Chapter 5.1 — Exercise 5.1 — Introduction
Introduction to coordinate geometry. This is Lesson 1 of 3 in Chapter 5: Co-ordinate Geometry.
One Number Isn't Enough
Locating something precisely almost always takes more than a single number. Exercise 5.1 builds that idea through three everyday situations — a ball on a shelf, a student's seat in a classroom, and a house on a street — before Co-ordinate Geometry gives the idea a formal, two-number shape in the exercises that follow.
The Ball on the Shelf
A teacher holds up a red ball and asks where it is. One student says "6th place." Another says "2nd place." Both are right — one counted from the left end, the other from the right — and neither answer is wrong so much as incomplete, because a single position number only means something once everyone agrees, in advance, exactly where the counting is supposed to start from. Fixing a starting point removes the ambiguity entirely; without one, the same object has as many different "correct" position numbers as there are places someone might choose to start counting from. Notice that both students were using the exact same shelf and the exact same ball — nothing about the physical situation was in dispute. The disagreement was entirely about the counting convention being used, not about the ball's actual physical location, which is precisely why mathematics eventually insists on one single, universally agreed starting point rather than leaving that choice open to whoever happens to answer first.
Locating Hari in a Grid
A classroom seating grid pushes the same idea further. Asked where Hari sits, one student answers "2nd row," another answers "4th column." Neither alone pins down a single seat — an entire row has many columns, and an entire column has many rows. Only the two pieces stated together — "2nd row and 4th column" — identify exactly one single seat. This is the first genuinely two-dimensional version of the ball puzzle: one number locates a position along a single line, the way the shelf did, but a grid — spreading out in two directions at once — needs two numbers, one for each direction, before a single point is fixed rather than merely narrowed down to an entire row or an entire column. It's worth noticing that the two numbers aren't interchangeable, either — "4th row, 2nd column" describes a different seat than "2nd row, 4th column" in any grid wider than it is tall. Which number comes first, and what direction each one counts along, has to be agreed on just as firmly as where zero sits — an agreement Co-ordinate Geometry eventually locks in permanently with the x-coordinate always listed first.
Reading a Locality Map
Question 1 applies the same two-number logic to a real map: a locality with a main road running north–south, crossed by four numbered streets (1 to 4) running east–west, with buildings sitting on either side of the main road along each street. Locating any single building on this map needs exactly the same two pieces of information as locating Hari's seat did earlier — which street (comparable to which row), and where along that street relative to the main road (comparable to which column) — just relabelled for a street map instead of a classroom grid, with "street number" standing in for "row" and "side plus count" standing in for "column."
Five Questions, One Counting Rule
| Question | Answer |
|---|---|
| 3rd object, left side, Street-3 | Water Tank |
| 2nd house, right side, Street-2 | House J |
| Location of Mr. K's house | Street-2, 3rd house on the right |
| Position of the Post Office | Street-4, 1st building on the right |
| Location of the Hospital | Street-4, 3rd building on the left |
Every one of the five answers above has exactly the same three-part shape: a street number, then a side, then a count outward starting from the main road and moving away from it. Drop any one of those three pieces and the location stops being unique — "the 3rd building" alone could mean the 3rd on either side of either the same street or a different one entirely.
The "side" piece deserves a closer look, since it's easy to treat as an afterthought. Street-3 has buildings on its left and its right, and each side gets counted separately, starting fresh from the main road. School, Park, and Water Tank are 1st, 2nd, and 3rd specifically among left-side buildings — whatever sits on the right side of that same street has no bearing on that count at all, since the two sides never share a single counting sequence — a left-side building and a right-side building could both legitimately be called "1st" on the exact same street without any contradiction, precisely because "1st" is only ever meaningful relative to its own side's local counting rule, never as some kind of overall, street-wide ranking that spans both sides together. This is exactly why Question 1's five answers each mention a side explicitly rather than leaving it implied.
From Streets to Coordinates
Look again at what each answer actually specifies: a street number (how far along the north–south direction) and a side-plus-count (how far along the east–west direction, and which way). That is exactly the shape a coordinate takes. Instead of "Street-2, 3rd house on the right," mathematics writes a single pair of numbers, one for each direction, and reads both from one shared reference point rather than from a street's own local counting rule.
The map version also exposes a limitation the ball and grid examples didn't have to deal with: "side" is really just a compressed way of saying "positive distance in one direction versus positive distance in the other," described in words instead of with a sign. The main road plays exactly the role that zero plays on an ordinary number line — every building's position is measured as some distance to one side of it or to the other — except here that same idea has to work in two directions at once, one along each street and one along the road itself. Formalising exactly that limitation, replacing the word "side" everywhere with a proper plus or minus sign instead, is what the next exercise exists to do.
Building the Formal System Next
Everything here — the ball, the classroom grid, the locality map — is the same idea told three times with increasingly formal language. Exercise 5.2 replaces "main road" and "street number" with the x-axis and y-axis, and "side plus count" with a signed number — turning this chapter's map-reading intuition into the Cartesian plane used throughout the rest of Class 9 and 10 geometry. The main road becomes one perpendicular reference line, the streets become the other, and "side" becomes a plus or minus sign rather than a word — the underlying skill of reading two pieces of positional information together, built here through the ball, the classroom, and the map, doesn't change at all; only the notation used to write it down does.