Practice Shortest Path Intuition in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Worksheet PDF Free Answer-key PDF Family

Quick 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.

Showing a random 20 of 50 problems.

Example 1

medium
A fly is at one corner of a closed 3ร—4ร—53 \times 4 \times 5 box. What is the shortest surface path to the diagonally opposite corner?

Example 2

medium
On a cylinder, the shortest path between two points (wrapping around) becomes a straight line when you do what?

Example 3

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

Example 4

hard
On a graph with vertices A,B,C,DA, B, C, D and edges AA-BB (weight 11), AA-CC (weight 44), BB-CC (weight 22), CC-DD (weight 11), BB-DD (weight 55), find the shortest weighted path from AA to DD.

Example 5

hard
An ant on a cylinder of radius 11 goes from (1,0,0)(1, 0, 0) to (โˆ’1,0,5)(-1, 0, 5) (the diametrically opposite point, 55 units up). Find the shortest surface path length.

Example 6

easy
Light travels between two points by the shortest path in uniform air. What shape is that path?

Example 7

medium
Snell's law / refraction analogy: a swimmer at (0,4)(0, 4) on land wants to reach a drowner at (8,โˆ’3)(8, -3) in water. The land/water boundary is the xx-axis. The swimmer runs at 55 m/s on land and swims at 33 m/s. What entry point P=(x,0)P = (x, 0) minimizes total time?

Example 8

medium
On a sphere, is the equator a shortest path between two points on it?

Example 9

easy
On a globe, two cities at the same latitude (not the equator). Is the parallel of latitude the shortest path between them?

Example 10

medium
A delivery robot must visit point A, then B, then return home, all on a flat grid. Within that order, what makes each leg shortest?

Example 11

easy
On a 2D grid, you can only move along streets (horizontal or vertical, no diagonals). What is the shortest 'taxicab' distance from (0,0)(0,0) to (3,4)(3,4)?

Example 12

medium
Reflection trick: a point A=(1,4)A = (1, 4) must connect to a point B=(7,2)B = (7, 2) via the line y=0y = 0. Minimize AP+PBAP + PB where PP is on the xx-axis.

Example 13

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

Example 14

medium
On a sphere of radius RR, the great-circle distance between two points subtending central angle ฮธ\theta (radians) is what?

Example 15

easy
True or false: the shortest path on a flat plane satisfies the triangle inequality with equality.

Example 16

easy
On the surface of a sphere, the curves of shortest length are called what?

Example 17

medium
Why can the shortest path differ between a flat map and the real curved Earth?

Example 18

challenge
A cow at (2,5)(2, 5) must drink from a straight river along the xx-axis, then reach the barn at (10,3)(10, 3). Find the minimum total distance.

Example 19

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

Example 20

challenge
Why is the straight-line shortest path on a plane equivalent to the statement of the triangle inequality?
โ† Back to Shortest Path Intuition Worked Examples

Related Concepts

Distance