Straight Lines - Distance of a Point From a Line

The distance of a point from a line is defined as the length of the perpendicular segment drawn from the point to the line. Visually, if you have a line $L$ and a point $P$ not on the line, the shortest path to the line is the straight segment $PM$ that intersects $L$ at a $90^{\circ}$ angle at the foot of the perpendicular $M$.

To calculate distance, the line must be expressed in the general form $Ax + By + C = 0$. In a coordinate plane, the coefficients $A$ and $B$ relate to the slope of the line, while $C$ determines its position relative to the origin.

The distance is always considered a non-negative scalar quantity. Mathematically, this is handled by using the modulus (absolute value) symbol $|...|$ in the numerator of the formula, ensuring that even if the coordinates produce a negative result, the final distance is positive.

The distance from the origin $(0, 0)$ to a line is a simplification where the point coordinates are zero. Geometrically, this represents the length of the perpendicular segment from the center of the coordinate system $(0, 0)$ to the nearest point on the line $Ax + By + C = 0$.

Parallel lines are lines that share the same slope and never intersect. On a graph, they appear as two lines following the same direction with a constant gap between them. Their equations can be written as $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$, where the $x$ and $y$ coefficients are identical.

The distance between two parallel lines is the perpendicular distance from any point on one line to the other line. This distance remains constant regardless of which point is chosen on the first line, as the lines never converge or diverge.

If the calculated distance of a point from a line is zero, it implies that the point $(x_1, y_1)$ satisfies the equation $Ax_1 + By_1 + C = 0$, meaning the point lies directly on the line itself.