Class 9 Mathematics – Coordinate Geometry | Interactive Visual Revision Notes

📝 Section 1 — Classroom Notes

🌎 Real-World Hook: Have you ever used Google Maps to find a place? When you search for a location, the app uses two numbers — latitude and longitude — to pinpoint the exact spot on Earth. A city cinema might be at (latitude 28.6, longitude 77.2). This is coordinate geometry in action! In maths, we use the same idea on a grid called the coordinate plane, and every point has its own unique address written as (x, y). Once you learn this, you understand how GPS, video games, computer screens, and maps all work!

📍 What is a Coordinate System?

A coordinate system helps us find places.
It uses numbers to show where things are.
Think of a map with grid lines.
The grid has a horizontal line (left-right).
The grid has a vertical line (up-down).
These two lines cross each other.
We use two numbers to find any point.

➡️ The X-axis and Y-axis

The x-axis goes left and right (horizontal).
The y-axis goes up and down (vertical).
They make a + shape on paper.
Numbers go in both directions on each line.
Right of center = positive numbers.
Left of center = negative numbers.
Up from center = positive numbers.
Down from center = negative numbers.

🎯 The Origin

The origin is where both lines cross.
We call it O.
Its coordinates are (0, 0).
It is the starting point for counting.
Everything is measured from here.

📍 Coordinates of a Point

Every point has two numbers.
We write them as (x, y).
The first number (x) = distance from y-axis.
The second number (y) = distance from x-axis.
Example: (4, 6) means go 4 right, then 6 up.
Example: (−3, 5) means go 3 left, then 5 up.
x comes first, then y. Order matters!
(4, 6) is NOT the same as (6, 4).

🔳 The Four Quadrants

The two axes divide the plane into 4 parts.
Each part is called a quadrant.
Quadrant I (top right): x is +, y is +
Quadrant II (top left): x is −, y is +
Quadrant III (bottom left): x is −, y is −
Quadrant IV (bottom right): x is +, y is −

📏 Points on the Axes

A point on the x-axis has y = 0.
So it looks like (x, 0).
Example: (6, 0) is on the x-axis.
A point on the y-axis has x = 0.
So it looks like (0, y).
Example: (0, 4) is on the y-axis.

📐 Ordered Pair Rule

The pair (x, y) is called an ordered pair.
The order of the numbers matters.
(2, 7) and (7, 2) are different points.
Always write x first, then y.
Think: x = across, y = up.

📏 Plotting a Point

Start at the origin O (0, 0).
Look at the first number (x).
Move right if x is positive.
Move left if x is negative.
Then look at the second number (y).
Move up if y is positive.
Move down if y is negative.
Mark the spot with a dot.

📐 Distance Between Two Points

To find how far apart two points are:
We use a special formula.
It comes from the Pythagoras theorem.
Take the difference of x values.
Take the difference of y values.
Square both differences.
Add the two squares.
Take the square root of the sum.

d = √[ (x₂ − x₁)² + (y₂ − y₁)² ]

💡 Why Coordinate Geometry Matters

Maps and GPS use coordinates.
Computer screens show pixels by (x, y).
Games use coordinates for character positions.
Engineers plan buildings on a grid.
It connects algebra and geometry.

🎨 Section 2 — Easy Illustrations

Picture 1: The Coordinate Plane

The coordinate plane with x-axis (red) and y-axis (blue) crossing at origin O.

Picture 2: How to Plot a Point

To plot R(4, 3): go 4 right, then 3 up from origin.

Picture 3: Quadrant Sign Chart

The 4 quadrants and their sign rules for (x, y).

Picture 4: Finding the Distance (Animated)

Click "Play" to see how the distance formula works step by step.

Picture 5: Points in All Four Quadrants

One point in each quadrant showing the sign rules.

✏️ Section 3 — Worked Examples

Example 1: How to Plot a Point on a Graph

Question: Plot the point M (5, 4) on the graph.

1 Draw two lines that cross — the x-axis and y-axis.

2 Start at the origin O (0, 0).

3 Look at the first number: 5. Move 5 steps right.

4 Look at the second number: 4. Move 4 steps up.

5 Put a dot where you stopped. Write "M (5, 4)" next to it.

Answer: Point M is at (5, 4) — 5 right and 4 up from the origin.

Example 2: Which Quadrant is the Point In?

Question: In which quadrant is the point (−6, 2)?

1 Look at x: it is −6. That is negative.

2 Look at y: it is 2. That is positive.

3 Check: x is negative, y is positive = Quadrant II.

Answer: Point (−6, 2) is in Quadrant II (top-left area).

Example 3: Find the Distance Between Two Points

Question: Find the distance between R (1, 1) and S (9, 7).

1 x difference: 9 − 1 = 8.

2 y difference: 7 − 1 = 6.

3 Square both: 8² = 64 and 6² = 36.

4 Add: 64 + 36 = 100.

5 Square root of 100 = 10.

Answer: Distance = 10 units.

d = √[(x₂−x₁)² + (y₂−y₁)²]

Topic	What it Means	Golden Rule
Coordinate System	Grid to find points	Uses (x, y)
X-axis	Horizontal line	Goes left-right
Y-axis	Vertical line	Goes up-down
Origin O	Where axes cross	O = (0, 0)
Ordered Pair	Address of a point	(x, y) x first!
Quadrant I	Top-right	(+, +)
Quadrant II	Top-left	(−, +)
Quadrant III	Bottom-left	(−, −)
Quadrant IV	Bottom-right	(+, −)
On x-axis	y is always 0	(x, 0)
On y-axis	x is always 0	(0, y)
Distance	Gap between 2 points	d = √[(Δx)²+(Δy)²]

📄 Section 5 — Take-Home Handout

📍 A. Coordinate System

A coordinate system uses numbers to find points.
It has two number lines that cross each other.
We use two numbers to describe any spot.

Example:

1 Point R is at (4, 3).

2 That means: go 4 right, then 3 up.

➡️ B. X-axis and Y-axis

X-axis = the horizontal line (left-right).
Y-axis = the vertical line (up-down).
Right and Up = positive (+).
Left and Down = negative (−).

🎯 C. The Origin and Coordinates

The origin is where both axes meet: (0, 0).
Every point has an address: (x, y).
Order matters! (4, 6) is not (6, 4).

🔳 D. Four Quadrants

Q-I (top-right): (+, +)
Q-II (top-left): (−, +)
Q-III (bottom-left): (−, −)
Q-IV (bottom-right): (+, −)

Practice: Which quadrant is (−3, 5) in? __________

📐 E. Distance Formula Practice

Find the distance between A(2, 3) and B(6, 6).

Show your work here

✅ Detailed Solutions — Take-Home Handout

✓ D. Practice Answer: Which quadrant is (−3, 5) in?

1 Look at x: x = −3. This is negative.

2 Look at y: y = 5. This is positive.

3 Check the rule: x negative + y positive = Quadrant II (top-left area).

🏆 Answer: (−3, 5) is in Quadrant II.

✓ E. Distance Formula Practice: Distance between A(2, 3) and B(6, 6)

1 Write down the formula: d = √[(x₂ − x₁)² + (y₂ − y₁)²]

2 Identify: x₁ = 2, y₁ = 3, x₂ = 6, y₂ = 6.

3 Find x difference: x₂ − x₁ = 6 − 2 = 4.

4 Find y difference: y₂ − y₁ = 6 − 3 = 3.

5 Square both: 4² = 16 and 3² = 9.

6 Add the squares: 16 + 9 = 25.

7 Square root: √25 = 5.

🏆 Answer: Distance between A(2, 3) and B(6, 6) = 5 units.

💡 Quick check: 4² + 3² = 16 + 9 = 25. This is a 3-4-5 Pythagorean triple!

🃏 Section 6 — Revision Flashcards

👉

📍

What is a Coordinate System?

A grid that uses two number lines to find where points are.

It uses numbers called coordinates.

➡️

What is the X-axis and Y-axis?

X-axis = horizontal (left-right)

Y-axis = vertical (up-down)

They cross at the origin.

🎯

What is the Origin?

Where x-axis and y-axis cross.

O = (0, 0)

📍

What do coordinates (x, y) mean?

x = how far right (+) or left (−)

y = how far up (+) or down (−)

x comes first! (4,6) ≠ (6,4)

🔳

What are the signs in each Quadrant?

Q-I: (+, +) Q-II: (−, +)

Q-III: (−, −) Q-IV: (+, −)

📏

What do points on the axes look like?

On x-axis: (x, 0)

On y-axis: (0, y)

The other number is always 0.

📐

How to find the distance between two points?

1. Find x difference

2. Find y difference

3. Square both, add, square root

d = √[(Δx)² + (Δy)²]

📝

Why does order matter in (x, y)?

(4, 6) and (6, 4) are different points!

Always write x first.

📍 Class 9 Mathematics — Coordinate Geometry Interactive Visual Revision Notes Aligned with the Class 9 Mathematics syllabus. Independently created study material. Not affiliated with or endorsed by NCERT.

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