Shortest Path Intuition Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Shortest Path Intuition.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The minimum-length route connecting two points, whose form depends on the geometry of the underlying space.

On a flat surface the straight line is always the shortest path between any two points.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Shortest paths depend on the geometry of the space—curved spaces have curved shortest paths.

Common stuck point: On curved surfaces, 'straight' doesn't mean what you expect.

Sense of Study hint: Try stretching a string tightly between two points on the surface. The path the string takes is the shortest route.

Worked Examples

Example 1

medium
A river runs along the x-axis. Town A is at (2, 3) and town B is at (8, 5), both north of the river. A pumping station on the river at point P(x, 0) connects to both towns. Find x that minimises the total pipe length AP + PB.

Solution

  1. 1
    Step 1: Reflect B across the river (x-axis) to get B'(8, -5).
  2. 2
    Step 2: The shortest total path from A to P to B (with P on the x-axis) equals the straight line AB' by the reflection principle.
  3. 3
    Step 3: Line AB': slope = \dfrac{-5-3}{8-2} = \dfrac{-8}{6} = -\dfrac{4}{3}. Equation: y - 3 = -\dfrac{4}{3}(x-2) \Rightarrow y = -\dfrac{4}{3}x + \dfrac{17}{3}.
  4. 4
    Step 4: Find P: set y = 0: 0 = -\dfrac{4}{3}x + \dfrac{17}{3} \Rightarrow x = \dfrac{17}{4} = 4.25.

Answer

P = (4.25,\; 0); minimum pipe length = |AB'| = \sqrt{(8-2)^2+(-5-3)^2} = \sqrt{100} = 10 units.
The reflection principle converts the two-segment path problem into a single straight-line problem. Reflecting one endpoint across the barrier and drawing a straight line to the other endpoint gives the optimal relay point.

Example 2

hard
What is the shortest path between two points in the Euclidean plane, and why? Then explain why the shortest path on the surface of a sphere is a great circle arc.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
What is the shortest distance from point P(4, 3) to the origin? Justify that the straight line gives the minimum.

Example 2

hard
An ant on the surface of a unit cube wants to travel from vertex A = (0,0,0) to the opposite vertex B = (1,1,1) along the surface. What is the shortest surface path, and what is its length?
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Related Concepts

Distance

Background Knowledge

These ideas may be useful before you work through the harder examples.

distance formal