Orienting Yourself: The Use of Coordinates – Class 9 Maths Chapter 1 Notes & NCERT PDF

Introduction

A system of coordinates is a structured framework — like the grid lines on a map or on graph paper — that lets us use numbers to describe the exact position of a point or object. The chapter opens with the long history of this idea:

The chapter then follows Reiaan, who cannot see, and his sister Shalini, who has just finished Grade 9. When the family moves to a new city, Shalini builds a pin-and-thread model of Reiaan’s room on a grid (scale 1 cm : 1 foot) so he can feel where things are.

Key Ideas

1. The 2-D Cartesian coordinate system

The coordinates of a point P are written (x, y): x is its perpendicular distance from the y-axis, and y is its perpendicular distance from the x-axis.

2. Quadrants

The axes divide the Cartesian plane (xy-plane) into four quadrants:

Order matters: (x, y) = (y, x) only when x = y.

3. Distance between two points

For points on a line parallel to an axis, the distance is simply the absolute difference: |x₂ – x₁| or |y₂ – y₁|.

For any two points, move along the x-direction and the y-direction and use the Baudhāyana–Pythagoras Theorem:

Distance between (x₁, y₁) and (x₂, y₂) = √[(x₂ – x₁)² + (y₂ – y₁)²]

It does not matter whether the differences are positive or negative, because they are squared.

4. Reflection preserves length

Reflecting triangle ADM in the y-axis changes the sign of each x-coordinate — A (3, 4) → A′ (–3, 4) — but every side length stays the same.

Worked Examples

Example 1 — Side AD of triangle ADM. A (3, 4), D (7, 1).

Example 2 — Sides DM and MA. D (7, 1), M (9, 6), A (3, 4).

Example 3 — Finding an endpoint from a midpoint. M (–7, 1) is the midpoint of A (3, –4) and B (x, y).

Example 4 — Points on a circle. Do A (1, –8), B (–4, 7), C (–7, –4) lie on one circle centred at O?

Example 5 — Is ABCD a square? A (2, 1), B (–1, 2), C (–2, –1), D (1, –2).

Figure it Out — Hints

Maths Words

Historical and Indian Connections

The grid streets of the Sindhu-Sarasvatī cities, Baudhāyana’s Śulba constructions, the Ujjayinī meridian, Āryabhaṭa’s sines and Brahmagupta’s negative numbers are all steps on the road to the coordinate plane that Descartes formalised in 1637.

Common Mistakes

Maths in Daily Life

Maps and GPS, seat numbers in a theatre, pixels on a phone screen (origin at a corner), floor plans for furniture, and accessible design — for example, checking whether a door is wide enough for a wheelchair, as the chapter asks.