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.