Question

Reflect the plane in the x-axis, then in the line y=k. Show that the resulting isometry sends (x,y) to (x,y+2k), so it is the translation t₀.₂k-

Answer 1

To reflect a point (x, y) in the x-axis, we keep the x-coordinate the same and change the sign of the y-coordinate. To reflect a point (x, y) in the line y=k, we keep the y-coordinate the same and find the distance between the line and the point in the y-direction. If we reflect a point (x, y) in the x-axis, and then in the line y=k, the resulting isometry sends the point to (x, y+2k).

Step-by-step explanation:

To reflect a point (x, y) in the x-axis, we keep the x-coordinate the same and change the sign of the y-coordinate. So, the reflected point would be (x, -y).

To reflect a point (x, y) in the line y=k, we keep the y-coordinate the same and find the distance between the line and the point in the y-direction. We then subtract this distance from the y-coordinate of the line. So, the reflected point would be (x, 2k-y).

Now, if we reflect a point (x, y) in the x-axis, and then in the line y=k, the resulting isometry would send the point to (x, -y), and then to (x, 2k-(-y)), which simplifies to (x, y+2k). Therefore, the resulting isometry is the translation t₀.₂k.