Coordinates, distance between two points and midpoint

The coordinates of a point

To describe a position on the plane, draw two perpendicular lines that meet at a point $O$, the origin. The horizontal one is the x-axis and the vertical one is the y-axis. Each axis has a scale: the two may be the same or different, but on a given scale the same distance on the drawing always represents the same quantity.

The order matters: $P(5,2)$ and $Q(2,5)$ are different points. The axes divide the plane into four regions, the quadrants, which are numbered anticlockwise starting from the top right.

[FIGURA mai-t4-a1-cuadrantes: Cartesian plane with the x-axis (from -5 to 6) and the y-axis (from -5 to 5) and a soft grid; origin O labelled. The four quadrants labelled with Roman numerals in a faint tone: I at the top right (x>0, y>0), II at the top left (x<0, y>0), III at the bottom left (x<0, y<0), IV at the bottom right (x>0, y<0). Four marked points with their labels: P(4, 3) in quadrant I, Q(-3, 2) in II, R(-4, -3) in III and S(2, -4) in IV. For P, dashed lines run to both axes, labelling the 4 on the x-axis and the 3 on the y-axis.]

A point on an axis has a coordinate equal to zero: $(3,0)$ lies on the x-axis and $(0,-2)$ on the y-axis.

Distance between two points

Given two points $A(x_1,y_1)$ and $B(x_2,y_2)$, the segment $AB$ is the hypotenuse of a right-angled triangle whose legs are horizontal and vertical. The horizontal leg measures $|x_2-x_1|$ and the vertical leg $|y_2-y_1|$, so you simply apply Pythagoras' theorem.

[FIGURA mai-t4-a1-distancia: Cartesian axes with point A(x1, y1) at the bottom left and point B(x2, y2) at the top right, joined by a sloping segment labelled d. From A, a horizontal line to the vertical dropped from B, forming a right-angled triangle with the right angle at the bottom right corner (marked with a small square). Horizontal leg labelled x2 - x1 and vertical leg labelled y2 - y1. Along the axes, dashed guide lines to x1, x2, y1 and y2, labelled.]

Because the differences are squared, it does not matter in which order you subtract: $d(A,B)=d(B,A)$. Nor do the quadrants matter: if a coordinate is negative, the subtraction takes it into account by itself. The only thing to watch is writing the signs with brackets, for example $(-3-2{)}^2$.

The result is in the units of the scale on the plane (kilometres, metres, squares…). Unless the question asks otherwise, round to 3 significant figures.

Midpoint of a segment

The midpoint $M$ of $AB$ is the point that is the same distance from $A$ as from $B$. To go from $A$ to $B$ you travel $x_2-x_1$ horizontally; the midpoint is halfway along that journey, and the same applies vertically. That is why each coordinate of $M$ is the average of the coordinates of the endpoints.

[FIGURA mai-t4-a1-punto-medio: Cartesian axes with A(x1, y1) at the bottom left and B(x2, y2) at the top right, joined by a segment. The point M at the centre of the segment, labelled M((x1+x2)/2, (y1+y2)/2). Two identical small tick marks (double stroke) on the two halves AM and MB. Dashed lines from A, M and B to the axes, marking x1, (x1+x2)/2, x2 on the x-axis and y1, (y1+y2)/2, y2 on the y-axis.]

This formula works in both directions. If you know the midpoint and one endpoint, the other coordinate can be found: $x_2=2x_M-x_1$ and $y_2=2y_M-y_1$.

[FIGURA mai-t4-a1-mapa-senderos: Plane with a soft grid, x-axis (from -4 to 6) and y-axis (from -9 to 5), 1 unit = 1 km. Four marked and labelled points: refuge A(-3, 4) at the top left, fountain C(5, 4) at the top right, viewpoint B(5, -8) at the bottom right and lake M(1, -2) at the centre, on segment AB. Sloping segment AB from A to B passing through M, with a double equality mark on each half. Horizontal segment AC labelled 8 km and vertical segment CB labelled 12 km, with a small right-angle square at C. Segment AB labelled with its length, the square root of 208, approximately 14.4 km.]