Coordinate Geometry

Why This Chapter Matters

On a number line, one number is enough to locate a point because there is only one direction to move. In a plane, one number is not enough. You need to know how far to move horizontally and how far to move vertically.

Coordinate geometry creates a bridge between algebra and geometry. A point becomes an ordered pair of numbers, while a geometric picture can be described numerically. This idea later allows equations to become graphs and graphs to reveal algebraic relationships.

The chapter looks simple, but accuracy matters. Nearly every mistake comes from one of three habits: reversing coordinates, ignoring signs or misreading the scale.

Bridge from Earlier Classes

Before starting, be comfortable with:

• Number line
• Positive and negative integers
• Perpendicular lines
• Basic graph reading

These are the tools this chapter reuses and deepens.

1. The Cartesian Plane

The Cartesian plane is formed by two perpendicular number lines. The horizontal line is the x-axis and the vertical line is the y-axis. Their intersection is the origin.

The axes divide the plane into four regions called quadrants. They are numbered anticlockwise beginning with the upper-right region.

Coordinates measure displacement from the axes. The x-coordinate controls horizontal position; the y-coordinate controls vertical position.

2. Ordered Pairs

A point is written as (x,y), and the order matters. To plot (4,-3), start at the origin, move 4 units right because x is positive, then 3 units down because y is negative.

The first coordinate is also called the abscissa and the second the ordinate. These names are useful, but the essential rule is simpler: horizontal first, vertical second.

3. Sign Pattern of Quadrants

Quadrant I has (+,+), Quadrant II (-,+), Quadrant III (-,-) and Quadrant IV (+,-).

You do not need to memorise this as an isolated table. Right of the y-axis means x positive; left means x negative. Above the x-axis means y positive; below means y negative. The sign pattern follows directly from direction.

4. Points on the Axes

A point on the x-axis has no vertical displacement, so y=0 and its coordinates look like (a,0). A point on the y-axis has x=0 and looks like (0,b).

The origin (0,0) lies on both axes and in no quadrant.

5. Reading and Plotting Carefully

Always read the scale before plotting. One grid square may represent 1 unit, 2 units or another value.

When reading a point, project vertically to the x-axis and horizontally to the y-axis. When plotting, reverse the process: move according to x first and y second. This consistent routine prevents most errors.

6. Why Coordinate Geometry Matters

Coordinates turn location into numbers. In the next chapter, every solution of a linear equation in two variables will be an ordered pair. Plotting those pairs produces a line.

So this chapter is really the language needed for graphical algebra.

Visualise the Core Idea

The sign pattern of each quadrant comes directly from the directions of the axes.

Formula / Theorem Sheet

Worked Examples — Follow the Reasoning

How to Think in the Exam

Common Mistakes

Quick Revision

Focused Practice

Final Chapter Summary