Coordinates in a Plane

Combining Algebra and GeometryGraphs of Equations in Two Variables

Coordinates in a Plane

The Cartesian coordinate system lets us describe every point in the plane with an ordered pair of numbers , called its coordinates. This system, introduced by René Descartes in 1637, is the foundation of analytic geometry, linking geometric shapes to algebraic equations.

Quick Reference

Formula Name Notes
Distance Formula Distance between and
Midpoint Formula Midpoint of segment

History

Before the 17th century, geometry and algebra were two distinct branches of mathematics. In 1637, René Descartes, a French mathematician, scientist, and philosopher, made a huge impact on mathematical knowledge by unifying these two fields. His approach is now called analytic geometry. Its key feature is the use of a coordinate system: by means of coordinates, we can apply algebraic methods to geometry and geometrically represent algebraic equations. Analytic geometry is now fundamental to mathematics, science, and engineering.

Points and Coordinates

In the study of sets of numbers, we showed how the real number system assigns coordinates to points on a line.

Coordinates on a Line


On a straight line, choose a unit length and a fixed origin point . Any point on the line corresponds to a real number: positive numbers lie to the right of , negative numbers to the left. Only one number is needed to specify a point's position on a line.

In the plane, two real numbers are needed to specify a point . We draw two perpendicular axes, the -axis (horizontal) and the -axis (vertical), intersecting at the origin. Using the same unit length on both axes:

  • Distances to the right on the -axis are positive; to the left, negative.
  • Distances upward on the -axis are positive; downward, negative.

The position of a point is described by two signed distances: along the -axis and along the -axis. These are called, respectively, the abscissa (or -coordinate) and the ordinate (or -coordinate) of . Together they are called the rectangular coordinates or Cartesian coordinates of , written .

The Cartesian coordinate plane showing the x-axis and y-axis intersecting at the origin O. A point P is plotted in the first quadrant with coordinates (a, b). The four quadrants are labeled I, II, III, and IV.
The Cartesian coordinate plane with point P(a, b)

The point lies 4 units right and 3 units down, while lies 3 units left and 4 units up. These are clearly different points:

A Cartesian coordinate plane plotting (4, −3) in the fourth quadrant and (−3, 4) in the second quadrant, illustrating that ordered pairs depend on order.
The points (4, −3) and (−3, 4) are different ordered pairs

For this reason, is called an ordered pair.

Key facts:

  • The axes divide the plane into four regions called quadrants, numbered I, II, III, and IV counterclockwise from the top-right.
  • Instead of "the point with coordinates ," we say simply "the point ."
  • The symbol can also denote an open interval; context determines the meaning.
  • The set of all ordered pairs is .

Distance Between Points

Since we use the same unit length on both axes, we can apply the Pythagorean theorem to find the distance between two points. Given and :

Distance Formula


The distance between points and is:

Diagram showing points P and Q connected by a line segment, with a right triangle formed by legs |x₂ − x₁| and |y₂ − y₁|, illustrating the distance formula.
The Distance Formula follows from the Pythagorean theorem

Example 4. Find the distance between and .

Solution.

The Midpoint Formula

Midpoint Formula


The coordinates of the midpoint of the line segment joining and are:

In words: the midpoint's coordinates are the averages of the endpoints' coordinates.

A coordinate plane showing a line segment between P and Q with midpoint M marked at the center.
The midpoint M of segment PQ

Example 5. Find the midpoint of the segment with endpoints and .

Solution.

Frequently Asked Questions

What is an ordered pair and why does order matter?

An ordered pair specifies a point in the plane using two coordinates in a fixed order: is the horizontal position and is the vertical position. Order matters because and refer to completely different locations in the plane.


How do I find the distance between two points? Use the Distance Formula: for points and , This is a direct application of the Pythagorean theorem.

What are the four quadrants? The axes divide the plane into four quadrants, numbered I through IV counterclockwise: Quadrant I (), Quadrant II (), Quadrant III (), and Quadrant IV ().

Can the distance between two points be negative? No. The distance formula always gives a non-negative result. If the distance is zero, the two points are identical.
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