CLASS 9 • MATHS • CHAPTER 1
Orienting Yourself:
Orienting Yourself:
The Use of Coordinates
Short, colourful, handwritten-style notes — by EduGrown
1What is a Coordinate System?
Suppose your friend says, “Meet me in the school ground.” That is not enough — the ground is huge! But if he says “3 steps right and 4 steps up from the gate”, you can find the exact spot.
📝 Definition
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 an object.
- A number line is 1-dimensional — only one number is needed to fix a point.
- A plane (flat surface) is 2-dimensional — we need two numbers to fix a point.
2A Quick Look at History
Grid-based thinking has very deep roots in Bhārat. Here is the whole story in one glance:
| Who / When | Contribution |
|---|---|
| Sindhu–Sarasvatī Civilisation | City streets built in N–S and E–W directions, about 10 m apart — the first grid used on a huge scale. |
| Baudhāyana | Used E–W and N–S lines for geometry; gave the Baudhāyana–Pythagoras Theorem. |
| Ujjayinī (4th century BCE) | Treated as the central longitude meridian from which other places were measured. |
| Āryabhaṭa (c. 499 CE) | Replaced Greek ‘chords’ with sines; mapped the sky using celestial coordinates. |
| Brahmagupta (c. 628 CE) | Formalised zero and negative numbers — without these the four quadrants would be impossible. |
| Al-Bīrūnī (c. 1000 CE) | Used Indian methods to find coordinates of cities; perfected the astrolabe. |
| Ömar Khayyām (c. 1100 CE) | First to solve algebra problems using geometry through coordinates. |
| Fermat & René Descartes (1636–37 CE) | Stated that any point in a plane can be fixed by just two numbers — birth of the Cartesian plane. |
💡 Remember this!
The word “Cartesian” comes from Descartes. And the origin being zero is Brahmagupta’s gift to mathematics.
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3The Story: Shalini Helps Reiaan
Reiaan’s family shifts to a new city. Reiaan is not able to see, so his sister Shalini uses what she learnt in Coordinate Geometry to help him.
- She made a rectangular grid of the floor of his room.
- Scale used: 1 cm : 1 foot.
- Key points were marked with pins, and corners of objects were joined with thick wool so that Reiaan could feel the positions with his fingers.
🤔 Think about it
This sketch shows only the floor of the room. That is why windows cannot be marked — windows are on the walls, not on the floor. A 2-D map can show only 2 dimensions!
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4The 2-D Cartesian Coordinate System
Take two number lines at right angles to each other. That is all a coordinate system is!
📝 The three basic words
- x-axis — the horizontal line (sleeping line).
- y-axis — the vertical line (standing line).
- Origin (O) — the point where both axes cut each other. Its coordinates are (0, 0).
✔ Sign convention — the golden rule
| Direction from O | Sign |
|---|---|
| Right → | Positive (+) |
| Up ↑ | Positive (+) |
| Left ← | Negative (–) |
| Down ↓ | Negative (–) |
Distances from O are marked in equal units on both the axes.
5Points Lying on the Axes
🔢 Two shortcuts
- Any point on the x-axis looks like (x, 0) → its y-coordinate is always 0.
- Any point on the y-axis looks like (0, y) → its x-coordinate is always 0.
- For P (x, 0): if x is positive → P lies to the right of O; if x is negative → P lies to the left of O.
- For P (0, y): if y is positive → P lies above O; if y is negative → P lies below O.
🔍 Reading the graph above
- B (4.5, 0) — on the x-axis, 4.5 units to the right of O.
- E (–2.9, 0) — on the x-axis, 2.9 units to the left of O.
- H (0, 4) — on the y-axis, 4 units above O.
- G (0, –4.5) — on the y-axis, 4.5 units below O.
✍️ Writing style
While plotting on a graph we usually drop the ‘=’ sign — instead of writing P = (x, y) we simply write P (x, y).
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6The Cartesian Plane & the Four Quadrants
The flat plane in which the two axes lie is called the Cartesian plane, the coordinate plane or the xy-plane.
The two axes cut this plane into four parts called quadrants, numbered anti-clockwise starting from the top-right.
| Quadrant | Position | Sign of (x, y) | Example |
|---|---|---|---|
| I | Top right | (+ , +) | (3, 5) |
| II | Top left | (– , +) | (–5, 3) |
| III | Bottom left | (– , –) | (–4, –2) |
| IV | Bottom right | (+ , –) | (3, –5) |
💡 Easy trick
Start from Quadrant I (top-right) and move anti-clockwise — exactly the way you turn a steering wheel to the left. Signs go: (+,+) → (–,+) → (–,–) → (+,–).
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7Meaning of the Coordinates (x, y)
📝 The most important idea
For a point P (x, y):
- x = perpendicular distance of P from the y-axis, measured along the x-axis. It is called the x-coordinate (or abscissa).
- y = perpendicular distance of P from the x-axis, measured along the y-axis. It is called the y-coordinate (or ordinate).
⚠️ Order matters!
(x, y) is an ordered pair — x is always written first.
- If x ≠ y, then (x, y) ≠ (y, x) e.g. (2, 7) and (7, 2) are two different points.
- (x, y) = (y, x) happens only when x = y, e.g. (4, 4).
Here is the same idea used in real life — Reiaan’s room drawn on a coordinate grid, with the corner of the room taken as the origin:
8Distance Between Two Points
✔ Case 1: The line segment is parallel to an axis
🔢 Simple subtraction
- Points (x1, y) and (x2, y) — same y → distance = | x2 – x1 |
- Points (x, y1) and (x, y2) — same x → distance = | y2 – y1 |
✔ Case 2: The segment is slanting (not parallel to any axis)
Drop a perpendicular and make a right-angled triangle, then use the Baudhāyana–Pythagoras Theorem.
✏️ Worked example
Find the distance between A (3, 4) and D (7, 1).
- Horizontal shift: CD = 7 – 3 = 4
- Vertical shift: AC = 4 – 1 = 3
- By Pythagoras: AD = √4² + 3² = √16 + 9 = √25 = 5 units
✔ The general Distance Formula
⭐ Must-learn formula
The distance between the points (x1, y1) and (x2, y2) is
d = √(x2 – x1)² + (y2 – y1)²
💡 Why signs don’t matter
It makes no difference whether (x2 – x1) and (y2 – y1) come out positive or negative — we are only measuring the shift along each axis, and squaring removes the minus sign anyway. So you may take the points in any order.
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9Reflection in the Axes
What happens when a figure is reflected (mirror image) in an axis? Look at ΔAMD reflected in the y-axis:
🧭 Mirror rules
- Reflection in the y-axis: (x, y) → (–x, y) e.g. A (3, 4) → A′ (–3, 4)
- Reflection in the x-axis: (x, y) → (x, –y)
🔍 What changes, what stays
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- Stays the same: the lengths of all sides, and therefore the size and shape of the triangle.
- Changes: the position (coordinates) of the points — the figure flips over.
10Common Mistakes to Avoid
❌ Don’t do this
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- Writing (y, x) in place of (x, y) — x always comes first.
- Thinking the x-coordinate is the distance from the x-axis. It is actually the distance from the y-axis.
- Mixing up Quadrant II (–, +) with Quadrant IV (+, –).
- Forgetting to square both brackets in the distance formula, or forgetting the square root at the end.
- Writing a distance with a minus sign — distance is never negative.
- Using unequal scales on the two axes while plotting.
11Chapter Summary — Quick Revision
🏆 All of Chapter 1 in 10 lines
- To fix a point in a plane we need two perpendicular lines — one horizontal, one vertical.
- The plane is the Cartesian / coordinate / xy-plane; the lines are the coordinate axes.
- Horizontal line = x-axis, vertical line = y-axis, their meeting point = origin O (0, 0).
- The axes divide the plane into four quadrants.
- x-coordinate = distance from the y-axis; y-coordinate = distance from the x-axis; together they form the coordinates (x, y).
- On the x-axis points look like (x, 0); on the y-axis they look like (0, y).
- Signs by quadrant: I (+,+) II (–,+) III (–,–) IV (+,–).
- (x, y) = (y, x) only if x = y; otherwise they are different points.
- Distance along a horizontal line = |x2 – x1|; along a vertical line = |y2 – y1|.
- Baudhāyana–Pythagoras distance formula: d = √(x2 – x1)² + (y2 – y1)²
✨ Happy Learning — @edugrown ✨