The point exactly halfway
A rope is tied between two poles. Where exactly is its middle?
- Two points, joined by a segment: (2, 3) and (8, 11).
- The midpoint: (5, 7).
- Both halves measure the same distance: 5.
Try it first. Slide the point M along the segment. Stop when both halves match.
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The rule
To find the middle, average each coordinate on its own. Add the two across counts, then halve the total. Add the two up counts, then halve that total. The result is the midpoint, the point halfway between the ends.
Take (2, 3) and (8, 11). Across: (2 + 8) ÷ 2 = 5. Up: (3 + 11) ÷ 2 = 7. The midpoint is (5, 7).
Check it: both halves measure 5. From (2, 3) to (5, 7): gaps of 3 and 4. So 3² + 4² = 25, and the distance is 5. From (5, 7) to (8, 11): the same gaps. That distance is 5 too. Two halves of 5 make the whole length, 10.
| The two ends | Across averaged | Up averaged | Midpoint |
|---|---|---|---|
| (2, 3) and (8, 11) | (2 + 8) ÷ 2 = 5 | (3 + 11) ÷ 2 = 7 | (5, 7) |
| (−3, 2) and (7, 6) | (−3 + 7) ÷ 2 = 2 | (2 + 6) ÷ 2 = 4 | (2, 4) |
| (1, 1) and (4, 6) | (1 + 4) ÷ 2 = 2.5 | (1 + 6) ÷ 2 = 3.5 | (2.5, 3.5) |
A midpoint can sit between two gridlines. It is still exact.
You can also work backwards. Suppose one end is (1, 2) and the midpoint is (4, 5). The midpoint is the average, so 2 × midpoint = known end + other end. So the other end is 2 × midpoint − known end. Across: 2 × 4 − 1 = 7. Up: 2 × 5 − 2 = 8. The other end is (7, 8).