Shortest Distance of a Point to a Line Revision Content

The shortest distance from a point to a line is the length of the perpendicular drawn from the point to the line.
Taking note that the shortest distance is always perpendicular to the given line is crucial, as this eliminates any slant or diagonal distances which may be longer.
This shortest, or most direct, distance is also referred to as the perpendicular distance.

To calculate the shortest distance, d, from a point with coordinates (x₁, y₁) to a line with equation Ax + By + C = 0, use the formula d = Ax₁ + By₁ + C / √(A² + B²).
Keep in mind that the absolute value symbol, , in the formula ensures the distance calculated is positive, which aligns with the idea that distance cannot be negative.
The denominator, √(A² + B²), essentially normalises the distance.

Pay attention to the signs of each component in the formula. Remember that the distance itself is always positive.
Keep units consistent throughout the problem. Distance ratio problems may involve different units of measurement. Always make sure you are working with like units.
Some questions might appear complex but will be based on applying the pathagorean theorem and principles of geometry. Balancing your understanding of geometry and algebra could prove beneficial.
Do not assume the shortest distance is along a straight path between the point and any point on the line. Always use the perpendicular distance.

The concept of shortest distance from a point to a line has numerous practical uses in physics, engineering, computer graphics, etc.
For instance, in Physics, it is applicable in applications of reflections and refractions, where light travels through the path of least distance.
Understanding and applying these principles correctly can greatly aid in problem-solving situations and relate mathematics to real-world phenomena.