Okay, so I need to figure out the reflection of the point (0,0) over the vertical line x=1, and then reflect that image over the horizontal line y=2. Hmm, reflections. Let me recall how reflections over vertical and horizontal lines work.

First, reflecting over a vertical line. The general formula for reflecting a point (x, y) over the vertical line x = a is (2a - x, y). Is that right? Let me check. Suppose I have a vertical line x=1. If I take a point (x, y), its reflection across x=1 should be the same distance from x=1 but on the opposite side. So the distance from x to 1 is |x - 1|, so the reflected x-coordinate would be 1 + (1 - x) = 2 - x. So yes, the formula is (2a - x, y) where a is the x-coordinate of the vertical line. So in this case, a=1, so the reflection of (0,0) over x=1 would be (2*1 - 0, 0) = (2, 0). Wait, that seems right. Let me visualize. The original point is at (0,0). The vertical line x=1 is to the right of the y-axis. The distance from x=0 to x=1 is 1 unit. So reflecting over x=1 would place the image 1 unit to the right of x=1, which is x=2. So yes, (2,0). Got that.

Now, the next step is reflecting this image over the horizontal line y=2. The formula for reflecting a point (x, y) over the horizontal line y = b is (x, 2b - y). Let me verify. If the original point is (2,0), and we reflect it over y=2, the distance from y=0 to y=2 is 2 units. So the reflection should be 2 units above y=2, which is y=4. So the reflected point would be (2, 4). Let me check with the formula. For a horizontal line y=b, the formula is (x, 2b - y). Here, b=2, so 2*2 - 0 = 4. So yes, (2,4). That seems correct.

Wait, let me make sure I didn't mix up vertical and horizontal reflections. For vertical lines, the x-coordinate changes, and for horizontal lines, the y-coordinate changes. So reflecting over x=1 affects the x-coordinate, and reflecting over y=2 affects the y-coordinate. So the first reflection is (2,0), then the second reflection is (2,4). So the final point is (2,4). Hmm, that seems straightforward, but let me double-check.

Alternatively, maybe I can think geometrically. Starting at (0,0), reflecting over x=1. The line x=1 is vertical, so the reflection should be symmetric with respect to that line. The distance from (0,0) to x=1 is 1 unit in the x-direction. So reflecting over x=1 would place the image 1 unit on the other side, which is x=2. So (2,0). Then reflecting over y=2. The distance from (2,0) to y=2 is 2 units in the y-direction. Reflecting over y=2 would place the image 2 units above y=2, which is y=4. So (2,4). Yeah, that makes sense.

Alternatively, maybe using coordinate transformations. For reflection over x=1, the transformation can be written as:

x' = 2*1 - x = 2 - x

y' = y

So applying that to (0,0):

x' = 2 - 0 = 2

y' = 0

So (2,0). Then reflecting over y=2:

x'' = x' = 2 (since reflection over y doesn't affect x)

y'' = 2*2 - y' = 4 - 0 = 4

So (2,4). Yep, same result.

I think that's solid. The key steps are applying the reflection formula twice, once for each axis. Since the reflections are over vertical and then horizontal lines, each reflection only affects one coordinate. So the final answer should be (2,4).

Wait, but let me check if I misread the problem. The original point is (0,0). The first reflection is over x=1, then the image is reflected over y=2. So the order is important. If I had reflected over y=2 first, then over x=1, the result would be different. But the problem specifies the order: first over x=1, then over y=2. So the steps are correct as per the problem statement.

Another way to think about it: imagine the original point. Reflecting over x=1 moves it to (2,0). Then reflecting that over y=2 moves it up to (2,4). So the x-coordinate remains the same after the second reflection, and the y-coordinate goes from 0 to 4. That seems right.

Alternatively, maybe using vectors. The reflection over x=1 can be seen as a translation. Wait, reflection is not a translation. But perhaps breaking it down. Let me recall that reflecting over a vertical line x=a is equivalent to reflecting over the y-axis (x=0) and then translating. But maybe that's complicating things. The formula is straightforward, so I think the first method is the simplest.

Alternatively, using coordinate geometry. The reflection of a point over a vertical line x=a is such that the midpoint between the original point and its reflection is (a, y). So for the first reflection over x=1, the midpoint between (0,0) and its reflection (x,0) should be (1,0). So (0 + x)/2 = 1 => x=2. So the reflection is (2,0). Then reflecting over y=2, the midpoint between (2,0) and its reflection (2,y) should be (2,2). So (0 + y)/2 = 2 => y=4. Therefore, the reflection is (2,4). Yep, same result.

All methods converge to (2,4). Therefore, I'm confident the answer is (2,4).
Thus, the resulting point after reflecting the original point (0,0) over the vertical line $x=1$ and then over the horizontal line $y=2$ is $\boxed{(2, 4)}$.
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