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 (2)
The reflection of the point ( 0 , k ) across the y-axis will produce an image at the same coordinates ( 0 , k ) for all values of k . This is distinct from other reflections, which only match the original coordinates when k = 0 . Thus, the chosen option is: a reflection of the point across the y-axis.
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Reflecting the point ( 0 , k ) across the x-axis results in ( 0 , − k ) , which is the same as ( 0 , k ) only if k = 0 .
Reflecting the point ( 0 , k ) across the y-axis results in ( 0 , k ) , which is the same as ( 0 , k ) for all values of k .
Reflecting the point ( 0 , k ) across the line y = x results in ( k , 0 ) , which is the same as ( 0 , k ) only if k = 0 .
Reflecting the point ( 0 , k ) across the line y = − x results in ( − k , 0 ) , which is the same as ( 0 , k ) only if k = 0 .
Therefore, the reflection across the y-axis will produce an image at the same coordinates ( 0 , k ) .
The answer is: a re f l ec t i o n o f t h e p o in t a cross t h ey − a x i s
Explanation
Analyze the problem Let's analyze each option to determine which reflection results in the same coordinates (0, k).
Reflection across the x-axis
Reflection across the x-axis: When reflecting a point (x, y) across the x-axis, the new coordinates become (x, -y). So, reflecting (0, k) across the x-axis results in (0, -k). For this to be the same as (0, k), we need -k = k, which means k = 0. This is only true for k = 0.
Reflection across the y-axis
Reflection across the y-axis: When reflecting a point (x, y) across the y-axis, the new coordinates become (-x, y). So, reflecting (0, k) across the y-axis results in (-0, k), which simplifies to (0, k). This is true for all values of k.
Reflection across the line y = x
Reflection across the line y = x: When reflecting a point (x, y) across the line y = x, the new coordinates become (y, x). So, reflecting (0, k) across the line y = x results in (k, 0). For this to be the same as (0, k), we need k = 0. This is only true for k = 0.
Reflection across the line y = -x
Reflection across the line y = -x: When reflecting a point (x, y) across the line y = -x, the new coordinates become (-y, -x). So, reflecting (0, k) across the line y = -x results in (-k, -0), which simplifies to (-k, 0). For this to be the same as (0, k), we need -k = 0 and 0 = k, which means k = 0. This is only true for k = 0.
Conclusion Comparing the results, only the reflection across the y-axis results in the same coordinates (0, k) for all values of 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 left and right sides are balanced and aesthetically pleasing. Similarly, in computer graphics, reflections are used to create realistic images of objects in mirrors or water.
