Coordinates

Coordinates are a set of numbers (one, two or more of them) that determine the position of a point or a geometric body within a coordinate system. The order of the coordinates is important and it indicates the axis of reference for that specific coordinate. In other words, it tells you which number refers to which axis.

There are several coordinate systems that are widely in use: the number line, the Cartesian coordinate system, the polar coordinate system, cylindrical and spherical coordinate systems. Since the others require some more advanced mathematics that was not covered on this site yet, we are going to turn our attention two the two most commonly used systems – the number line and the Cartesian system.

The number line is probably the first coordinate system you will come across during your education. It is basically a line divided in number of equal parts with a randomly chosen point of origin (O). The coordinate in this system shows the distance of the point in question from the point of origin. If the coordinate is a negative number, it is found on the left side of the point of origin. If it is a positive number, it lays on the right side of the point of origin.

The Cartesian coordinate system is the most well known coordinate system. It was invented in the 17th century by the French philosopher and mathematician René Descartes and it started a revolution in mathematics, as it was the first systematic link between Euclidean geometry and algebra. It can be used to represent two- or three-dimensional space and, depending on that, it consists of two or three mutually perpendicular lines called axes (the singular form is axis).

coordinates

In the two-dimensional version of the system, the lines are commonly designated as the x-axis (usually the horizontal one) and the y-axis (the vertical one). The point in which the two axes meet is called the point of origin. The coordinates are usually given in pairs, whereas the first number is the distance from the point of origin on the x-axis and the second number is the distance from O on the y-axis. The upper part of the y-axis (above the point of origin) contains the positive values, while the lower part contains the negative values. The positive part of the x-axis is the one on the right of the point of origin and the negative part is on the left. The system is divided into four quadrants. The quadrants are most often designated by roman numerals, starting with the upper-right quadrant (usually designated as the first quadrant). From that numbering goes counter-clockwise. In the first quadrant are coordinates with all positive values, such as (1,1), (3,7) and similar. In the second one the coordinates on the x-axis are negative, while the coordinates on the y-axis are positive and so on.

In the three-dimensional version, there are three perpendicular planes, three axes (x,y and z) and the position of a point is determined by three coordinates. Instead of quadrants, the system is divided into octants. The octant in which all three coordinates are positive is considered the first octant, but there are no established rules of nomenclature for the other octants.

The Midpoint formula

The midpoint is defined as the middle point of a line segment. It is equally distant (or equidistant) form both endpoints of the line segment. Depending on the number of dimensions, the position of the midpoint is determined by calculating each of its coordinates one at a time. If you are looking for the midpoint of a line segment on a number line, with endpoints (x1) and (x2), you will use the formula (x1 + x2)/2. The general form of that formula can be applied for all other coordinates in all other dimensions in search of the midpoint.

In a two-dimensional Cartesian system the endpoints will have coordinates (x1, y1) and (x2, y2) which means that to calculate the midpoint of that line segment you will have to calculate its x and y coordinates separately. The formulas you will use will be (x1 + x2)/2 for the x coordinate and (y1 +y2)/2 for the y coordinate.

If you wish to practice finding and plotting coordinates, as well as using the midpoint formula, please use the free worksheets below.

Coordinates exams for teachers

Exam Name File Size Downloads Upload date
Coordinates
Defining quadrant
Coordinates – Define quadrant number – very easy 662.2 kB 1743 October 13, 2012
Coordinates – Define quadrant number – easy 787.6 kB 862 October 13, 2012
Coordinates – Define quadrant number – medium 1.4 MB 1223 October 13, 2012
Coordinates – Define quadrant number – hard 3.7 MB 799 October 13, 2012
Coordinates – Define quadrant number – very hard 4.1 MB 991 October 13, 2012
Plot a point
Coordinates – Plot a point – very easy 374.2 kB 1254 October 13, 2012
Coordinates – Plot a point – easy 495.3 kB 983 October 13, 2012
Coordinates – Plot a point – medium 1.3 MB 1807 October 13, 2012
Coordinates – Plot a point – hard 3.6 MB 1087 October 13, 2012
Coordinates – Plot a point – very hard 4 MB 1155 October 13, 2012
Detect a point
Coordinates – Detect each point – very easy 364.2 kB 729 October 13, 2012
Coordinates – Detect each point – easy 443.5 kB 639 October 13, 2012
Coordinates – Detect each point – medium 1.2 MB 1052 October 13, 2012
Coordinates – Detect each point – hard 3.6 MB 740 October 13, 2012
Coordinates – Detect each point – very hard 3.9 MB 812 October 13, 2012
The Midpoint Formula
Integers
Midpoint Formula – Integers – very easy 613.1 kB 665 October 13, 2012
Midpoint Formula – Integers – easy 618.5 kB 466 October 13, 2012
Midpoint Formula – Integers – medium 630.2 kB 556 October 13, 2012
Midpoint Formula – Integers – hard 642.4 kB 424 October 13, 2012
Midpoint Formula – Integers – very hard 679.8 kB 480 October 13, 2012
Decimals
Midpoint Formula – Decimals – very easy 652.3 kB 416 October 13, 2012
Midpoint Formula – Decimals – easy 649.1 kB 369 October 13, 2012
Midpoint Formula – Decimals – medium 659.1 kB 395 October 13, 2012
Midpoint Formula – Decimals – hard 665.1 kB 377 October 13, 2012
Midpoint Formula – Decimals – very hard 696.3 kB 360 October 13, 2012
Fractions
Midpoint Formula – Fractions – very easy 672 kB 414 October 13, 2012
Midpoint Formula – Fractions – easy 674.8 kB 379 October 13, 2012
Midpoint Formula – Fractions – medium 694.5 kB 381 October 13, 2012
Midpoint Formula – Fractions – hard 738.9 kB 343 October 13, 2012
Midpoint Formula – Fractions – very hard 822.8 kB 390 October 13, 2012
Line segments
Midpoint Formula – Line segment – easy 1.4 MB 508 October 13, 2012
Midpoint Formula – Line segment – medium 1.4 MB 424 October 13, 2012
Midpoint Formula – Line segment – hard 1.4 MB 430 October 13, 2012


Coordinates worksheets for students

Worksheet Name File Size Downloads Upload date
Coordinates – Define quadrant number 37.2 MB 1000 October 14, 2012
Coordinates – Plot a point 36 MB 1299 October 14, 2012
Coordinates – Detect each point 34.8 MB 784 October 14, 2012
Midpoint Formula – Integers 260.5 kB 535 October 14, 2012
Midpoint Formula – Decimals 326 kB 444 October 14, 2012
Midpoint Formula – Fractions 517.3 kB 422 October 14, 2012
Midpoint Formula – Line segment 7.6 MB 521 October 14, 2012


Coordinates knowledge test