Coordinate Geometry Practice

What a straight line actually tells you

A straight line has two numbers that describe it completely: a direction, called the gradient, and a starting height, called the intercept. Written as y = mx + c, m is the gradient and c is where the line crosses the y-axis. Everything else on this page is a method for recovering one of those numbers from points on the line.

Gradient is rise over run: the change in y divided by the change in x. A midpoint is the average of the two x values and the average of the two y values, which places it exactly halfway along. Distance between two points uses the right triangle under them, with the horizontal and vertical gaps as the two shorter sides, so Pythagoras supplies the third. A positive gradient rises as you move from left to right; a negative one falls. A horizontal line has gradient 0 because its height does not change. A vertical line has no defined gradient because its horizontal change is zero, and division by zero is not allowed.

What this ladder covers

Plotting a point starts the ladder, and the equation of a line through two points finishes it.

• Plotting and reading points: move along x first, then up or down by y, and name a point as an ordered pair.
• Midpoints: average the x values and the y values separately to sit halfway between two points.
• Distance: build a right triangle from the two gaps and apply Pythagoras to its hypotenuse.
• Gradient: divide the vertical change by the horizontal change, keeping the order of the points consistent.
• The intercept: read where a line meets the y-axis, and find it by substituting a known point.
• Equations and parallel lines: write y = mx + c from a gradient and one point, and spot lines that share a gradient.

Worked examples

Find the midpoint of (2, 4) and (6, 10). Average each coordinate: x = (2 + 6) ÷ 2 = 4 and y = (4 + 10) ÷ 2 = 7, so the midpoint is (4, 7). Both averages are needed; one alone names a line rather than a point.

Find the gradient through (1, 2) and (5, 10). The change in y is 10 − 2 = 8 and the change in x is 5 − 1 = 4, so m = 8 ÷ 4 = 2. The line climbs two units for every one it moves right.

Find the distance from (0, 0) to (3, 4). The gaps are 3 and 4, so the distance is √(3² + 4²) = √(9 + 16) = √25 = 5. Squaring first and adding afterwards is what turns two separate gaps into one length.

Write the equation of the line through (0, 1) with gradient 3. The point on the y-axis gives the intercept directly, so c = 1, and the equation is y = 3x + 1. No substitution is needed when one of the points already sits on the axis.

Find the line through (2, 5) and (4, 9). The gradient is (9 − 5) ÷ (4 − 2) = 4 ÷ 2 = 2. Using (2, 5): 2 × 2 + c = 5, so c = 1 and the equation is y = 2x + 1. Substituting (4, 9) gives the same intercept, which is the check worth doing.

Mistakes worth watching for

Gradient means the change in y divided by the change in x, in that order.

• Putting the x-difference on top. For (1, 2) and (5, 10), 4 ÷ 8 = 0.5 is the reciprocal of the right answer. The rise goes above the run, giving 8 ÷ 4 = 2.
• Averaging only one coordinate for a midpoint. The midpoint of (2, 4) and (6, 10) is (4, 7), not (4, 4) and not (2, 7). Halfway in x and halfway in y are two separate calculations.
• Adding 3 and 4 for a distance. The distance from (0, 0) to (3, 4) is 5, not 7, because the gaps form a right triangle and the direct route is its hypotenuse. Adding gives the distance along the two legs instead.
• Reading y = 2x + 1 as gradient 1 and intercept 2. The coefficient of x is the gradient and the constant is the intercept, so the line climbs 2 and crosses at 1. Swapping them tilts the line the wrong way.
• Subtracting in a different order on each axis. The two differences must follow the same point order, or the gradient comes out with the wrong sign. Take the second point minus the first for both.
• Calling a vertical line's gradient zero. Zero belongs to a horizontal line, where the rise is zero. A vertical line has no gradient value because its run is zero, so the rise-over-run division cannot be done.

How the ladder works

There are six rungs here, opening with five Warm-Up questions and ending with eight or nine at Mastery. Clearing one takes 80% of first attempts, all six are open from the first visit, and a retry supplies fresh points.

Keep going

Draw a set of grid questions from the coordinate geometry worksheet generator, plot two points and read the line off the graphing calculator, or return to linear equations to practise rearranging the same relationship without a grid.

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