Cartesian Coordinate System: Axes, Quadrants, and Plotting Points

A number line gives every number an address. The Cartesian coordinate system does the same thing for a flat surface: cross two number lines at right angles and every point in the plane gets a unique address of its own, written as a pair of numbers.

That sounds like bookkeeping, and at first it is. What makes it one of the most useful ideas in mathematics is what follows from it — once points have numerical addresses, questions that look geometric turn into arithmetic. How far apart are these two things? Is this triangle a right triangle? Where is the exact centre of this line segment? All of it becomes calculation, no measuring involved.

The Cartesian Coordinate System Formula

A point is named by an ordered pair — horizontal position first, vertical second. The distance between two points is the Pythagorean theorem applied to the gaps between their coordinates.

The Two Axes and the Origin

The horizontal number line is the -axis. The vertical one is the -axis. They cross at a single point called the origin, which has the address .

The axes carry the same conventions as an ordinary number line, applied in two directions:

• On the -axis, values increase to the right and decrease to the left.
• On the -axis, values increase upward and decrease downward.

Together the two axes divide the plane into four regions, and the plane as a whole is often called the coordinate plane or the Cartesian plane.

Ordered Pairs: Why the Order Matters

Every point is written , and the sequence is fixed: the first number is always horizontal, the second always vertical. The numbers are called the point’s coordinates — the -coordinate, sometimes called the abscissa, and the -coordinate, sometimes called the ordinate.

This is why they are called ordered pairs. Swap the numbers and you get a different point:

The first sits three units right and five units up. The second sits five right and three up. They are only the same point when the two coordinates happen to be equal, as in .

If you ever lose track of which comes first, the alphabet settles it: before , horizontal before vertical.

The Four Quadrants

The axes cut the plane into four regions, numbered with Roman numerals anticlockwise starting from the top right. Which quadrant a point lands in is decided entirely by the signs of its two coordinates:

The numbering direction is worth committing to memory, because it runs anticlockwise while almost everything else in everyday life runs clockwise.

Points That Belong to No Quadrant

A quadrant is one of the four open regions between the axes, so a point sitting exactly on an axis is not inside any of them. This catches people out because the question “which quadrant?” seems to demand one of four answers.

• lies on the -axis.
• lies on the -axis.
• is the origin, on both.

The rule is simply that a zero in either coordinate puts the point on an axis. “On the -axis” is a complete and correct answer.

Plotting a Point

Plotting is two moves, always in the same order:

1. Start at the origin.
2. Move horizontally by the -coordinate — right if positive, left if negative.
3. Move vertically by the -coordinate — up if positive, down if negative.
4. Mark the point and label it.

Reading a point off an existing graph is the same process in reverse: drop straight down to the -axis to read the first coordinate, then straight across to the -axis for the second.

Distance Between Two Points

Two points and the right angle between their horizontal and vertical gaps form a right triangle. The horizontal leg has length , the vertical leg , and the segment joining the points is the hypotenuse. The Pythagorean theorem does the rest:

The absolute value signs disappear because each difference gets squared, and squaring destroys the sign. That also means the order of the two points is irrelevant — subtracting the other way round flips the sign of both differences, and both get squared away.

A distance is never negative, so the answer is always the positive square root. If the number under the root is not a perfect square, leave it exact — — unless the question asks for a decimal. Simplifying Radicals covers that step in full.

The Midpoint of a Segment

The midpoint is the point exactly halfway between two others, and it is found by averaging each coordinate on its own:

Notice the structural difference from the distance formula: this one adds and halves, that one subtracts and squares. Mixing them up is common, and the sanity check is easy — a midpoint must land between the two original points, so if your answer sits outside them, you have subtracted somewhere you should have added.

Worked Example A: Plotting a Point

Plot .

Step 1 — read the coordinates. The -coordinate is ; the -coordinate is .

Step 2 — move horizontally. From the origin, go units right, because is positive.

Step 3 — move vertically. From there, go units down, because is negative.

Right and down puts the point in Quadrant IV, which matches the sign pattern .

Worked Example B: Naming Quadrants

State the quadrant, or the axis, for each point: , , , , .

Read the signs, nothing else:

• is → Quadrant II
• is → Quadrant I
• is → Quadrant III
• has → on the -axis, no quadrant
• is → Quadrant IV

Worked Example C: Distance Between Two Points

Find the distance between and .

Step 1 — substitute into the formula:

Step 2 — evaluate inside the root:

Step 3 — take the positive root:

The gaps of and with a hypotenuse of is the 3–4–5 right triangle, which is a useful sign the arithmetic went right.

Worked Example D: A Distance That Is Not a Whole Number

Find the distance between and .

, so the exact distance is , roughly .

Note the double negative in the first bracket: , not . Subtracting a negative coordinate is where most sign errors in this formula happen.

Worked Example E: Finding a Midpoint

Find the midpoint of the segment joining and .

Average each coordinate separately:

Check it against the sanity rule: sits between and , and sits between and . Both hold, so the point is plausibly the middle.

Worked Example F: Classifying a Triangle

The points , and are the corners of a triangle. Is it isosceles?

Find all three side lengths:

Two sides measure , so the triangle is isosceles. This is what the coordinate system buys you — a question about shape, answered with nothing but subtraction and square roots.

Common Mistakes to Avoid

• Reversing the coordinates. and are different points. Horizontal always comes first.
• Numbering the quadrants clockwise. They run anticlockwise from the top right, so Quadrant II is the top left.
• Forcing an axis point into a quadrant. A zero in either coordinate means the point is on an axis and in no quadrant.
• Losing a sign when subtracting a negative. In the distance formula, is , not .
• Mixing up the two formulas. Distance subtracts and squares; midpoint adds and halves. If a “midpoint” lands outside the two points, they have been swapped.
• Reporting a negative distance. The square root here is the principal one, so a distance is always zero or positive.
• Averaging the wrong pairs for a midpoint. Average the two -values together and the two -values together — never an with a .

Where This Shows Up Later

• Graphing equations. Every line, parabola and curve you draw is a set of ordered pairs that satisfy an equation, plotted on this plane.
• Slope and linear equations. Slope is the vertical change divided by the horizontal change between two points, built from the same coordinate differences as the distance formula.
• Functions. A function’s graph is the set of points , so the vocabulary here is the vocabulary of every function you will meet.
• Circles and conic sections. The equation of a circle is the distance formula rearranged: every point a fixed distance from a fixed centre.
• Vectors and later mathematics. Coordinates extend to three dimensions and beyond with the same logic and an almost identical distance formula.

Practice Problems

Work each problem yourself before opening the answer.

Problem 1. In which quadrant does lie?

Problem 2. In which quadrant does lie?

Problem 3. Where is the point ?

Problem 4. Are and the same point?

Problem 5. Find the distance between and .

Problem 6. Find the distance between and .

Problem 7. Find the distance between and .

Problem 8. Find the distance between and , leaving the answer exact.

Problem 9. Find the midpoint of the segment joining and .

Problem 10. Find the midpoint of the segment joining and .

Problem 11. Find the exact distance between and , then say which quadrant each point is in.

Problem 12. One endpoint of a segment is and its midpoint is . Find the other endpoint.

Quick Reference

The square roots in the distance formula are simplified using Radicals and Simplifying Radicals. Once points are plotted, Linear Equations are what get graphed on this plane first. More Algebra lessons are available too.