Name the shape from its distances
Take A(1, 1), B(5, 1), C(5, 4) and D(1, 4). Is the shape a rectangle? Distances can tell.
- The points are A(1, 1), B(5, 1), C(5, 4) and D(1, 4), taken in order.
- The four sides are 4, 3, 4 and 3. Opposite sides are equal.
- The diagonals AC and BD are both 5. They are equal.
- Opposite sides equal and diagonals equal: a rectangle.
The rule
Yes. Here the opposite sides are equal and the diagonals are equal. That makes a rectangle.
Find every distance. Compare them. The pattern names the shape.
Compare squared distances for equal sides and right angles. Each squared distance is the sum of two squared gaps. Then no root is needed. For a straight line, add the distances themselves.
Test for a straight line first. Three points on one line make no triangle. This stays true if two distances are equal. In the table, a, b and c are the three distances. The longest is c.
| Points | Pattern of the distances | The shape |
|---|---|---|
| 3 | the two shorter add to the longest | not a triangle: one line |
| 3 | not a line, and exactly two equal | isosceles triangle |
| 3 | all three equal | equilateral triangle |
| 3 | a² + b² = c² for the longest c | right triangle |
| 4 | four sides equal, diagonals equal | square |
| 4 | four sides equal, diagonals not equal | rhombus |
| 4 | opposite sides equal, not all four; diagonals equal | rectangle |
| 4 | opposite sides equal, not all four; diagonals not equal | parallelogram |
| 4 | none of these patterns | none of these four |
A triangle can match two rows. Squared distances 20, 20 and 40 make a right isosceles triangle.
For four points, take them in order around the shape. Then AB, BC, CD and DA are the sides. AC and BD are the diagonals.
Example. P(1, 1), Q(5, 1) and R(3, 5). PQ² = 4² + 0² = 16. QR² = 2² + 4² = 20. PR² = 2² + 4² = 20. The squares are 16, 20 and 20. The distances are 4, √20 and √20. The shorter two, 4 and √20, do not add to √20. So it is not a line. Two are equal, so the triangle is isosceles.
For a right angle, two squares must add to the longest. 16 + 20 = 36, not 20. So it is not right-angled.
Now try it. Read the points. Name the shape.
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