What are the abscissa and the ordinate?

The abscissa is the first coordinate of a point, measured along the horizontal x axis. The ordinate is the second one, measured along the vertical y axis. In A = (3, 2) the abscissa is 3 and the ordinate is 2. The order is part of the notation: (3, 2) and (2, 3) are two different places on the plane.

Which quadrant does (−3, 5) lie in?

The second. Quadrants are numbered counter-clockwise starting from the top right: in the first both coordinates are positive, in the second the abscissa is negative and the ordinate positive, in the third both are negative, and in the fourth the abscissa is positive and the ordinate negative. The point (−3, 5) has a negative abscissa and a positive ordinate, so it falls in quadrant II.

How do you find the midpoint of a segment?

Average the abscissas and the ordinates separately: the midpoint of a segment with endpoints A = (x_A, y_A) and B = (x_B, y_B) is ((x_A + x_B)/2, (y_A + y_B)/2). For A = (−4, −1) and B = (2, 3) that gives (−1, 1). The formula works for any endpoints, negative ones included — averaging is ordinary addition and division by two.

Does the point (0, 4) lie in a quadrant?

No. Points on an axis belong to no quadrant — the axes are the boundaries between them. The point (0, 4) has abscissa zero, so it lies on the y axis. Likewise (5, 0) lies on the x axis, and (0, 0) is the origin.

How do you measure a segment that is not parallel to either axis?

Subtraction is enough only for horizontal and vertical segments. For a slanted one the length comes from the Pythagorean theorem applied to the differences of the coordinates — that formula is introduced by the lesson on analytic geometry. This lesson measures only segments parallel to an axis, because those are what reading a plan, a map or a tile grid actually needs.

Basic level

The coordinate plane

Two perpendicular axes turn the plane into a map on which every point has an address made of two numbers. The order of those numbers is part of the address, their signs name the quadrant, and a segment parallel to an axis is measured by subtraction. Plus the midpoint of a segment, which is the average of the endpoints.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

Negative numbersNegative numbers are the numbers below zero — frost on a thermometer, an overdraft, a level underground. Learn the number line running both ways, opposite numbers, comparing negatives, and the sign rules for addition, subtraction, multiplication and division.

Where this is used

Real situations where you count exactly the way this lesson teaches:

Screens, windows and the centre of a button
Every pixel on a screen has an address in a coordinate system. A button with corners (320, 180) and (480, 240) has its centre at (400, 210) — the average of the two corners. An interface that "centres itself" is computing exactly that formula, and the width of the button is 480 − 320 = 160 pixels, which is a subtraction of abscissas.
Surveying and setting out a point
A surveyor works in a national coordinate system where every boundary marker is a pair of numbers in metres. Given A(6210, 4380) and B(6250, 4460), the middle of the boundary falls at (6230, 4420) — and that is where the intermediate marker goes. The distance along the northing axis alone is 4460 − 4380 = 80 m.
Games and the position of a character
A character in a 2D game stands at (12, 5) on the tile grid and the exit is at (12, 11). The segment is vertical, so the walk is simply 11 − 5 = 6 tiles, and the engine puts the halfway checkpoint at (12, 8).
A room plan on squared paper
Drawing a room on squared paper, you take a corner as the origin and one square as 25 cm. A socket at (6, 0) is 6 · 25 = 150 cm from the corner, and a lamp hung halfway between walls at 0 and 14 lands on square 7, which is 175 cm.

All formulas

Coordinates of a point
$A = (x_A,\ y_A)$
the abscissa first, then the ordinate
Signs in the quadrants
$\text{I}: (+,\, +), \quad \text{II}: (-,\, +), \quad \text{III}: (-,\, -), \quad \text{IV}: (+,\, -)$
numbered counter-clockwise
Segment parallel to the x axis
$|AB| = |x_B - x_A|$
both ordinates are equal
Segment parallel to the y axis
$|AB| = |y_B - y_A|$
both abscissas are equal
Midpoint of a segment
$S = \left( \frac{x_A + x_B}{2},\ \frac{y_A + y_B}{2} \right)$
the average of the endpoints, coordinate by coordinate
The other endpoint
$B = (2x_S - x_A,\ 2y_S - y_A)$
the same formula read backwards

To say where a number sits on a line, one number is enough. To say where a point sits on a plane, one is not — two are needed, given in a fixed order. That idea is called a coordinate system, and it underpins everything we draw on squared paper: the graph of a function, a map, a floor plan and a phone screen.

Two axes and the origin

Take two perpendicular number lines and cross them at zero. The horizontal one is the x axis, the vertical one the y axis, and the point where they meet is the origin, written $O$ . Both axes have a direction: on the x axis numbers grow to the right, on the y axis upwards.

The axes cut the plane into four regions called quadrants. They are numbered $\text{I}$ to $\text{IV}$ counter-clockwise, starting from the top right.

Four points, one in each quadrant. The quadrant follows from the signs of the coordinates alone.

The coordinates of a point

The address of a point is a pair of numbers: the first says how far to go along the x axis, the second how far along the y axis. The first is the abscissa, the second the ordinate, and the pair is written in brackets:

A = (x_A,\ y_A)

The order is not a convention to remember for a minute — it is part of the address. The points $(3, 2)$ and $(2, 3)$ are in different places: three units right and two up versus two right and three up. That is why the pair is called ordered.

Give the coordinates of the point lying 4 units to the left of the y axis and 3 units below the x axis.

The quadrant follows from the signs

You do not need the drawing to name the quadrant — the signs of the two numbers are enough, because they are exactly what says which side of each axis the point is on. Negative numbers are the only earlier idea this needs.

Quadrant	Abscissa $x$	Ordinate $y$	Example
$\text{I}$	positive	positive	$(3,\ 2)$
$\text{II}$	negative	positive	$(-4,\ 3)$
$\text{III}$	negative	negative	$(-2,\ -3)$
$\text{IV}$	positive	negative	$(4,\ -2)$

Points on an axis belong to no quadrant

The axes are boundaries, not parts of the quadrants. A point lies on the x axis exactly when its ordinate is zero, and on the y axis exactly when its abscissa is:

(a,\ 0) \in OX, \qquad (0,\ b) \in OY, \qquad (0,\ 0) = O

The distinction comes back with every graph: a zero of a function is where the graph crosses the x axis, and the constant term is where it crosses the y axis.

A segment parallel to an axis is measured by subtracting

If two points share the same ordinate, the segment between them is horizontal and its length is the distance between the abscissas:

|AB| = |x_B - x_A|

Likewise for a vertical segment, where the abscissas are equal:

|AB| = |y_B - y_A|

The modulus is there so that the result does not depend on the order of subtraction — a length is never negative. It is the same distance on the number line, carried over to the plane.

A slanted segment needs the Pythagorean theorem, and its length is computed in the lesson on analytic geometry.

A(−3, 4) and B(5, 4) are the endpoints of a segment. How long is it?

The midpoint of a segment

The midpoint lies exactly halfway — and "halfway" for each coordinate separately means the average:

S = \left( \frac{x_A + x_B}{2},\ \frac{y_A + y_B}{2} \right)

The segment AB and its midpoint. The abscissa of S is the average of −4 and 2, its ordinate the average of −1 and 3.

Read backwards, the same formula answers the opposite question: given one endpoint and the midpoint, find the other endpoint. Since $x_S$ is an average, $x_B = 2x_S - x_A$ , and the same for the ordinates:

B = (2x_S - x_A,\ 2y_S - y_A)

S(2, −1) is the midpoint of AB and A = (−3, 4). Find B.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

which quadrant: A(4, 1)

Common mistakes

Swapping the coordinates — $(3, 2)$ is not $(2, 3)$ . The first number always runs along the horizontal axis.
Assigning a quadrant to a point on an axis — $(0, 4)$ and $(5, 0)$ lie in no quadrant.
Subtracting coordinates of a slanted segment — the difference of abscissas is the length only when the segment is horizontal; otherwise the Pythagorean theorem is needed.
Adding instead of averaging for the midpoint — the sum of the coordinates still has to be divided by two.
Averaging when looking for the other endpoint — there the reversed formula $2x_S - x_A$ applies, not an average.
Losing a minus when subtracting a negative — $5 - (-3) = 8$ , not $2$ .

Formula card

Topic: The coordinate plane

Coordinates of a point
$A = (x_A,\ y_A)$
the abscissa first, then the ordinate
Signs in the quadrants
$\text{I}: (+,\, +), \quad \text{II}: (-,\, +), \quad \text{III}: (-,\, -), \quad \text{IV}: (+,\, -)$
numbered counter-clockwise
Segment parallel to the x axis
$|AB| = |x_B - x_A|$
both ordinates are equal
Segment parallel to the y axis
$|AB| = |y_B - y_A|$
both abscissas are equal
Midpoint of a segment
$S = \left( \frac{x_A + x_B}{2},\ \frac{y_A + y_B}{2} \right)$
the average of the endpoints, coordinate by coordinate
The other endpoint
$B = (2x_S - x_A,\ 2y_S - y_A)$
the same formula read backwards

The four quadrants with one point in each. A quadrant is recognised from the signs of the coordinates alone.

The midpoint of AB with A(−4, −1) and B(2, 3). Each coordinate of the midpoint is the average of that coordinate of the endpoints.

Frequently asked questions

A luminous right triangle with squares built on its sides drawn as a constellation, beside a circle with its radius, against a cosmic nebula in violet and magenta

Geometry

Why we measure area in squares

Perimeter is counted in metres, area in square metres — and that single exponent is behind most of the mistakes people make in plane geometry. It is why 1 m² is not 100 cm², why two plots with identical area can need wildly different amounts of fencing, and why the Pythagorean theorem is really a statement about the areas of squares rather than the lengths of sides.

11 min