Chapter 5.3 — Exercise 5.3 — Plotting Points

Six Points, Six Positions

Plotting each one is entirely mechanical once the sign-reading habit built in the previous exercise is solid: count along the x-axis first (right for positive, left for negative), then straight up or down by the y-value from wherever that lands. (0, 11) and (−9, 0) are worth a second look precisely because they don't fit that two-step description as neatly — each has already done half the journey for free, since one of the two coordinates is 0 and needs no counting at all, landing the point directly on an axis rather than out somewhere in open quadrant space.

Same Digits, Different Point

Question 2 asks directly whether (5, −8) and (−8, 5) mark one and the same position on the plane. They don't — swapping two coordinates almost never leaves a point where it started. (5, −8) lies in the fourth quadrant (positive x, negative y); (−8, 5) lies in the second quadrant (negative x, positive y). Same two digits, same two signs even, just attached to the opposite coordinate each time, and the result is two points in entirely different quadrants rather than two names for one location. The only case where swapping the two coordinates does leave a point unmoved is when both coordinates are already equal to begin with, like (3, 3) — reflecting a point across the line y = x, which is really what coordinate-swapping always does geometrically, only returns the same point when that point was already sitting exactly on that line of reflection.

A Family of Points With One Coordinate Fixed

Plotting (1, 2), (1, 3), (1, −4), (1, 0), and (1, 8) — every one with x = 1 — produces five points scattered vertically but all sitting on one single line: the vertical line exactly 1 unit to the right of the y-axis. Fixing y instead of x produces the mirror-image pattern: (5, 4), (8, 4), (3, 4), (0, 4), (−4, 4), and (−2, 4) all sit on one horizontal line, 4 units above the x-axis. In both cases, whichever coordinate is held constant across every point in the set determines a line parallel to one axis, sitting exactly that many units away from it. Which axis the line runs parallel to, and which axis it's measured away from, are opposite: a fixed x gives a line parallel to the y-axis but offset from it, while a fixed y gives a line parallel to the x-axis but offset from that one instead — it's always the axis matching the coordinate that isn't fixed that the resulting line ends up parallel to.

Area From a Handful of Vertices

Plotting (0, 0), (0, 3), (4, 3), and (4, 0) and joining them in order gives a rectangle 4 units wide and 3 units tall — area 12 square units, no different at all from finding area any other way once the two side lengths are simply read straight off the coordinates themselves.

The triangle formed by (2, 3), (6, 3), and (4, 7) works the same way. The base runs along y = 3 from x = 2 to x = 6 — 4 units. The height is the vertical distance from that base up to (4, 7) — also 4 units, read directly from the difference in y-values (7 − 3). Area = ½ × base × height = ½ × 4 × 4 = 8 square units, without needing to measure anything off a drawn grid at all. This trick — reading a base and a height straight from coordinates rather than a ruler — only works this cleanly because two of the triangle's vertices happened to share the same y-value, giving a perfectly horizontal base to measure along. A triangle whose vertices don't line up so conveniently along one axis needs a more general area formula, which is exactly the tool a later coordinate geometry chapter introduces to handle that harder case.

Six Points Sharing One Property

Asked for six points whose coordinates sum to 5, any pair that adds to 5 qualifies: (−2, 7), (1, 4), (0, 5), (3, 2), (5, 0), and (6, −1) all satisfy x + y = 5, and plotting them reveals they fall on a single straight line — the condition "coordinates sum to a fixed value" always describes a line, the same way "one coordinate fixed" did earlier in this exercise, just tilted rather than parallel to an axis. Every point on that tilted line trades y for x at a fixed rate — increase x by 1 and y must drop by exactly 1 to keep the sum at 5, which is why the six sample points step evenly down and to the right of each other rather than clustering or jumping unpredictably around the plane.

Reading a Finished Graph Back Into Coordinates

The reverse skill closes the loop: given a graph with labelled points already plotted, reading off each one's coordinates by tracing to both axes. A(−3, 4), B(0, 5), C(3, 4), and so on through a full set of labelled points — the exact same tracing technique introduced in Exercise 5.2, just applied to more points at once, and then joined together, in the order given, by straight line segments to see what overall shape the full set of points ends up tracing out. A grid with a dozen or more labelled points can look intimidating at first glance, but nothing about the underlying task changes with scale — each point still only needs its own two distances read off independently, in any order, without reference to any of the others sharing the same page.

What Comes After the Plane Itself

Every technique from this exercise — plotting, comparing swapped coordinates, spotting a fixed-coordinate line, computing area from vertices — is exactly what every later geometry proof involving coordinates assumes can already be done fluently, without needing to re-derive any of it from scratch each time it comes up again. The most immediate use is in Linear Equations in Two Variables, the next chapter, where every solution to an equation is itself a coordinate pair, plotted on exactly this same plane to see the line it traces out. Describing distances and lines with actual formulas rather than reading them off a grid by eye comes later still, once Class 10 picks Co-ordinate Geometry back up with the distance and section formulas.