Gradient, Midpoint and Distance
Three questions about two points
Once two points are pinned to the Cartesian plane, three questions almost always follow: how steep is the line joining them, where is its exact middle, and how far apart are they. Year 9 answers all three with short formulas that are really just careful bookkeeping of horizontal and vertical change. The beauty is that every one of them reduces to the same two raw ingredients: the change in x and the change in y between the points.
Gradient: rise over run
Gradient measures steepness as the ratio of vertical change to horizontal change, often remembered as rise over run. For two points, you subtract the y-values to get the rise, subtract the x-values in the same order to get the run, and divide. Joining one comma two to four comma eight, the rise is eight minus two, which is six, and the run is four minus one, which is three, so the gradient is six over three, equal to two. The order matters only in that you must be consistent: subtract the coordinates of the same point first in both differences. A positive gradient rises to the right, a negative gradient falls, and a horizontal line has a gradient of zero because its rise is nothing at all, as for two comma three joined to six comma three.
Midpoint: averaging the coordinates
The midpoint is gentler: it is simply the average of the two x-coordinates paired with the average of the two y-coordinates. To average, add and halve. The midpoint of one comma two and four comma eight is the average of the x-values, two and a half, with the average of the y-values, five, giving the point two and a half comma five. This always lands exactly halfway along the segment, and because it is just averaging, it never produces anything more exotic than a fraction.
Distance: Pythagoras on the grid
Distance is where the plane reveals its hidden right angle. Between two points, the horizontal and vertical changes are the two short sides of a right-angled triangle, and the segment itself is the hypotenuse, so Pythagoras applies directly. Square the change in x, square the change in y, add them, and take the square root. From (3, 4) to the origin, the changes are 3 and 4, so the distance squared is 9 + 16 = 25, and the distance is √25, exactly 5. This is the familiar 3, 4, 5 right triangle hiding on the grid.
When the distance is irrational
Distances are not always whole numbers, and this is where earlier work on real numbers pays off. Joining (1, 2) to (4, 8), the distance squared is 9 + 36 = 45, so the distance is √45. That value is irrational; it simplifies to 3√5, and is approximately 6.71, but it is never exactly 6.71. Leaving the answer as √45, or as 3√5, is the exact form; the decimal is only an approximation, and it is important not to write an equals sign between them. Similarly the distance from (−2, 3) to (4, −3) is √72, or 6√2, about 8.49.
Watching signs and order
A few habits keep this reliable. Always compute the change in x and the change in y once, carefully, and reuse them for all three calculations, since gradient, midpoint and distance all feed on the same differences. Watch signs when points sit in negative regions, because a change such as −1 to 2 is a rise of 3, not 1. Keep gradient as rise over run, never the other way up, and remember that distance can never be negative because it comes from a square root of a sum of squares.
Three views of the same two points
With those differences computed once and handled cleanly, the three formulas stop feeling like separate rules and start looking like three views of the same pair of points. Gradient reads their slope, the midpoint marks their balance point, and distance measures the straight line between them, all from one pair of subtractions.