The straight path is shortest
A detour through a third point is always longer.
Tent, tree and pole
Picture a small plot of land with a tent, a tree and a pole. You are at the entrance of the tent and want to go to the tree. You can walk straight to the tree, or walk to the pole first and then on to the tree.
Which path is shorter?
What this lesson covers
The idea
The direct straight-line path between two points is the shortest path, so it is shorter than any roundabout path via a third point.
Tent, tree and pole
Picture a small plot of land with a tent, a tree and a pole. You are at the entrance of the tent and want to go to the tree. You can walk straight to the tree, or walk to the pole first and then on to the tree.
Which path is shorter?
- Straight from the tent to the tree
- From the tent to the pole, then to the tree
- It depends on where the pole is
Move the pole
The red path goes straight to the tree. The yellow path goes via the pole. Drag the pole anywhere on the plot.
A detour never wins
Wherever you put the pole, the path via the pole is longer than the straight path. It ties only when the pole stands on the straight path, because then there is no detour at all.
The same is true for any point, not just a pole. The straight line between two points is the shortest way to go.
The direct straight-line path between two points is the shortest path, so it is shorter than any roundabout path via a third point.
Notes
The direct straight-line path between two points is the shortest path, so it is shorter than any roundabout path via a third point.
Check yourself
The tent is 12 m from the tree. The pole is 5 m from the tent and 8 m from the tree. How many metres longer is the path via the pole than the direct path?
Answer: 1
Via the pole: 5 + 8 = 13 m. The direct path is 12 m. The detour is 13 − 12 = 1 m longer.
A school is 300 m from a temple, and the temple is 400 m from a park. Which could be the straight-line distance from the school to the park?
Where must the pole stand so that the path via the pole is exactly as long as the straight path?
Meena says: “The pole is 6 m from the tent and 3 m from the tree, and the tent and the tree are 10 m apart.” Can Meena be right?
- 500 m — correct. Yes! The direct distance must be less than the roundabout 300 + 400 = 700 m.
- 800 m. The direct path cannot be longer than the roundabout path via the temple, which is 300 + 400 = 700 m.
- 1,000 m. The direct path cannot be longer than the roundabout path via the temple, which is 300 + 400 = 700 m.
- Close to the tent, but off the straight path. Even a small step off the path makes the way via the pole a little longer.
- On the straight path, between the tent and the tree — correct. Yes! Then walking via the pole is walking straight, so there is no detour.
- Far away from the straight path. The farther from the straight path, the bigger the detour.
- Yes. The pole can be anywhere on the plot.. Not any distances: the path via the pole (9 m) would beat the direct path (10 m), and that is impossible.
- Yes. The pole is nearer the tree than the tent is.. Which one is nearer does not matter. The path via the pole (6 + 3 = 9 m) would beat the direct 10 m, which is impossible.
- No. Via the pole the path would be 6 + 3 = 9 m, shorter than the direct 10 m. That cannot happen. — correct. Yes! A detour can never be shorter than the straight path.