11.2 Coordinate Geometry

The cartesian plane is where geometry meets algebra, and ICAS tests it at every level from Paper D upwards. Younger papers ask you to read or plot a point; Papers E-F expect midpoints, distances and gradients; senior papers fold coordinates into transformation and trigonometry problems. The good news is that almost every coordinate question reduces to counting squares or applying one of three small formulas — none of which needs a calculator.

Plotting and Reading Points in All Four Quadrants

Every point is written as an ordered pair (x, y): the x-coordinate first (movement across), then the y-coordinate (movement up or down). Remember it as "you walk along the corridor before you climb the stairs." The axes divide the plane into four quadrants, numbered anticlockwise starting from the top right.

Points on an axis are in no quadrant: (0, 4) lies on the y-axis and (0, 0) is the origin. The classic ICAS trap is swapping the order — the point (2, 5) is 2 across and 5 up, not the other way round. Read the signs before you plot: a negative y means move DOWN from the x-axis.

Midpoint of a Segment

The midpoint of the segment joining (x₁, y₁) and (x₂, y₂) is found by averaging each coordinate:

M = ( (x₁ + x₂) / 2 , (y₁ + y₂) / 2 )

Worked example: find the midpoint of A(−3, 2) and B(5, 8). Average the x-coordinates: (−3 + 5)/2 = 1. Average the y-coordinates: (2 + 8)/2 = 5. The midpoint is (1, 5). Check on a sketch: it should sit exactly halfway between the two points. A reverse question asks for a missing endpoint: if M(1, 5) is the midpoint of A(−3, 2) and B, then B = (2×1 − (−3), 2×5 − 2) = (5, 8). Double the midpoint, subtract the known endpoint.

Distance Between Two Points

For a horizontal or vertical segment, just subtract: the distance from (2, 3) to (7, 3) is 7 − 2 = 5 units, and from (1, −2) to (1, 4) is 4 − (−2) = 6 units.

For a slanted segment, build a right-angled triangle and use Pythagoras (see section 11.3):

d = √( (x₂ − x₁)² + (y₂ − y₁)² )

Worked example: find the distance from P(1, 2) to Q(4, 6). Horizontal change: 4 − 1 = 3. Vertical change: 6 − 2 = 4. Distance = √(3² + 4²) = √(9 + 16) = √25 = 5 units. Recognising the 3-4-5 triple saves time — no calculator required.

Gradient

The gradient (slope) of a line measures its steepness:

gradient = rise / run = (y₂ − y₁) / (x₂ − x₁)

Worked example: the line through (1, 2) and (5, 10) has gradient (10 − 2)/(5 − 1) = 8/4 = 2. For every 1 unit across, the line rises 2 units. Four cases to know:

• Positive gradient: line rises left to right
• Negative gradient: line falls left to right, e.g. (2 − 8)/(5 − 1) = −6/4 = −3/2
• Zero gradient: horizontal line (y values equal)
• Undefined gradient: vertical line (x values equal, so you would divide by zero)

Subtract the coordinates in the SAME order for both points; mixing (y₂ − y₁) with (x₁ − x₂) flips the sign and is a very common error. Parallel lines have equal gradients — ICAS uses this fact often.

Using Coordinates to Describe Transformations

Coordinates make transformations precise. Rules worth knowing about the origin and axes:

• Reflect (a, b) in the x-axis → (a, −b)
• Reflect (a, b) in the y-axis → (−a, b)
• Reflect (a, b) in the line y = x → (b, a)
• Rotate (a, b) 90° clockwise about the origin → (b, −a)
• Rotate (a, b) 180° about the origin → (−a, −b)
• Translate by h right and k up → (a + h, b + k)
• Enlarge about the origin with scale factor k → (ka, kb)

Worked example: triangle with vertex (2, 3) is rotated 90° clockwise about the origin. The image is (3, −2). Check: the point moved from Quadrant I to Quadrant IV, which is exactly where a quarter-turn clockwise takes it.

Coordinate Problems Involving Shapes

The signature ICAS problem: "Three vertices of a parallelogram are (1, 1), (6, 1) and (7, 4). Find the fourth vertex." In a parallelogram the diagonals bisect each other, so opposite vertices share a midpoint. Label the points A(1, 1), B(6, 1), C(7, 4) and the unknown D. Since AB is horizontal with length 5, the side DC must also be horizontal with length 5: D = (7 − 5, 4) = (2, 4). Verify with the midpoint rule: midpoint of diagonal AC is ((1+7)/2, (1+4)/2) = (4, 2.5); midpoint of BD is ((6+2)/2, (1+4)/2) = (4, 2.5). They match, so D(2, 4) is correct.

Strategy summary for shape questions:

1. Sketch the points first — a rough grid drawing exposes wrong answers immediately.
2. Use horizontal/vertical counting before reaching for formulas.
3. Use equal gradients to check parallel sides, and the midpoint rule for parallelograms.
4. Estimate: if your calculated vertex lands in the wrong quadrant, recheck your signs.