The missing vertex
A farmer marks three corners of a field shaped like a parallelogram.
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- Corners A(1, 1), B(6, 2), C(9, 6) are fixed.
- Diagonal AC has its midpoint M(5, 7/2).
- Diagonal BD must have the same midpoint M.
- The fourth corner is D(4, 5).
The rule
A parallelogram is a four-sided shape with opposite sides parallel. The two diagonals of a parallelogram share one midpoint. A diagonal joins two opposite corners. The corners are named in order A, B, C, D. So the diagonals are AC and BD.
Midpoint of AC: ((1 + 9)/2, (1 + 6)/2) = (5, 7/2). Let D be (x, y). Midpoint of BD is ((6 + x)/2, (2 + y)/2). Set it equal to (5, 7/2). Across: (6 + x)/2 = 5, so 6 + x = 10 and x = 4. Up: (2 + y)/2 = 7/2, so 2 + y = 7 and y = 5.
Shortcut: D = A + C − B. Add A and C, the ends of the known diagonal. Take away B, the corner opposite D. Here (1 + 9 − 6, 1 + 6 − 2) = (4, 5).
Why: A + C and B + D both equal twice the midpoint.
Check the sides. From A to B is 5 across and 1 up. From D to C is also 5 across and 1 up. So AB and DC are equal and parallel.