Problem with rectangles

In summary, my professor gave a problem which stated that there must be at-least one pair of points in a 15x20 rectangle which is less than or equal to 5 units away from each other. My explanation was that one of the smaller squares should be split into quarters and each quarter should have two points. My professor's explanation was that the diagonal is the longest distance and that the other configurations lead to a possibility of placing a third point more than 5 units away from both of the previous points.
  • #1
sam400
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<<Mentor note: Missing template due to originally being posted in other forum.>>

So, my professor gave a problem which stated:

Given a 15 x 20 rectangle, prove that if 26 points are chosen, at least one pair will be at most five units away.

What I said was to split the rectangle into 12 5x5 squares. Then, at least 1 square will have at least 3 points, since if each of them had 2 points, it would only add up to 24. from there, I went on to say that the longest distance between 2 points in one of the smaller squares is 5 root 2 units. But since one of them will have a third point, the next longest distance possible for that point is 5 units, which is the length of one side.

My professor's explanation was much more eloquent, since he just divided the rectangle into 25 3 x 4 rectangles, then by pigeonhole principle, one of the rectangles will have two points, and the diagonal length will be 5.

While his explanation is much smoother than mine, he said my explanation was entirely wrong as well, but I forgot to ask him why that is. I could understand it being bad since it's so much longer, but I'm not sure what else there is.
 
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  • #2
sam400 said:
from there, I went on to say that the longest distance between 2 points in one of the smaller squares is 5 root 2 units. But since one of them will have a third point, the next longest distance possible for that point is 5 units, which is the length of one side.

This is wrong. You can have three points inside a square with side five without either being closer than five. Consider the points: (0,0), (1,5), (5,1)
 
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  • #3
oh, so the diagonal is the longest distance only for a rectangle?
 
  • #4
sam400 said:
oh, so the diagonal is the longest distance only for a rectangle?

The diagonal is still the longest distance, but since the diagonal is quite a bit longer than 5, you do not need to put two points in opposite corners. There are other configurations, some of which will lead to a possibility of placing a third point more than 5 units away from both of the previous points.
 
  • #5
sam400 said:
at least one pair will be at most five units away.
This statement is a little ambiguous, (or perhaps confusing is a better word), I think your question should be phrased like this:- If 26 points are chosen at random inside a 15##\times##20 rectangle, than there exists at-least one pair of points the distance between which is less than or equal to 5 units.
 

1. What is a rectangle?

A rectangle is a two-dimensional shape with four straight sides and four right angles. It is a quadrilateral and has opposite sides that are equal in length.

2. What is the area of a rectangle?

The area of a rectangle is calculated by multiplying the length of its base by its height. The formula for area is A = bh, where A is the area, b is the base, and h is the height.

3. How do you find the perimeter of a rectangle?

The perimeter of a rectangle is the distance around its outer edges. It is calculated by adding the lengths of all four sides. The formula for perimeter is P = 2l + 2w, where P is the perimeter, l is the length, and w is the width.

4. What is the difference between a square and a rectangle?

A square is a type of rectangle where all four sides are equal in length and all angles are right angles. A rectangle can have different length and width, but still has four right angles.

5. What is the Pythagorean Theorem and how does it relate to rectangles?

The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. This theorem can be used to find the diagonal of a rectangle, which is the hypotenuse of a right triangle formed by the two sides of the rectangle.

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