Question

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

Explanation:

Step1: Recall reflection rules

• Reflection over the line \(y = x\): The rule is \((x,y)\to(y,x)\).
• Reflection over the line \(y=-x\): The rule is \((x,y)\to(-y, -x)\).

Step2: Check coordinates

Let's assume a point. For example, if we consider a general point \((x,y)\) in the original figure and its image \((x',y')\) in the reflected figure.
If we take a point \(A(-1,0)\) in the original triangle (assuming coordinates based on the grid - let's assume \(A(-1,0)\), \(B(-3,4)\), \(C(-4,1)\)) and its image \(A'(0, - 1)\)
If we use the rule \((x,y)\to(-y,-x)\) for \(A(-1,0)\): \((-1,0)\to(0,1)\) (incorrect).
If we use the rule \((x,y)\to(y,x)\) for \(A(-1,0)\): \((-1,0)\to(0,-1)\) (incorrect for \(y = x\) reflection).
If we use the rule \((x,y)\to(-y,-x)\) for reflection over \(y=-x\).
Let's take another approach. The formula for reflection over the line \(y =-x\) is a transformation.
We know that for a point \((x,y)\) reflected over \(y=-x\), the new coordinates \((x',y')\) satisfy the following relationship.
The line \(y=-x\) has a slope of \(- 1\). The transformation matrix for reflection over the line \(y=-x\) is \(

$$\begin{pmatrix}0&-1\\-1&0\end{pmatrix}$$

\). When we multiply \(

$$\begin{pmatrix}0&-1\\-1&0\end{pmatrix}$$

\) with the column vector \(

$$\begin{pmatrix}x\\y\end{pmatrix}$$

\), we get \(

$$\begin{pmatrix}-y\\-x\end{pmatrix}$$

\), so the rule is \((x,y)\to(-y,-x)\)

Answer:

\(r_{y =-x}(x,y)\to(-y,-x)\)