Shapes on coordinate axes

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.

1. A(−2, 1) and B(4, 9)
2. across: 4 − (−2) = 6, up: 9 − 1 = 8
3. AB² = 6² + 8² = 36 + 64 = 100
4. 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.

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.

1. A(1, 2), B(4, 3), C(6, 7)
2. B to C: 2 across, 4 up
3. D = A + the same step = (1 + 2, 2 + 4) = (3, 6)
4. check: the midpoint of AC is (3.5, 4.5) and the midpoint of BD is (3.5, 4.5)

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.

1. box from x = 2 to 8 and y = 1 to 8: 6 × 7 = 42
2. A to B: ½ × 6 × 3 = 9
3. B to C: ½ × 4 × 4 = 8
4. C to A: ½ × 2 × 7 = 7
5. area = 42 − 9 − 8 − 7 = 18 square units

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.

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.