To determine which of the given lines would be a line of reflection that maps the shape ABCD onto itself, we need to analyze the reflection properties of each line with respect to symmetrical shapes.
1. Line [tex]\( x = 1 \)[/tex]:
- This line is vertical and parallel to the [tex]\( y \)[/tex]-axis. Reflecting a shape over this line would mean that points on one side of the line map to points directly opposite on the other side. Unless ABCD is symmetrically positioned relative to this line, this reflection will not map ABCD onto itself.
2. Line [tex]\( -x + y = 2 \)[/tex]:
- This line can be rewritten in slope-intercept form as [tex]\( y = x + 2 \)[/tex]. It has a slope of 1, which means it runs diagonally from bottom-left to top-right. A reflection over this line would involve an intersection with the line and a symmetrical mapping along this diagonal. For a symmetrical shape like ABCD to map onto itself over this line, it would need to be symmetrically aligned with this line, which is not generally the case.
3. Line [tex]\( x - y = 2 \)[/tex]:
- This line can be rewritten in slope-intercept form as [tex]\( y = x - 2 \)[/tex]. Like the previous line, it has a slope of 1, but it runs parallel to and offset from the line [tex]\( y = x \)[/tex]. Reflection over this line would also involve an intersection and symmetrical mapping, which similarly is not assumed to hold for a generic symmetrical shape like ABCD relative to this line.
4. Line [tex]\( 2x + y = 3 \)[/tex]:
- This line can be rewritten in slope-intercept form as [tex]\( y = -2x + 3 \)[/tex]. It has a slope of -2, meaning it runs diagonally and quite steeply. Reflecting across this line involves a more complex intersection and symmetry that does not typically align with the simplicity expected for a shape like ABCD to map onto itself without specific consideration of the shape's positioning and symmetry.
After examining each of the lines, none of them appear to map the shape ABCD onto itself through reflection. Therefore, based on the typical symmetry considerations and analysis, we conclude that:
None of the lines [tex]\( x = 1 \)[/tex], [tex]\( -x + y = 2 \)[/tex], [tex]\( x - y = 2 \)[/tex], [tex]\( 2x + y = 3 \)[/tex] would map the shape ABCD onto itself through reflection.
