The midpoint, in numbers
A wire is stretched between two nails on a board.
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- A(−2, 1) and B(6, 7).
- Average the coordinates: M(2, 4).
- Distance from A to M: 5.
- Distance from M to B: 5.
The rule
Take m = n in the section formula. Then m cancels, and the formula averages each coordinate on its own. To average two numbers, add them and halve the sum. The result is the midpoint. The midpoint is the same distance from both ends.
Midpoint formula: ((x₁ + x₂)/2, (y₁ + y₂)/2).
For A(−2, 1) and B(6, 7): across is (−2 + 6)/2 = 4/2 = 2. Up is (1 + 7)/2 = 8/2 = 4. The midpoint is M(2, 4).
Check with distances. From A to M the gaps are 4 and 3. So AM = √(16 + 9) = 5. From M to B the gaps are 4 and 3 again. So MB = 5 too. The two halves are equal.
A diameter runs across a circle through its centre. The centre of a circle is the midpoint of any diameter. Take the centre O(3, 1) and one end B(7, −2). Let the other end be A(a, b). Then 3 = (a + 7)/2, so a + 7 = 6 and a = −1. Also 1 = (b − 2)/2, so b − 2 = 2 and b = 4. The other end is A(−1, 4).