A reflection of a point \( A \) across a line \( l \) involves finding the point \( A' \) such that \( A' \) is the mirror image of \( A \) with respect to the line \( l \). The steps to achieve this geometrically and mathematically are as follows:

### Geometric Process:
1. **Draw perpendicular**: Start by drawing a perpendicular line from point \( A \) to line \( l \). The foot of the perpendicular is the point of intersection, say \( P \).
   
2. **Measure the distance**: Measure the distance from \( A \) to \( P \).
   
3. **Mark the reflection**: From \( P \), move the same distance along the perpendicular line, but in the opposite direction to locate the reflected point \( A' \).

### Mathematical Process (using coordinates):
If \( A = (x_1, y_1) \) and line \( l \) is represented by the equation \( ax + by + c = 0 \), the coordinates of the reflected point \( A' = (x', y') \) can be calculated using the formula:

\[
x' = \frac{x_1(b^2 - a^2) - 2y_1ab - 2ac}{a^2 + b^2}
\]
\[
y' = \frac{y_1(a^2 - b^2) - 2x_1ab - 2bc}{a^2 + b^2}
\]

This formula ensures that the line through \( A \) and \( A' \) is perpendicular to \( l \), and \( A \) and \( A' \) are equidistant from \( l \).

Would you like further details on the reflection process? Here are five related questions to deepen your understanding:

1. How can you determine the line of reflection if given two points and their reflected images?
2. How does reflection across the x-axis or y-axis work?
3. Can the reflection process be applied to geometric shapes instead of single points?
4. What happens when the line of reflection is diagonal, such as \( y = x \)?
5. How does reflection relate to other transformations like rotation or translation?

**Tip:** Reflections preserve the distance between points and maintain the shape of geometric figures, making them an example of an **isometry**.

Explore how to reflect a point A across a line l geometrically and mathematically. This problem uses the reflection theorem and coordinate geometry formulas to find the mirror image of A. Learn step-by-step instructions suitable for Grades 9-12.

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