To determine which reflection will produce an image of triangle \( \Delta RST \) with a vertex at \( (2, -3) \), we need to explore the coordinates of the vertices after each reflection.
-
Reflection across the x-axis:
- The x-coordinates remain the same, and the y-coordinates are negated.
- Reflecting each vertex:
- \( R(-1, 3) \) becomes \( R'(-1, -3) \)
- \( S(3, -2) \) becomes \( S'(3, 2) \)
- \( T(1, -4) \) becomes \( T'(1, 4) \)
So, the image after reflecting across the x-axis is \( R'(-1, -3) \), \( S'(3, 2) \), and \( T'(1, 4) \). There is no vertex at \( (2, -3) \).
-
Reflection across the y-axis:
- The y-coordinates remain the same, and the x-coordinates are negated.
- Reflecting each vertex:
- \( R(-1, 3) \) becomes \( R'(1, 3) \)
- \( S(3, -2) \) becomes \( S'(-3, -2) \)
- \( T(1, -4) \) becomes \( T'(-1, -4) \)
The image after reflecting across the y-axis is \( R'(1, 3) \), \( S'(-3, -2) \), and \( T'(-1, -4) \). No vertex at \( (2, -3) \).
-
Reflection across the line \( y = x \):
- The x and y coordinates of each point are swapped.
- Reflecting each vertex:
- \( R(-1, 3) \) becomes \( R'(3, -1) \)
- \( S(3, -2) \) becomes \( S'(-2, 3) \)
- \( T(1, -4) \) becomes \( T'(-4, 1) \)
The image after reflecting across the line \( y = x \) is \( R'(3, -1) \), \( S'(-2, 3) \), and \( T'(-4, 1) \). No vertex at \( (2, -3) \).
-
Reflection across the line \( y = -x \):
- The x and y coordinates are negated and swapped.
- Reflecting each vertex:
- \( R(-1, 3) \) becomes \( R'(-3, 1) \)
- \( S(3, -2) \) becomes \( S'(2, -3) \)
- \( T(1, -4) \) becomes \( T'(4, -1) \)
The image after reflecting across the line \( y = -x \) is \( R'(-3, 1) \), \( S'(2, -3) \), and \( T'(4, -1) \). Here, we have a vertex at \( (2, -3) \).
Therefore, the correct answer is:
A reflection of \( \Delta RST \) across the line \( y = -x \) produces an image of the triangle with a vertex at \( (2, -3) \).
1 answer