Chapter 7 · Introduction

Chapter 7 — Coordinate Geometry

A point on a map or a playground can be described by two numbers. Once every position is a pair of numbers, geometry turns into arithmetic: one formula gives the straight-line distance between any two points, and another finds the point that divides a segment in any ratio.

On sports day your school marks the ground with a grid of white lines. The starting flag stands at the corner, and every other flag is described by two numbers: how many metres to the right, and how many metres up. A flag at $(24, 7)$ is $24$ m along and $7$ m up from the corner. Now a question that a referee has to answer a hundred times a day: how far is that flag from the corner, in a straight line? You could walk it with a tape measure. Or you could notice that the $24$ m, the $7$ m and the straight line form a right-angled triangle, and use Pythagoras: $\sqrt{24^2+7^2}=\sqrt{625}=25$ m. No tape needed.

That is the idea of coordinate geometry: describe points by numbers, and let algebra do the measuring. In this chapter you build two tools. The distance formula gives the length of the segment joining any two points, and lets you test whether three points lie on one line or whether four points make a square. The section formula finds the point that divides a segment in a given ratio — the midpoint, the two trisection points, or the spot $\tfrac35$ of the way along a road — and tells you where a segment crosses an axis. Both tools are used by every map app, every game engine and every surveyor.

Points as pairs $(x, y)$→Right triangle on the segment→Distance $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$→Divide in the ratio $m_1:m_2$→Midpoint, trisection, axis crossing
🎯 By the end of this chapter you can…
  • read and plot points $(x, y)$ and find the distance between two points on a horizontal line, on a vertical line, or anywhere, using $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$;
  • find the distance of a point from the origin and decide which of several points is nearest or farthest;
  • test whether three points are collinear, and whether four points form a square, rectangle, rhombus or parallelogram, by comparing distances;
  • find a point on an axis that is equidistant from two given points, and the missing coordinate when a distance is given;
  • use the section formula to find the point dividing a segment internally in the ratio $m_1:m_2$, and write down the midpoint and the two trisection points;
  • find the ratio in which a point, or an axis, divides a segment, and find an unknown endpoint from a midpoint.
Teaser — Distance and Midpoint from Two Points

Each situation gives two points $A$ and $B$ on a grid. Press Play to build the picture step by step: first the points, then the horizontal and vertical legs of a right triangle, then the straight line $AB$ as its hypotenuse, whose length comes from Pythagoras, and finally the midpoint. Pick a situation and predict the distance before it appears.

legs—
distance $AB$—
midpoint—

Sections in this chapter

7.1

The Distance Formula

The straight-line distance between two points, with the right triangle behind it; tests for collinear points and for the shape of a triangle or quadrilateral.

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$
Open section →7.2

The Section Formula

The point that divides a segment internally in the ratio $m_1:m_2$: midpoint, trisection points, where an axis cuts a segment, and missing endpoints.

$P=\left(\dfrac{m_1x_2+m_2x_1}{m_1+m_2},\ \dfrac{m_1y_2+m_2y_1}{m_1+m_2}\right)$
Open section →
✅ Before you start
From Class 9 you need to plot points $(x, y)$ in the four quadrants, to know the Pythagoras relation $c^2=a^2+b^2$ in a right-angled triangle, and to recognise a square, rectangle, rhombus and parallelogram by their sides and diagonals. You need to solve linear equations (Section 3.2) and simple quadratic equations such as $(x-3)^2=16$ (Section 4.2). The proof of the section formula uses similar triangles, so a quick look at Section 6.3 helps. Warm up: what is the distance between $(1, 2)$ and $(4, 6)$? (Answer: $5$ — a $3$–$4$–$5$ triangle.)
← 6.3 Criteria for Similarity of Triangles7.1 The Distance Formula →