What are the coordinates after a reflection over x=1

To find the coordinates after reflecting over the line \( x = 1 \), follow these steps:

1. **Identify the original coordinates** of the point, let’s call it \((x, y)\).

2. **Calculate the distance of \( x \)-coordinate** from the line \( x = 1 \). This distance is \( |x - 1| \).

3. **Determine the reflected \( x \)-coordinate** by moving the same distance to the opposite side of the line \( x = 1 \). 
   - If \( x > 1 \), the new \( x \)-coordinate will be \( x - 2 \times |x - 1| \), which simplifies to \( 2 - x \).
   - If \( x < 1 \), the new \( x \)-coordinate will be \( x + 2 \times |x - 1| \), which also simplifies to \( 2 - x \).

Thus, for any point \((x, y)\), the coordinates after reflecting over the line \( x = 1 \) are:

\[
(2 - x, y)
\]

### Example
If the original point is \((3, 4)\):
- The distance from \( x = 1 \) is \( 3 - 1 = 2 \).
- Moving the same distance to the other side, the new \( x \)-coordinate is \( 1 - 2 = -1 \).

So, the reflected coordinates of \((3, 4)\) over \( x = 1 \) are \((-1, 4)\).

Would you like further details or have any questions about this reflection process?

Here are some related questions for practice:
1. What are the coordinates of \((5, -3)\) after reflecting over \( x = 1 \)?
2. How would the coordinates of \((2, 7)\) change if reflected over \( x = -1 \)?
3. Can you generalize the formula for reflections over any vertical line \( x = k \)?
4. How would the transformation differ for a reflection over the y-axis?
5. What are the properties of reflections over non-horizontal or non-vertical lines, such as \( y = x \)?

**Tip:** To confirm your reflection results, plot the points on a coordinate plane to visualize the distance symmetry relative to the line.

What are the coordinates after a reflection over x=1?

Learn how to find the coordinates of a point after reflecting over the line x=1. This guide explains the steps, key formulas, and examples, such as reflecting (3, 4) to (-1, 4). Suitable for Grades 8-10 students studying coordinate geometry.

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