What is the shortest distance to a line?

Understanding the shortest distance from a point to a line is a fundamental concept in geometry. This distance is always the length of the perpendicular segment from the point to the line.

Perpendicular Distance

To find the shortest distance, we need to determine the perpendicular distance from a point to the line. This is the most direct path, forming a right angle with the line.

Formula for the Shortest Distance

The formula to calculate the shortest distance from a point
$(x_1, y_1)$ to a line given by the equation
$Ax + By + C = 0$ is:

$d = frac{|Ax_1 + By_1 + C|}{sqrt{A^2 + B^2}}$

Example

Let’s say we want to find the shortest distance from the point
$(3, 4)$ to the line
$2x – 3y + 5 = 0$

1. Identify the coefficients:
   $A = 2$,
   $B = -3$, and
   $C = 5$
2. Substitute these values into the formula:

$d = frac{|2(3) – 3(4) + 5|}{sqrt{2^2 + (-3)^2}}$

3. Simplify the expression:

$d = frac{|6 – 12 + 5|}{sqrt{4 + 9}}$

$d = frac{|-1|}{sqrt{13}}$

$d = frac{1}{sqrt{13}}$

4. The distance is approximately
   $0.28$ units.

Understanding Perpendicularity

The concept of perpendicularity is crucial here. When a line is perpendicular to another, they intersect at a right angle
$(90^text{o})$. This ensures the shortest path between the point and the line.

Applications

Knowing how to find this distance is useful in various fields such as physics, engineering, and computer graphics. For instance, in navigation systems, determining the shortest distance to a route can optimize travel paths.

Conclusion

The shortest distance from a point to a line is always along the perpendicular segment. By using the formula
$d = frac{|Ax_1 + By_1 + C|}{sqrt{A^2 + B^2}}$, we can easily compute this distance, aiding in practical and theoretical applications alike.

Citations

1. 1. Khan Academy – Distance between Point and Line
2. 2. Math is Fun – Distance from a Point to a Line
3. 3. Paul’s Online Math Notes – Distance Between Point and Line