To reflect the given triangle over the line \( y = -x \), let's follow these steps:
### 1. Understand the Geometry of Reflection over \( y = -x \)
Reflecting a point \((x, y)\) over the line \(y = -x\) switches the coordinates and changes their signs, producing the image \((-y, -x)\).
### 2. Identify the Points of the Triangle
The given triangle has three vertices represented by points in the coordinate plane:
- \( A = (0, 0) \)
- \( B = (7, -3) \)
- \( C = (-1, -4) \)
### 3. Apply the Reflection Rule to Each Point
We will apply the rule for reflection over \( y = -x \) to each coordinate:
- Reflecting point \( A \):
[tex]\[ (0, 0) \Rightarrow (0, 0) \][/tex]
- Reflecting point \( B \):
[tex]\[ (7, -3) \Rightarrow (3, -7) \][/tex]
- Reflecting point \( C \):
[tex]\[ (-1, -4) \Rightarrow (4, 1) \][/tex]
### 4. Compile the Reflected Points
After reflecting each point, the coordinates of the reflected triangle will be:
- \( A' = (0, 0) \)
- \( B' = (3, -7) \)
- \( C' = (4, 1) \)
### 5. Verify the Complete Reflection
Putting the original points and their reflected points together:
- Original points of the triangle: [tex]\[ \left[ \begin{array}{ccc} 0 & 7 & -1 \\ 0 & -3 & -4 \end{array} \right] \][/tex]
- Reflected points of the triangle over \( y = -x \): [tex]\[ \left[ \begin{array}{ccc} 0 & 3 & 4 \\ 0 & -7 & 1 \end{array} \right] \][/tex]
### Conclusion
The original points and the reflected points of the triangle over the line \( y = -x \) are as follows:
- Original points:
[tex]\[ \left[ \begin{array}{ccc} 0 & 7 & -1 \\ 0 & -3 & -4 \end{array} \right] \][/tex]
- Reflected points:
[tex]\[ \left[ \begin{array}{ccc} 0 & 3 & 4 \\ 0 & -7 & 1 \end{array} \right] \][/tex]
