17.5. Question 8

Try these first

1. 1.

   Define the term 'midpoint' in your own words.

   Hint

   Think about how you would represent the middle of a line.

2. 2.

   What do we mean by 'distinct points'?

   Hint

   Consider if two points can be the same.

3. 3.

   According to the pigeonhole principle, if there are five points and four pairs possible, what must be true?

   • At least two points are the same
   • All points are distinct
   • At least two points belong to the same pair

   Hint

   Think about how many types of bins you have versus items.

4. 4.

   If the midpoint of two points is (2, 2), what can you say about the coordinates of those points?

   Hint

   Remember how the midpoint is calculated.

5. 5.

   Provide five distinct integer-coordinate points and demonstrate their midpoints have integer values.

   Hint

   Use the midpoint formula for your calculations.

6. 6.

   Mathematically prove that among n distinct points defined on integer coordinates, where n > 4, there exists an integer midpoint.

   Hint

   How does the pigeonhole principle apply here with various n?