Naming a shape from its corners alone
Four corners, given as numbers, are enough to name a shape. No ruler needed.
- Four corners, joined in order.
- One diagonal: corner 1 to corner 3.
- The other diagonal: corner 2 to corner 4.
- Both diagonals share the same midpoint.
Try it first. Slide corner D. Stop when ABCD is a parallelogram.
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The rule
Find the four sides, the two diagonals and their midpoints. A diagonal joins two opposite corners. To bisect is to cut in half.
A shape has every name whose tests it passes. The tests are these. Parallelogram: diagonals that bisect each other. Rhombus: four equal sides. Rectangle: equal diagonals that bisect each other. Square: the rhombus test and the rectangle test.
Take (0, 0), (4, 0), (5, 3) and (1, 3). The side squares are 16, 10, 16 and 10. The diagonal squares are 34 and 18. Both diagonals share the midpoint (2.5, 1.5). The diagonals bisect each other. So the shape is a parallelogram.
| Four corners | Side squares | Diagonal squares | Same midpoint? | Name |
|---|---|---|---|---|
| (0,0), (4,0), (5,3), (1,3) | 16, 10, 16, 10 | 34, 18 | yes | parallelogram |
| (0,0), (3,0), (3,3), (0,3) | 9, 9, 9, 9 | 18, 18 | yes | square |
| (0,0), (5,0), (5,2), (0,2) | 25, 4, 25, 4 | 29, 29 | yes | rectangle |
| (0,0), (6,0), (4,3), (1,3) | 36, 13, 9, 10 | 25, 34 | no | trapezium |
The last row's diagonals miss each other's midpoint. Two of its sides lie on y = 0 and y = 3. Both lines are horizontal, so the sides are parallel. One parallel pair is enough for a trapezium. A square passes the rectangle test and the rhombus test. So it is also a parallelogram.
The diagonals of a parallelogram share a midpoint. So you can find a missing corner. Take A(0, 0), B(4, 0) and C(6, 3). The midpoint of AC is (3, 1.5). The midpoint of BD must be (3, 1.5) too. So D = 2 × (3, 1.5) − B = (6 − 4, 3 − 0) = (2, 3).