Calculator

Enter the values for the line equation (Ax + By + C = 0) and the coordinates of the point:

Understanding Distance from a Point to a Line

The distance from a point to a line is the shortest distance between them, which is always the length of the perpendicular segment from the point to the line.

Distance = |Ax₁ + By₁ + C| / √(A² + B²)

Where (x₁, y₁) are the coordinates of the point, and the line is represented by the equation Ax + By + C = 0.

Derivation of the Formula

Consider a line represented by Ax + By + C = 0 and a point P(x₁, y₁). The perpendicular distance from P to the line can be derived using the concept of projection and the properties of right triangles.

The normal vector to the line is (A, B). The distance formula is essentially the length of the projection of the vector from any point on the line to P onto the unit normal vector.

Visual Representation

Geometric Representation

The red point represents the given point, the blue line is the line equation, and the green line shows the perpendicular distance.

Formula Components

Numerator (|Ax+By+C|)
Denominator (√(A²+B²))
Other Factors

Solved Example

Problem: Find the distance from point (1, 2) to the line 2x + 3y – 6 = 0.

Solution:

Identify values: A=2, B=3, C=-6, x₁=1, y₁=2
Calculate numerator: |2×1 + 3×2 – 6| = |2 + 6 – 6| = |2| = 2
Calculate denominator: √(2² + 3²) = √(4 + 9) = √13 ≈ 3.606
Distance = 2 / √13 ≈ 0.5547 units

Applications

This concept has numerous applications in:

  • Computer graphics and game development
  • Geographic information systems (GIS)
  • Robotics and path planning
  • Architecture and civil engineering
  • Physics and engineering calculations