A coordinate $(x,y)$ locates a point on a flat grid using two numbers: the $x$-coordinate gives the horizontal distance from the origin $(0,0)$, and the $y$-coordinate gives the vertical distance. Together, the pair fixes one exact location, in exactly the same way a map reference or a seat number combines two pieces of information into a single unique position.

The grid is divided into four quadrants by the two axes, and the signs of the coordinates alone tell you which one a point sits in:

                 y-axis
                   │
     Quadrant II   │   Quadrant I
      (-x, +y)     │    (+x, +y)
                   │
 ──────────────────┼──────────────────► x-axis
   Origin (0,0)    │
     Quadrant III  │   Quadrant IV
      (-x, -y)     │    (+x, -y)
                   │

State the quadrant of $(-3,1)$: $x$ is negative, $y$ is positive, so it lies in Quadrant II.

A point on the $x$-axis always has $y=0$ (e.g. $(4,0)$); a point on the $y$-axis always has $x=0$ (e.g. $(0,4)$) — mixing these two up is a very easy way to lose a mark. Neither of these points belongs to any of the four quadrants at all, since the quadrants are defined as strictly positive or negative regions, and the axes themselves sit exactly on the boundary.

Use the fact that opposite sides of a rectangle (or other special quadrilateral) share a coordinate — one side is horizontal (same $y$-value at both ends), the other vertical (same $x$-value).

$ABCD$ is a rectangle with $A(2,5)$, $B(-3,5)$ and $C(-3,-2)$. Find $D$.

1. Side $AB$ is horizontal (both points have $y=5$); side $BC$ is vertical (both points have $x=-3$).
2. $D$ completes the rectangle by sharing its $x$-coordinate with $A$ and its $y$-coordinate with $C$:

Sketching a rough grid — even without exact scale — and labelling the three known vertices in order before hunting for the fourth makes it far easier to see which coordinates must match, rather than trying to picture the rectangle purely in your head.

$PQRS$ is a rectangle with $P(-4,3)$, $Q(2,3)$ and $S(-4,-1)$. Find the coordinates of $R$.

Solution

Side $PQ$ is horizontal ($y=3$ at both ends); side $PS$ is vertical ($x=-4$ at both ends). $R$ must share its $x$-coordinate with $Q$ and its $y$-coordinate with $S$: