Which reflection will produce an image of [tex]$\triangle RST$[/tex] with a vertex at (2,-3)?
A. a reflection of [tex]$\triangle RST$[/tex] across the x-axis
B. a reflection of [tex]$\triangle RST$[/tex] across the y-axis
C. a reflection of [tex]$\triangle RST$[/tex] across the line y=x
D. a reflection of [tex]$\Delta$[/tex]RST across the line y=-x
Answer (1)
Reflection across the x-axis transforms a point ( x , y ) to ( x , − y ) .
Reflection across the y-axis transforms a point ( x , y ) to ( − x , y ) .
Reflection across the line y = x transforms a point ( x , y ) to ( y , x ) .
Reflection across the line y = − x transforms a point ( x , y ) to ( − y , − x ) .
The reflection that produces an image of △ RST with a vertex at ( 2 , − 3 ) is a reflection across the x-axis. a re f l ec t i o n o f △ RST a cross t h e x − a x i s
Explanation
Analyzing Reflection Transformations Let's analyze how different reflections transform a point (x, y) on the coordinate plane. We'll consider each option to see which one results in the point (2, -3).
Examining Each Reflection Option
Reflection across the x-axis: This transformation changes the sign of the y-coordinate, so (x, y) becomes (x, -y). If the image is (2, -3), then the original point was (2, 3) because (2, 3) reflected across the x-axis is (2, -3).
Reflection across the y-axis: This transformation changes the sign of the x-coordinate, so (x, y) becomes (-x, y). If the image is (2, -3), then the original point was (-2, -3) because (-2, -3) reflected across the y-axis is (2, -3).
Reflection across the line y = x: This transformation swaps the x and y coordinates, so (x, y) becomes (y, x). If the image is (2, -3), then the original point was (-3, 2) because (-3, 2) reflected across the line y = x is (2, -3).
Reflection across the line y = -x: This transformation changes the signs of both coordinates and swaps them, so (x, y) becomes (-y, -x). If the image is (2, -3), then the original point was (3, -2) because (3, -2) reflected across the line y = -x is (2, -3).
Determining the Correct Reflection From the analysis above, we see that if the original point was (2, 3), reflecting across the x-axis results in the point (2, -3). Therefore, a reflection of △ RST across the x-axis will produce an image with a vertex at (2, -3) if the original vertex was at (2, 3).
Examples
Reflections are used in various real-world applications, such as creating symmetrical designs in art and architecture. For example, an architect might reflect a building design across a central axis to ensure a balanced and aesthetically pleasing facade. Similarly, in computer graphics, reflections are used to create realistic mirror images and lighting effects, enhancing the visual appeal of virtual environments.
