Point to Line Distance Calculator

Perpendicular distance, foot, and formula — live on the graph

Enter a line and a point to instantly calculate the shortest distance between them. The calculator shows the perpendicular foot, the right angle, and the step-by-step formula on an interactive graph.

The shortest distance from a point to a line is always the perpendicular distance — the length of the segment that meets the line at 90°. This calculator computes it using the formula d = |ax₀ + by₀ + c| / √(a² + b²) and shows every step.

Want to understand why the perpendicular is always shortest? Ask the AI tutor — it will walk you through the geometric proof (Pythagorean theorem) and the algebraic proof (the distance is a quadratic with a unique minimum).

Related Topics

Two-Point Geometry CalculatorPythagorean TheoremSlope of a LineLinear Equations

Suggested Lessons

Slope Quadratic Equations Unit Circle Word Problems
What is the distance from a point to a line?
The distance from a point to a line is the length of the shortest path between them. That shortest path is always the perpendicular — the line segment from the point to the line that forms a 90° angle. Any other path would be longer.
What is the point-to-line distance formula?
For a line ax + by + c = 0 and a point (x₀, y₀), the distance is d = |ax₀ + by₀ + c| / √(a² + b²). For slope-intercept form y = mx + b, rewrite as mx − y + b = 0 and apply the formula.
How do I find the foot of the perpendicular?
If the line has slope m, the perpendicular has slope −1/m (the negative reciprocal). Write the perpendicular line through your point, then solve the system to find where it meets the original line. That intersection is the foot.
Why is the perpendicular the shortest distance?
Geometric proof: Pick any other point Q on the line. The triangle PFQ (where F is the foot) has a right angle at F, making PQ the hypotenuse. The hypotenuse is always longer than a leg, so PQ > PF. Algebraic proof: The squared distance from P to any point on the line is a quadratic (parabola opening upward) — its unique minimum occurs exactly at the perpendicular foot.
What can it graph?
It can plot explicit, implicit, and parametric functions, add points and geometry, and animate sliders on the same graph.
Can I use voice or a photo?
Yes. You can talk to the tutor, upload a worksheet or handwritten problem, and let the graph update from that input.
Will it explain the steps?
Yes. The AI explains what it is drawing and why, so you see the answer on the graph instead of getting only a final number.