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Coordinate Geometry
Introduction
Distance Formula
Practice: Coordinate Distances
Section Formula
Practice: Dividing Segments in a Ratio
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Coordinate GeometryIntroduction

Reading time: ~15 min

The story goes that one morning, almost four hundred years ago, the French thinker René Descartes lay in bed watching a fly crawl across the tiled ceiling of his room. How could he tell a friend exactly where the fly was sitting?

He realised that two numbers were enough: how many tiles across from one corner, and how many tiles up. That simple idea — a point is a pair of numbers — is where coordinate geometry begins.

You already use it. In Class IX you learnt that to fix a point on a plane we need a pair of coordinate axes. The distance of a point from the is its x-coordinate, or abscissa. Its distance from the is its y-coordinate, or ordinate. A point on the x-axis always looks like (x, 0), and a point on the y-axis always looks like (0, y).

Tip

The abscissa comes first and runs across — both start with "a". The ordinate comes second and runs up. So the point (4, 3) means 4 across, then 3 up, exactly like Descartes counting ceiling tiles.

Join the Dots

Your textbook has a puzzle for you. On a pair of axes, join A(4, 8) to B(3, 9) to C(3, 8) to D(1, 6) to E(1, 5) to F(3, 3) to G(6, 3) to H(8, 5) to I(8, 6) to J(6, 8) to K(6, 9) to L(5, 8) and back to A. Then make three small triangles: P(3.5, 7), Q(3, 6), R(4, 6); X(5.5, 7), Y(5, 6), Z(6, 6); and S(4, 5), T(4.5, 4), U(5, 5). Finally, join S to (0, 5) and (0, 6), and join U to (9, 5) and (9, 6).

Plot the first four points yourself by tapping the grid — the first number tells you how far across, the second how far up. Then let the machine finish the job.

What picture have you got? A ! The tip of its nose is the point T, at (, ).

You have also seen pictures made by equations. A linear equation in two variables, ax + by + c = 0 (with a and b not both zero), gives a when you draw its graph. In Chapter 2 you met the graph of y = ax^2 + bx + c \; (a \neq 0), which is a parabola.

This is the whole point of coordinate geometry: it lets us study shapes with algebra, and understand algebra by looking at shapes. That two-way street is why it turns up everywhere — in physics, engineering, navigation, the study of earthquakes (seismology) and even art.

In this chapter you will learn two things: how to find the distance between two points whose coordinates you know, and how to find the point that divides a line segment joining two given points in a given ratio.

Speed Round: Where Does It Live?

Every point that is not on an axis lives in one of four quadrants, and the signs of its coordinates tell you which one: (+, +) is Quadrant I, (−, +) is Quadrant II, (−, −) is Quadrant III and (+, −) is Quadrant IV. A zero coordinate puts the point on an axis.

Get five right to move on.

Sina