Answer :
To determine the coordinates of the vertices of the image of \(\triangle ABC\) after rotating \(180^\circ\) clockwise about the origin and then reflecting across the line \(y = -x\), let’s follow each transformation step by step.
### Step 1: Rotate \(180^\circ\) clockwise about the origin
Rotating a point \((x, y)\) by \(180^\circ\) clockwise about the origin results in the point \((-x, -y)\).
- For vertex \(A(-3, 0)\):
[tex]\[ A \rightarrow A_{\text{rotated}} = (3, 0) \][/tex]
- For vertex \(B(-2, 3)\):
[tex]\[ B \rightarrow B_{\text{rotated}} = (2, -3) \][/tex]
- For vertex \(C(-1, 1)\):
[tex]\[ C \rightarrow C_{\text{rotated}} = (1, -1) \][/tex]
### Step 2: Reflect across the line \(y = -x\)
Reflecting a point \((x, y)\) across the line \(y = -x\) results in the point \((-y, -x)\).
- For vertex \(A_{\text{rotated}}(3, 0)\):
[tex]\[ A_{\text{rotated}} \rightarrow A' = (0, -3) \][/tex]
- For vertex \(B_{\text{rotated}}(2, -3)\):
[tex]\[ B_{\text{rotated}} \rightarrow B' = (3, -2) \][/tex]
- For vertex \(C_{\text{rotated}}(1, -1)\):
[tex]\[ C_{\text{rotated}} \rightarrow C' = (1, -1) \][/tex]
Thus, the coordinates of the vertices of the image \(\triangle A'B'C'\) are:
[tex]\[ A'(0, -3), B'(3, -2), C'(1, -1) \][/tex]
Therefore, the correct answer is:
B. [tex]\(A^{\prime}(0, -3), B^{\prime}(3, -2), C^{\prime}(1, -1)\)[/tex]
### Step 1: Rotate \(180^\circ\) clockwise about the origin
Rotating a point \((x, y)\) by \(180^\circ\) clockwise about the origin results in the point \((-x, -y)\).
- For vertex \(A(-3, 0)\):
[tex]\[ A \rightarrow A_{\text{rotated}} = (3, 0) \][/tex]
- For vertex \(B(-2, 3)\):
[tex]\[ B \rightarrow B_{\text{rotated}} = (2, -3) \][/tex]
- For vertex \(C(-1, 1)\):
[tex]\[ C \rightarrow C_{\text{rotated}} = (1, -1) \][/tex]
### Step 2: Reflect across the line \(y = -x\)
Reflecting a point \((x, y)\) across the line \(y = -x\) results in the point \((-y, -x)\).
- For vertex \(A_{\text{rotated}}(3, 0)\):
[tex]\[ A_{\text{rotated}} \rightarrow A' = (0, -3) \][/tex]
- For vertex \(B_{\text{rotated}}(2, -3)\):
[tex]\[ B_{\text{rotated}} \rightarrow B' = (3, -2) \][/tex]
- For vertex \(C_{\text{rotated}}(1, -1)\):
[tex]\[ C_{\text{rotated}} \rightarrow C' = (1, -1) \][/tex]
Thus, the coordinates of the vertices of the image \(\triangle A'B'C'\) are:
[tex]\[ A'(0, -3), B'(3, -2), C'(1, -1) \][/tex]
Therefore, the correct answer is:
B. [tex]\(A^{\prime}(0, -3), B^{\prime}(3, -2), C^{\prime}(1, -1)\)[/tex]
