A point has the coordinates $(0, k)$. Which reflection of the point will produce an image at the same coordinates, $(0, k)$?
A. a reflection of the point across the $x$-axis
B. a reflection of the point across the $y$-axis
C. a reflection of the point across the line $y=x$
D. a reflection of the point across the line $y=-x
Answer (1)
Reflecting a point (0, k) across the x-axis results in (0, -k), which is the same only if k=0.
Reflecting a point (0, k) across the y-axis results in (0, k), which is always the same.
Reflecting a point (0, k) across the line y=x results in (k, 0), which is the same only if k=0.
Reflecting a point (0, k) across the line y=-x results in (-k, 0), which is the same only if k=0.
Therefore, the reflection across the y-axis will always produce an image at the same coordinates. \boxed{a reflection of the point across the y -axis}
Explanation
Analyze the reflection options Let's analyze each reflection option to see which one results in the same coordinates (0, k).
Reflection across the x-axis
Reflection across the x-axis: When reflecting a point across the x-axis, the x-coordinate remains the same, and the y-coordinate changes its sign. So, the reflection of (0, k) across the x-axis is (0, -k). For the image to be at the same coordinates, we need (0, -k) = (0, k), which means -k = k. This is only true when k = 0. So, this option is not always true.
Reflection across the y-axis
Reflection across the y-axis: When reflecting a point across the y-axis, the y-coordinate remains the same, and the x-coordinate changes its sign. So, the reflection of (0, k) across the y-axis is (-0, k) = (0, k). Since the reflected point is (0, k), this reflection always results in the same coordinates.
Reflection across the line y=x
Reflection across the line y=x: When reflecting a point across the line y=x, the x and y coordinates are swapped. So, the reflection of (0, k) across the line y=x is (k, 0). For the image to be at the same coordinates, we need (k, 0) = (0, k), which means k = 0. So, this option is not always true.
Reflection across the line y=-x
Reflection across the line y=-x: When reflecting a point across the line y=-x, the x and y coordinates are swapped and their signs are changed. So, the reflection of (0, k) across the line y=-x is (-k, -0) = (-k, 0). For the image to be at the same coordinates, we need (-k, 0) = (0, k), which means -k = 0 and k = 0. So, this option is not always true.
Conclusion Based on the analysis above, only the reflection across the y-axis always results in the same coordinates (0, k).
Examples
Reflections are used in various real-world applications, such as creating symmetrical designs in art and architecture. For example, when designing a building, architects use reflections to ensure that the two halves of the building are mirror images of each other, creating a balanced and aesthetically pleasing structure. Understanding reflections helps in creating these symmetrical designs accurately.
