Chapter 3: Coordinate Geometry

Hundreds of years ago, René Descartes asked a question that changed mathematics forever: can geometry be done with algebra? His answer was yes. Cover the plane with two perpendicular number lines, label every point with a pair of numbers, and now every line, circle, triangle and curve can be described by an equation. This chapter introduces the coordinate plane , the bridge that makes Descartes's dream work.

The chapter is short and almost entirely about fluency. You will set up the $x$-axis and $y$-axis, learn how the plane splits into four quadrants, and master the convention of an ordered pair $(x, y)$. Once you can plot points and read coordinates without thinking, the geometry of triangles, lines, circles and curves opens up in a completely new way.

Coordinate geometry is everywhere outside school too. A map uses latitude and longitude , that is just a coordinate system on the Earth. A computer screen labels every pixel by a pair $(x, y)$ to draw an image. GPS does the same in three dimensions. Even the seats in a stadium use coordinates: row and column. The convention you learn here is the same convention used in physics, engineering, computer graphics, and data science.

What makes this chapter especially powerful is that you can now prove geometric statements algebraically. Want to show that a particular triangle is right-angled? Plot the vertices, compute the distances, and check the Pythagoras condition. Want to show three points are collinear? Show that one ordering of slopes is equal. We will use the simplest version of these ideas here; in Class X you will see them generalised.

You will also begin to think of equations as graphs. The equation $x + y = 5$ describes infinitely many points, but they all lie on a single straight line in the plane. We touch on this idea here and develop it fully in Chapter 4 on linear equations in two variables.

By the end of the chapter you should be able to plot any point at a glance, identify which quadrant it lives in, find the coordinates of any point shown on a grid, and understand why "$(3, 5)$" and "$(5, 3)$" are different points.

What's inside

• The Cartesian plane , setting up the two axes and the origin.
• Quadrants and signs , which signs of $x$ and $y$ go with which quadrant.
• Ordered pairs , coordinates, abscissa, ordinate, and why order matters.
• Plotting points , going from $(x, y)$ to a dot on the grid.
• Reading coordinates , going from a dot to $(x, y)$, with attention to the special positions on the axes.

Key results / Formula card

Memorise this card. Nearly every question in the chapter is read directly from one of these lines.