Question

Five points are drawn inside an equilateral triangle with a side length of 2 units. Justify that there must be a pair of points that are 1 unit in length or less from each other. Hint: Use the pigeonhole principle to explain this observation.

Answer 1

Using the pigeonhole principle and the properties of an equilateral triangle, we can deduce that among five points inside an equilateral triangle with side length 2, there must be at least one pair of points that are no more than 1 unit apart.

Step-by-step explanation:

The essence of this problem can be explained using the pigeonhole principle. Imagine that you are trying to fit more pigeons into a given number of pigeonholes than there are pigeonholes; at least one pigeonhole must contain more than one pigeon. Applying this concept to the problem, the equilateral triangle of side length 2 units can be divided into 4 smaller equilateral triangles, each with a side length of 1 unit.

Now, if there are 5 points inside the larger triangle, and only 4 smaller triangles to place them in, according to the pigeonhole principle, at least one of the smaller triangles must contain at least two of the points. Since the distance between any two points within the same small equilateral triangle is at most 1 unit (the length of a side), there must be at least one pair of these points that are 1 unit or less apart from each other.

This conclusion is guaranteed by the geometric properties of the equilateral triangle and the logic of the pigeonhole principle.