17.5.1. Midpoint of the Line Joining Points

Try these first

1. 1.

   Calculate the midpoint of the points (1, 1) and (3, 3).

   Hint

   Sum the x-coordinates and y-coordinates separately and divide by 2.

2. 2.

   What can you infer about the midpoint of (5, 7) and (5, 9)?

   Hint

   Examine the x-coordinates; they are identical.

3. 3.

   What is the midpoint of (3, 4) and (5, 8)?

   • (4
   • 6)
   • (3.5
   • 6)
   • (6
   • 10)

   Hint

   Remember the midpoint formula.

4. 4.

   Applying the pigeonhole principle, how many odd points do we need to guarantee at least two share the same characteristic?

   • 3
   • 4
   • 5

   Hint

   Think about the maximum characteristics available.

5. 5.

   If you select any five integer-coordinate points from (1,1) to (8,8), construct a proof showing at least one pair will have an integer midpoint.

   Hint

   Always categorize the points based on parities first.

6. 6.

   Consider points at the corners of a square. Calculate their midpoints and verify whether they yield integers.

   Hint

   Pairs to consider are (x1, y1) with (x2, y2) opposite.