Shapes on coordinate axes
A length is the long side of a right-angled triangle
Between two points, go across and then up.
Those two steps are the short sides of a right-angled triangle, and the line segment is its longest side. So square the two steps, add, and take the square root. Subtract the coordinates to get the steps; a negative step is fine, because squaring makes it positive.
- A(−2, 1) and B(4, 9)
- across: 4 − (−2) = 6, up: 9 − 1 = 8
- AB² = 6² + 8² = 36 + 64 = 100
- AB = √100 = 10
The slip to avoid: adding the steps (6 + 8 = 14). That is the walk along the grid lines, which is always longer than the straight line.
Midpoints, centres and missing corners
The midpoint of a segment is the mean of its end points: add the x coordinates and halve, then the same for y.
Two facts turn this into geometry:
- the centre of a circle is the midpoint of any diameter, and the radius is half the diameter's length;
- the diagonals of a parallelogram (and so of a rectangle, rhombus or square) cross at the midpoint of each one.
For a missing corner of a parallelogram ABCD, use equal steps: the step from B to C is the same as the step from A to D.
- A(1, 2), B(4, 3), C(6, 7)
- B to C: 2 across, 4 up
- D = A + the same step = (1 + 2, 2 + 4) = (3, 6)
- check: the midpoint of AC is (3.5, 4.5) and the midpoint of BD is (3.5, 4.5)
Areas: draw the box, take away the corners
When a shape's corners are on grid points but its sides slope, draw the smallest rectangle around it.
Each sloping side cuts off a right-angled triangle in a corner of the box. The shape's area is the box minus those triangles, and each triangle is half of across × up.
- box from x = 2 to 8 and y = 1 to 8: 6 × 7 = 42
- A to B: ½ × 6 × 3 = 9
- B to C: ½ × 4 × 4 = 8
- C to A: ½ × 2 × 7 = 7
- area = 42 − 9 − 8 − 7 = 18 square units
Naming a shape from its coordinates
Two calculations decide what a quadrilateral is.
Gradients tell you which sides are parallel (equal gradients) and where the right angles are. Squared lengths tell you which sides are equal; there is no need to square root them to compare.
- parallelogram: both pairs of opposite sides parallel
- rectangle: a parallelogram with right angles
- rhombus: four equal sides
- kite: two pairs of equal sides next to each other
- trapezium: exactly one pair of parallel sides
A triangle works the same way: work out the three squared lengths. Two equal means isosceles. A triangle with all its corners on grid points can never be equilateral.
Working backwards from a length
If you know a length and one step, Pythagoras gives the other step: subtract the squares.
A is (1, 2), B is (7, k) with k > 2, and AB = 10. Across is 6, so up² = 100 − 36 = 64, up = 8 and k = 2 + 8 = 10. Read the condition: k > 2 tells you B is above A, so you add the step.
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