Geometry and Road Design

Today I have a very interesting question involving geometry. Its solution is so simple that it will make you think "How didn't I think about it?!" when you see the answer. Here it is, try to solve yourself first (as I said, it's easy):

Two towns, town 1 and town 2, are located to the south of road A going in east-west direction. No roads are connecting the towns to each other or to road A. The people of both towns decided to build two roads: one from each town to road A, so they could travel between the towns. What is the shortest route to build these roads?

Well, the shortest route is to build these roads from town to town, but the people of the towns want a connection to road A, so let's give them that.

To solve this question, let's shift our thinking a little bit: let's imagine that town 1 moves to the north of road A, but still keeps the same distance from it (if it was 100 miles to the south, now it's 100 miles to the north).

The shortest distance between two points is on the straight line between them, and so is the road between the towns now. Notice that this road intersects road A.

Now move town 1 back to where it was, and create the road from town 1 to where the roads between the towns intersected road A. It's the same distance as when town 1 was at north, so it must be the shortest distance. Problem solved!

This principle is also known in physics in the field of optics and mirrors. Who said math has no use in real life?

Nadav

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