📐 Mathematics | Grade 9 | Part I
Chapter 1: Orienting Yourself
The Use of Coordinates
Master the Cartesian Coordinate System — locate points, understand quadrants, and calculate distances using the Baudhāyana–Pythagoras Theorem.
📚 Ganita Manjari
🗺️ Coordinate Geometry
📏 Distance Formula
🧭 Quadrants
✅ Exam Ready
📋 Contents
- Introduction — What is a Coordinate System?
- Historical Roots — India & the World
- The 2-D Cartesian Coordinate System
- Axes, Origin & Coordinates Explained
- Quadrants — The Four Regions
- Special Points — On the Axes
- Distance Between Two Points
- Reflection & Coordinates
- Chapter Summary
- Important Exam Questions
1. What is a Coordinate System?
A system of coordinates is a structured framework — like the grid lines on a map or graph paper — that lets us use numbers to describe the exact physical location of any point or object in space.
Remember This!
Think of a coordinate system like a city map — you need a Street (x-direction) and an Avenue (y-direction) to find any location precisely.
Think of a coordinate system like a city map — you need a Street (x-direction) and an Avenue (y-direction) to find any location precisely.
Without a coordinate system, we can only say “somewhere to the right” or “a bit upward.” With coordinates, we can say exactly where — using just two numbers!
🏙️
Real-World Coordinate System!The Sindhu-Sarasvatī Civilisation used city streets in perfectly north-south and east-west directions, 10 metres apart — this was a real coordinate system thousands of years ago!
2. Historical Roots of Coordinate Geometry
The idea of coordinates has deep roots in Bhārat — long before René Descartes is commonly credited with inventing coordinate geometry!
| Mathematician / Civilisation | Period | Contribution |
|---|---|---|
| 🏛️ Sindhu-Sarasvatī Civilisation | Thousands of years ago | First systematic grid-based city planning with N-S and E-W streets |
| 📐 Baudhāyana | c. 800 CE | Used East-West and North-South lines for geometry; developed the Baudhāyana–Pythagoras Theorem |
| 🌟 Ujjayinī (Ancient City) | 4th century BCE | Used as the central longitude meridian (zero reference) in ancient geography |
| 🔭 Āryabhaṭa | c. 499 CE | Replaced Greek ‘chords’ with ‘sines’; mapped the sky using Celestial Coordinates |
| 0️⃣ Brahmagupta | c. 628 CE | Formalised zero and negative numbers — making the 4-quadrant Cartesian plane possible |
| 🌍 Al-Bīrūnī | c. 1000 CE | Used Indian trigonometry to calculate city coordinates; perfected the astrolabe |
| 📊 René Descartes | 1637 CE | Formalised that any point in 2D can be defined by two numbers — the Cartesian system |
Did You Know?
Brahmagupta’s work was translated into Arabic as the Sindhind. The Ujjayinī meridian became ‘Arin’ in Arabic maps. Without Brahmagupta’s formalisation of zero and negative numbers, we could not have the four-quadrant Cartesian plane we study today!
Brahmagupta’s work was translated into Arabic as the Sindhind. The Ujjayinī meridian became ‘Arin’ in Arabic maps. Without Brahmagupta’s formalisation of zero and negative numbers, we could not have the four-quadrant Cartesian plane we study today!
3. The 2-D Cartesian Coordinate System
The 2-D Cartesian Coordinate System uses two number lines placed at right angles to each other to locate any point in a two-dimensional plane.
📌 Key Terms You Must Know
x-axisThe horizontal number line. Positive to the right, negative to the left of Origin.
y-axisThe vertical number line. Positive going up, negative going down from Origin.
Origin (O)The point where x-axis and y-axis meet. Its coordinates are always (0, 0).
Coordinate AxesThe x-axis and y-axis together are called the coordinate axes.
Cartesian PlaneThe plane formed by the two axes. Also called the coordinate plane or xy-plane.
Coordinates (x, y)An ordered pair of numbers that gives the exact location of a point in the plane.
Quick Rule for Directions:
- Right of O → Positive x-coordinate
- Left of O → Negative x-coordinate
- Above O → Positive y-coordinate
- Below O → Negative y-coordinate
4. Understanding Coordinates of a Point
The coordinates of a point P are written as P(x, y). Here:
- x-coordinate = perpendicular distance of P from the y-axis, measured along the x-axis.
- y-coordinate = perpendicular distance of P from the x-axis, measured along the y-axis.
Distance Formula for Coordinates: P = (x, y) → x is horizontal, y is vertical from Origin
📌 Reading Coordinates — Examples from the Book
| Point | Coordinates | Location Description |
|---|---|---|
| O | (0, 0) | Origin — where the two axes meet |
| B | (4.5, 0) | On x-axis, 4.5 units to the right of O |
| G | (0, −4.5) | On y-axis, 4.5 units below O |
| H | (0, 4) | On y-axis, 4 units above O |
| E | (−2.9, 0) | On x-axis, 2.9 units to the left of O |
| Q | (−5, 3) | 5 units left, 3 units up from O (Quadrant II) |
| S | (3, −5) | 3 units right, 5 units down from O (Quadrant IV) |
Order Matters!
(x, y) is NOT the same as (y, x) unless x = y. For example, (3, 5) and (5, 3) are completely different points!
(x, y) is NOT the same as (y, x) unless x = y. For example, (3, 5) and (5, 3) are completely different points!
A point P = (x, y) means → go x units along x-axis, then y units along y-axis from Origin
5. The Four Quadrants
The coordinate axes divide the Cartesian plane into four regions called quadrants, numbered I, II, III, and IV in anti-clockwise order starting from the top-right.
| Quadrant | x-coordinate Sign | y-coordinate Sign | Example Point |
|---|---|---|---|
| 🟢 Quadrant I (Top-Right) | Positive (+) | Positive (+) | (3, 4) |
| 🔵 Quadrant II (Top-Left) | Negative (−) | Positive (+) | (−5, 3) |
| 🔴 Quadrant III (Bottom-Left) | Negative (−) | Negative (−) | (−3, −4) |
| 🟠 Quadrant IV (Bottom-Right) | Positive (+) | Negative (−) | (3, −5) |
Memory Trick — Signs in Each Quadrant:
Think of a clock going anti-clockwise: I(+,+) → II(−,+) → III(−,−) → IV(+,−)
Think of a clock going anti-clockwise: I(+,+) → II(−,+) → III(−,−) → IV(+,−)
📌 What Lies on the Axes?
- A point on the x-axis has y-coordinate = 0. Form: (x, 0)
- A point on the y-axis has x-coordinate = 0. Form: (0, y)
- Points on the axes do NOT belong to any quadrant.
- The Origin O = (0, 0) is on both axes.
Common Mistake to Avoid!
Points on the axes (x-axis or y-axis) are NOT part of any quadrant. Only points strictly inside the four regions belong to quadrants.
Points on the axes (x-axis or y-axis) are NOT part of any quadrant. Only points strictly inside the four regions belong to quadrants.
6. Special Points & Important Rules
📌 Points on the x-axis
If P = (x, 0) is on the x-axis:
- If x > 0 → P lies to the right of Origin
- If x < 0 → P lies to the left of Origin
- If x = 0 → P is the Origin itself
📌 Points on the y-axis
If P = (0, y) is on the y-axis:
- If y > 0 → P lies above Origin
- If y < 0 → P lies below Origin
Key Rule: If x = y, then (x, y) = (y, x). But if x ≠ y, then (x, y) ≠ (y, x).
🏠 Story Context: Reiaan’s Room
Real-Life Application in the Book
Shalini used a grid model of Reiaan’s room (1 cm : 1 foot scale) with pins and threads to help her visually impaired brother understand coordinates. The four corners O, A, B, C of the bedroom were at (0,0), (12,0), (12,10), (0,10). This is a beautiful example of how coordinates help map real spaces!
Shalini used a grid model of Reiaan’s room (1 cm : 1 foot scale) with pins and threads to help her visually impaired brother understand coordinates. The four corners O, A, B, C of the bedroom were at (0,0), (12,0), (12,10), (0,10). This is a beautiful example of how coordinates help map real spaces!
📌 Important Observations about Coordinates
A point in any quadranthas specific sign combination for (x, y) as per its quadrant.
Negative coordinatesjust mean the point is in the left or lower region — they are valid and important!
Reading coordinatesAlways read x first (horizontal), then y (vertical). Never swap them!
Writing coordinatesP = (x, y) or P(x, y) are both acceptable notations.
7. Distance Between Two Points
To find the distance between two points in the xy-plane when the line joining them is not parallel to either axis, we use the Baudhāyana–Pythagoras Theorem.
Distance Formula: d = √[ (x₂ − x₁)² + (y₂ − y₁)² ]
Here, (x₁, y₁) and (x₂, y₂) are the two points. The differences (x₂ − x₁) and (y₂ − y₁) can be positive or negative — we square them, so it doesn’t matter!
📌 Step-by-Step: How to Calculate Distance
- Identify the two points: (x₁, y₁) and (x₂, y₂)
- Find the horizontal difference: (x₂ − x₁)
- Find the vertical difference: (y₂ − y₁)
- Square both differences and add them: (x₂−x₁)² + (y₂−y₁)²
- Take the square root: distance = √[ (x₂−x₁)² + (y₂−y₁)² ]
📌 Worked Example (from the Book)
Find the distances between points A(3, 4), D(7, 1), and M(9, 6) — triangle ADM.
Finding AD: A = (3, 4), D = (7, 1)
Horizontal shift (CD) = 7 − 3 = 4
Vertical shift (AC) = 4 − 1 = 3
AD = √(4² + 3²) = √(16 + 9) = √25 = 5 unitsFinding DM: D = (7, 1), M = (9, 6)
Horizontal shift = 9 − 7 = 2
Vertical shift = 6 − 1 = 5
DM = √(2² + 5²) = √(4 + 25) = √29 units
Horizontal shift (CD) = 7 − 3 = 4
Vertical shift (AC) = 4 − 1 = 3
AD = √(4² + 3²) = √(16 + 9) = √25 = 5 unitsFinding DM: D = (7, 1), M = (9, 6)
Horizontal shift = 9 − 7 = 2
Vertical shift = 6 − 1 = 5
DM = √(2² + 5²) = √(4 + 25) = √29 units
Finding MA: M = (9, 6), A = (3, 4)
Horizontal shift = 9 − 3 = 6
Vertical shift = 6 − 4 = 2
MA = √(6² + 2²) = √(36 + 4) = √40 units
📌 Special Cases of Distance
| Case | Points | Formula | Why? |
|---|---|---|---|
| Both on x-axis | (x₁, 0) and (x₂, 0) | |x₂ − x₁| | y-difference is zero |
| Both on y-axis | (0, y₁) and (0, y₂) | |y₂ − y₁| | x-difference is zero |
| Parallel to x-axis | (x₁, y) and (x₂, y) | |x₂ − x₁| | Same y-coordinate |
| Parallel to y-axis | (x, y₁) and (x, y₂) | |y₂ − y₁| | Same x-coordinate |
| General case | (x₁,y₁) and (x₂,y₂) | √[(x₂−x₁)²+(y₂−y₁)²] | Baudhāyana–Pythagoras |
Why do we square the differences?
Squaring removes the negative sign! Whether we go left or right, up or down, we only care about the magnitude (size) of the shift, not its direction. Squaring and then square-rooting gives the absolute distance.
Squaring removes the negative sign! Whether we go left or right, up or down, we only care about the magnitude (size) of the shift, not its direction. Squaring and then square-rooting gives the absolute distance.
8. Reflection & Coordinates
When a figure is reflected (mirrored) in one of the coordinate axes, its shape and size remain the same, but the signs of certain coordinates change.
Reflection in the y-axisPoint (x, y) becomes (−x, y). The x-coordinate changes sign. Example: A(3, 4) → A'(−3, 4)
Reflection in the x-axisPoint (x, y) becomes (x, −y). The y-coordinate changes sign. Example: A(3, 4) → A”(3, −4)
Reflection Preserves Distances!
When a triangle ADM is reflected in either axis to form A’D’M’, all corresponding side lengths remain equal. The shape is congruent — only the position changes. This is confirmed using the distance formula.
When a triangle ADM is reflected in either axis to form A’D’M’, all corresponding side lengths remain equal. The shape is congruent — only the position changes. This is confirmed using the distance formula.
Reflection of triangle ADM in the y-axis:
A(3, 4) → A'(−3, 4)
D(7, 1) → D'(−7, 1)
M(9, 6) → M'(−9, 6)Checking distance D’M’:
D’M’ = √((-2)² + 5²) = √(4 + 25) = √29 units ✓ (same as DM!)
A(3, 4) → A'(−3, 4)
D(7, 1) → D'(−7, 1)
M(9, 6) → M'(−9, 6)Checking distance D’M’:
D’M’ = √((-2)² + 5²) = √(4 + 25) = √29 units ✓ (same as DM!)
Checking distance M’A’:
M’A’ = √((-6)² + 2²) = √(36 + 4) = √40 units ✓ (same as MA!)
9. Midpoint of a Line Segment
The midpoint M of a segment joining S(x₁, y₁) and T(x₂, y₂) has coordinates:
Midpoint M = ( (x₁ + x₂)/2 , (y₁ + y₂)/2 )
In simple words: average the x-coordinates and average the y-coordinates!
📌 Checking Midpoints — Table from the Book
| S | M (given) | T | Is M midpoint? | Reason |
|---|---|---|---|---|
| (−3, 0) | (0, 0) | (3, 0) | ✅ Yes | (−3+3)/2=0, (0+0)/2=0 → (0,0) ✓ |
| (2, 3) | (3, 4) | (4, 5) | ✅ Yes | (2+4)/2=3, (3+5)/2=4 → (3,4) ✓ |
| (0, 0) | (0, 5) | (0, −10) | ❌ No | (0+0)/2=0, (0−10)/2=−5 → (0,−5) ≠ (0,5) |
| (−8, 7) | (0, −2) | (6, −3) | ❌ No | (−8+6)/2=−1, (7−3)/2=2 → (−1,2) ≠ (0,−2) |
Finding Trisection Points (Advanced)
To divide AB into 3 equal parts, find midpoint M of AB first, then find midpoint P of AM, and midpoint Q of MB. This uses the midpoint formula twice!
To divide AB into 3 equal parts, find midpoint M of AB first, then find midpoint P of AM, and midpoint Q of MB. This uses the midpoint formula twice!
10. Chapter Summary at a Glance
Coordinate SystemA framework using two perpendicular number lines to locate any point in 2D space using two numbers (x, y).
x-axis & y-axisHorizontal line = x-axis. Vertical line = y-axis. They intersect at Origin O(0,0).
Four QuadrantsI: (+,+) | II: (−,+) | III: (−,−) | IV: (+,−). Axes themselves don’t belong to any quadrant.
Coordinates of a PointP(x, y): x = horizontal distance from y-axis; y = vertical distance from x-axis.
Distance Formulad = √[(x₂−x₁)² + (y₂−y₁)²]. Derived from the Baudhāyana–Pythagoras Theorem.
Special Axes PointsOn x-axis: (x, 0). On y-axis: (0, y). Origin: (0, 0). Axes points have one coordinate = 0.
Reflection in y-axis(x, y) → (−x, y). Reflection preserves all distances and side lengths.
Midpoint FormulaM = ((x₁+x₂)/2, (y₁+y₂)/2). Average of x-coordinates and average of y-coordinates.
Order of Coordinates(x, y) ≠ (y, x) unless x = y. Always write x first, y second!
Historical NoteBrahmagupta’s zero & negatives → Arabic maps → Europe → Descartes’ formalisation in 1637.
11. Important Exam Questions & Solutions
Q1. What are the coordinates of the origin?
Ans: The coordinates of the origin are (0, 0). It is the point where the x-axis and y-axis intersect.
Q2. In which quadrant does the point (−4, 3) lie? What about (3, −5)?
Ans: (−4, 3) has negative x and positive y → Quadrant II. (3, −5) has positive x and negative y → Quadrant IV.
Q3. Find the distance between points A(3, 4) and D(7, 1).
Ans: d = √[(7−3)² + (1−4)²] = √[16 + 9] = √25 = 5 units
Q4. What is the x-coordinate of any point lying on the y-axis?
Ans: The x-coordinate of any point on the y-axis is always 0. Such points are of the form (0, y).
Q5. If M(−7, 1) is the midpoint of A(3, −4) and B(x, y), find B.
Ans: Using midpoint formula: (3+x)/2 = −7 → x = −17; (−4+y)/2 = 1 → y = 6. So B = (−17, 6).
Q6. Are the points M(−3, −4), A(0, 0) and G(6, 8) on the same straight line?
Ans: Calculate MA = √(9+16) = 5, AG = √(36+64) = 10, MG = √(81+144) = 15. Since MA + AG = 5+10 = 15 = MG, the points are collinear. ✓
Q7. Show that A(1,−8), B(−4,7) and C(−7,−4) lie on a circle centred at the origin.
Ans: OA = √(1+64) = √65; OB = √(16+49) = √65; OC = √(49+16) = √65. All three distances from origin are equal (= √65), so they all lie on a circle of radius √65 centred at O. ✓
Q8. Plot points A(2,1), B(−1,2), C(−2,−1), D(1,−2). Is ABCD a square?
Ans: AB = √(9+1) = √10; BC = √(1+9) = √10; CD = √(9+1) = √10; DA = √(1+9) = √10. All sides equal. Also diagonal AC = √(16+4) = √20 = AB√2. Since all sides are equal and diagonals are equal, ABCD is a square. Area = 10 sq. units.
Q9. A point W has x-coordinate = −5. In which quadrants can a point H on the vertical line through W lie?
Ans: H has x-coordinate = −5. So H = (−5, y). If y > 0 → Quadrant II; if y < 0 → Quadrant III; if y = 0 → on the x-axis (no quadrant). So H can lie in Quadrant II or Quadrant III.
Q10. What would a coordinate system be like without negative numbers?
Ans: Without negative numbers, we would only have one quadrant (Quadrant I) — only the top-right region. We could not locate points to the left of the y-axis or below the x-axis. We could only map one-quarter of the plane, not the entire 2-D plane!
Star (*) Questions are Higher-Order Thinking (HOT) Questions!
Questions marked with * in the textbook (like Q6, Q7, Q8, Q9, Q12, Q13) require deeper reasoning and may appear in exams as advanced/HOT questions. Practice them thoroughly!
Questions marked with * in the textbook (like Q6, Q7, Q8, Q9, Q12, Q13) require deeper reasoning and may appear in exams as advanced/HOT questions. Practice them thoroughly!
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