Coordinate Geometry
Using the distance formula
Use distances to identify shapes and find equidistant points.
With the distance formula you can test shapes without drawing them: work out the distances, then compare.
Three points:
- if two distances add up exactly to the third, the points lie on one straight line (they are collinear)
- if the squares fit Pythagoras, it is a right triangle
- (3, 2), (–2, –3), (2, 3): the squared sides are 50, 2 and 52. Since 50 + 2 = 52, it is a right triangle.
Four points: work out the four sides and the two diagonals.
| Sides | Diagonals | Shape |
|---|---|---|
| all four equal | equal | square |
| all four equal | not equal | rhombus |
| opposite sides equal | equal | rectangle |
| opposite sides equal | not equal | parallelogram |
'Equidistant' means at equal distances. To find such a point, set the two distances equal, and square both sides first so there are no roots to carry:
- On an axis, there's only one unknown. The point on the x-axis is (x, 0), so you solve for x alone.
- Anywhere in the plane, you get a rule linking x and y. Points equidistant from (7, 1) and (3, 5) all satisfy x – y = 2. That is a line: the line that cuts AB in half at right angles.
Read the whole question. 'The origin lies inside the triangle' or 'the point is below the x-axis' tells you which of two answers to keep.
Work out the distances, then compare them: equal sides, Pythagoras, or two distances adding exactly to the third (collinear points).
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