting a point $(x,y)$ over the line $y = -x$, the transformation rule is $(x,y)\to(-y,-x)$. When reflecting over the line $y=x$, the rule is $(x,y)\to(y,x)$. Analyze the orientation of the pre - image and image triangles.

Step2: Observe the transformation

The original triangle $\triangle ABC$ is reflected to $\triangle A'B'C'$. By looking at the coordinates of corresponding points, we can see that the transformation is a reflection over the line $y=-x$. For example, if we consider a general point $(x,y)$ in the pre - image, its image under the reflection over $y = -x$ has coordinates $(-y,-x)$.

Answer:

$r_{y=-x}(x,y)=(-y,-x)$