A complete visual guide — Cartesian plane, Distance Formula, Midpoint Formula & Collinearity. Fully illustrated with step-by-step solutions & 50 practice questions.
📌 Chapter 1 · 8 Marks
Cartesian Plane
★ Distance Formula — New
★ Midpoint Formula — New
Collinearity
6 Worked Examples
50 Practice Qs
Section 1.1
The Cartesian Plane
Named after René Descartes (1596–1650), the Cartesian system gives every point on a flat surface a unique address using two numbers. It is the foundation of coordinate geometry.
🗺️ How the plane is built
Two number lines called axes are drawn at right angles to each other. The horizontal one is the x-axis and the vertical one is the y-axis. They meet at a special point called the origin O(0, 0). Every point on the plane gets a unique address called its coordinates (x, y).
X-axis
↔
Horizontal number line. Positive direction → right, negative → left.
Y-axis
↕
Vertical number line. Positive direction → up, negative → down.
Origin
(0, 0)
Where x-axis and y-axis intersect. Distance from itself is 0.
Abscissa (x)
x
Horizontal distance from the y-axis. The first coordinate.
Ordinate (y)
y
Vertical distance from the x-axis. The second coordinate.
Coordinates
(x, y)
Ordered pair — x always first, y second. Order matters!
Fully labelled Cartesian plane — 4 sample points (one in each quadrant), plus points E on x-axis and F on y-axis
⚠️
Common Error: (3, 2) and (2, 3) are different points! The x-coordinate always comes first. Never swap them.
Section 1.2
The Four Quadrants
The two axes divide the plane into four regions. Learning their sign pattern is essential — many exam questions test just this.
Quadrant
Location
Sign of x
Sign of y
Example
Memory Aid
I
Top-Right
+
+
(4, 3)
Both positive ✓✓
II
Top-Left
−
+
(−3, 2)
Left side, above x-axis
III
Bottom-Left
−
−
(−2, −2)
Both negative ✗✗
IV
Bottom-Right
+
−
(3, −3)
Right side, below x-axis
📌 Special Positions — Points on Axes
• A point on the x-axis has ordinate = 0 → form is (x, 0). Example: (3, 0), (−5, 0)
• A point on the y-axis has abscissa = 0 → form is (0, y). Example: (0, 4), (0, −2)
• The origin (0, 0) lies on both axes. It does NOT belong to any quadrant.
Sign of coordinates in each quadrant — the most frequently tested basic concept
Section 1.3
Plotting Points — Step by Step
Every plotting problem follows the same four-step routine. Master this once and it works for every point.
Illustrated Method
Plot P(−3, 4) — which quadrant does it lie in?
1
Read the coordinates. x = −3 (negative → move LEFT), y = 4 (positive → move UP).
2
Start at origin O(0, 0).
3
Move along x-axis. x = −3 → move 3 units LEFT to reach (−3, 0).
4
Move parallel to y-axis. y = 4 → move 4 units UP from (−3, 0).
5
Mark and label the point. Write P(−3, 4) next to the dot.
P(−3, 4) lies in Quadrant II — because x is negative and y is positive.
📌
CBSE Exam Tip: Always draw dashed lines from the point to both axes and label each point clearly. This earns full method marks even if you miscount slightly.
Section 1.4 NEW 2026-27
Distance Formula
This formula was previously in Class 10 and is now part of Class 9 (2026-27). It finds the straight-line distance between any two points using the Pythagorean Theorem.
🔍 Where does the formula come from?
Consider two points P₁(x₁, y₁) and P₂(x₂, y₂). Draw a right-angled triangle by adding a third point R(x₂, y₁).
The midpoint of segment AB is M(3, −1). If A = (1, 5), find B. [2 Marks]
1
Let B = (x, y). Apply the midpoint formula with M = (3, −1) and A = (1, 5).
2
For x-coordinate:
(1 + x) / 2 = 3
1 + x = 6 → x = 5
3
For y-coordinate:
(5 + y) / 2 = −1
5 + y = −2 → y = −7
B = (5, −7)
⚠️
Common Mistake: Forgetting to divide by 2. The midpoint is the average, not the sum. Always write the ÷ 2 explicitly in your working.
Section 1.6
Collinearity of Three Points
Three points are collinear if they all lie on a single straight line. Use distances to check.
📐 Condition for Collinearity
Three points A, B, C are collinear if and only if: AB + BC = AC
where AC is the largest of the three distances. If this equation holds, B lies between A and C on a line. If it does NOT hold, the three points form a triangle.
Example 5
Show that A(1, 1), B(2, 2), C(3, 3) are collinear. [3 Marks]
1
Find AB:
AB = √[(2−1)²+(2−1)²] = √[1+1] = √2
2
Find BC:
BC = √[(3−2)²+(3−2)²] = √[1+1] = √2
3
Find AC (longest):
AC = √[(3−1)²+(3−1)²] = √[4+4] = √8 = 2√2
4
Check collinearity:
AB + BC = √2 + √2 = 2√2 = AC ✓ 💡 All three points lie on the line y = x.
A, B, C are collinear. They all lie on the line y = x.
NCERT Style · CBSE Pattern
Advanced Worked Examples
These examples combine multiple concepts and mirror the kind of 3–5 mark CBSE exam questions.
Example 6
Identify the type of triangle with vertices A(0,0), B(4,0), C(2, 2√3). [3 Marks]
Conclusion: AB = CD and BC = DA (opposite sides equal). Diagonals AC ≠ BD (not equal). This is a Rhombus (parallelogram with equal opposite sides, unequal diagonals)
Quick Reference
Formula & Fact Sheet
Everything you need to revise in 5 minutes before the exam.
Concept
Formula / Rule
Notes
Distance between two points
d = √[(x₂−x₁)²+(y₂−y₁)²]
Derived from Pythagoras
Distance from origin
OP = √(x²+y²)
Set x₁=0, y₁=0
Midpoint
M = ((x₁+x₂)/2 , (y₁+y₂)/2)
Average of coordinates
Finding unknown endpoint
x₂ = 2Mx − x₁, y₂ = 2My − y₁
Reverse midpoint
Collinearity check
AB + BC = AC
AC must be the longest
Point on x-axis
(x, 0)
y is always 0
Point on y-axis
(0, y)
x is always 0
Quadrant I
(+, +)
Top-right
Quadrant II
(−, +)
Top-left
Quadrant III
(−, −)
Bottom-left
Quadrant IV
(+, −)
Bottom-right
Smart Study
10 Study Tips
Based on common CBSE exam patterns and the mistakes most students make.
1
x always comes first
(3,2) and (2,3) are different points. The first number is always horizontal (x). Never swap them.
2
Memorise quadrant signs
Q1:(+,+) Q2:(−,+) Q3:(−,−) Q4:(+,−). Think of a compass: North-East, North-West, South-West, South-East.
3
Negative² is always positive
In distance formula, (x₂−x₁) can be negative. But squaring it always gives positive. Never worry about sign inside the square.