3. Coordinate geometry

Cambridge IGCSE Mathematics 0580 (2025-2027) · Core C3 · Extended E3 · official syllabus

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COORDINATE LAB

Learn it by playing it

Four short missions on one live grid. Grab the point sliders, watch the line, gradient, midpoint and length react, and figure out the rules. When a mission glows green, read why it worked.

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Mission 1 of 4

Climb the hill

Points A and B pin a straight line to the grid. Move the point sliders and make the line climb with gradient exactly 2 - that is 2 up for every 1 across.

A (-6, -4)B (-3, 0)
gradient 1.33 · midpoint (-4.5, -2) · length 5

Before you touch the sliders: point B sits 6 right and 3 up from point A. What is the gradient of AB?

Currently gradient 1.33 - too shallow. Add more rise or shorten the run.

Study notes

Coordinates C3.1 / E3.1

  • A point in two dimensions is fixed by an ordered pair (x, y): x steps across from the origin, then y steps up or down.
  • The x-axis and y-axis divide the grid into four quadrants; the signs of x and y tell you which quadrant a point sits in, e.g. (-3, 5) has x negative and y positive, so it is in the second quadrant.
  • A point on the x-axis has y = 0; a point on the y-axis has x = 0.
  • Read coordinates off a grid by moving along to the point and then reading across to each axis - never guess from the picture alone.

Drawing straight-line graphs C3.2 / E3.2

  • Every linear equation in x and y draws as a straight line; build it from a table of values by picking x-values, working out y, and plotting the pairs.
  • Three points are enough to draw and check a line: if the third point does not sit on the line, one of your calculations is wrong.
  • Core questions give the equation in the form y = mx + c, for example y = -2x + 5, or supply the table directly.
  • Extended questions may give the equation in another form, like 3x + 2y = 5; rearrange it or substitute x-values as it stands.
  • Vertical and horizontal lines are special cases: x = k is vertical and crosses the x-axis at k; y = c is horizontal and crosses the y-axis at c.

Gradient C3.3 / E3.3

  • The gradient measures steepness: gradient = change in y / change in x (rise over run) between two points on the line.
  • A line climbing left to right has a positive gradient; falling left to right, negative; a horizontal line has gradient 0; a vertical line has no defined gradient.
  • Core questions read the rise and run straight off a grid.
  • At Extended, calculate the gradient from the coordinates of two points: m = (y2 - y1) / (x2 - x1), keeping the subtraction order the same on top and bottom.
  • Steeper means a bigger size of gradient, whatever the sign: a gradient of -3 is steeper than a gradient of 2.

The equation y = mx + c C3.5 / E3.5

  • In y = mx + c, m is the gradient and c is the y-intercept - the y-value where the line crosses the y-axis.
  • Read an equation off a graph by picking out the y-intercept and the rise-over-run between two clear grid points.
  • Build an equation from facts you are given: gradient 3 through (0, -2) is y = 3x - 2, because c is the value of y when x is 0.
  • Given a gradient and one point (x1, y1), substitute to find c: c = y1 - m * x1.
  • At Extended, rearrange forms like ax + by = c into y = mx + c before reading off m and c, and always give the final equation fully simplified.

Parallel and perpendicular lines C3.6 / E3.6, E3.7

  • Parallel lines have equal gradients; their equations differ only in the intercept.
  • To find the line parallel to a given line through a given point, keep the gradient and substitute the point to fix c.
  • Perpendicular lines (Extended) have gradients that multiply to -1: each is the negative reciprocal of the other, so m = 3/2 becomes -2/3.
  • The perpendicular bisector of a segment passes through the segment's midpoint at the negative-reciprocal gradient.
  • A quick check: a line perpendicular to a shallow positive line must be steep and negative.

Length and midpoint (Extended) E3.4

  • The midpoint of a segment is the average of the coordinates: ((x1 + x2)/2, (y1 + y2)/2).
  • The length of a segment comes from Pythagoras on the horizontal and vertical steps: length = sqrt((x2 - x1)^2 + (y2 - y1)^2).
  • Sketch the right-angled triangle on the grid to see which differences are the two legs; the segment itself is the hypotenuse.
  • Answers in exact (surd) form are acceptable unless the question asks for a decimal.

EduTribe study notes - original material aligned to the Cambridge IGCSE Mathematics 0580 syllabus (2025-2027). Automated checks completed; not yet reviewed by an educator.

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