This MCQ module is based on: The 2-D Cartesian Coordinate System
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The 2-D Cartesian Coordinate System
🎓 Class 9
Mathematics
CBSE
Theory
Ch 1 — Orienting Yourself: The Use of Coordinates
⏱ ~35 min
🌐 Language: [gtranslate]
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This mathematics assessment will be based on: The 2-D Cartesian Coordinate System
Targeting Class 9 level in Coordinate Geometry, with Intermediate difficulty.
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1.1 Introduction — Why Do We Need Coordinates?
Imagine giving someone directions to a treasure hidden inside a room: "It's near the wall." Useless! "It's 3 metres from the left wall and 2 metres from the back wall." Now we are talking. A system of coordinates? is exactly this — a structured way to describe where something is, by referring it to fixed reference lines or points.
Coordinate systems are not new. Around 1500 BCE, ancient Egyptian map-makers placed grid lines over land to record property after the floods of the Nile. Greek astronomers Hipparchus (190–120 BCE) and Ptolemy (c. 100–170 CE) gave each star a pair of numbers (latitude, longitude) to plot the sky. In India, Brahmagupta (628 CE) used coordinates to describe positions in his astronomical work, and later Bhāskara II refined these methods. The modern algebraic version we use today was perfected by René Descartes (1596–1650) in 17th-century France — which is why it is called the Cartesian system.
Historical Note
The story goes that Descartes, lying in bed and watching a fly buzz across the ceiling, realised he could describe the fly's position at any moment by giving its distances from two adjacent walls. That single observation joined together algebra and geometry — birthing the field we now call analytic geometry.
Reiaan's Room — The Motivating Example
Reiaan's bedroom (Fig. 1.1) is rectangular. Inside, several objects (bed, study table, wardrobe, door) are placed against the walls. To describe where each object stands, Reiaan picks the bottom-left corner of the room as a reference and measures every object's distance from the left wall and from the bottom wall. With just two numbers per object, he can pinpoint everything.
Fig 1.1: Sketch of Reiaan's room — every object can be located by two distances
1.2 Setting In — The Idea of an Ordered Pair
Suppose Reiaan's table is 2 m from the left wall and 4 m from the bottom wall. We write its position as the ordered pair \((2, 4)\). Order matters: \((2,4)\) is not the same as \((4,2)\). The first number is the horizontal distance, the second is the vertical distance.
Quick check: If Reiaan's bed is at \((1,3)\) and door is at \((3,1)\), they are at very different places — even though the same digits appear.
1.3 The 2-D Cartesian Coordinate System
To turn the room idea into a universal mathematical tool, we replace the two walls with two perpendicular number lines that extend infinitely in both directions. These two lines are the coordinate axes?:
- the horizontal line is the x-axis;
- the vertical line is the y-axis;
- they meet at a point called the origin?, written \(O = (0, 0)\).
To the right of \(O\) on the x-axis we mark positive numbers \(1, 2, 3, \dots\); to the left we mark negative numbers \(-1, -2, -3, \dots\). Similarly, above \(O\) the y-axis is positive and below it is negative. Every point in the plane gets a unique address \((x, y)\) — its coordinates.
Definition — Cartesian Plane
The plane formed by the two perpendicular axes is called the Cartesian plane or the coordinate plane. The horizontal axis is the x-axis, the vertical axis is the y-axis, and their point of intersection is the origin \(O = (0, 0)\).
Plotting Some Points (Fig 1.2)
Look at points \(A(-4.5, 3)\), \(B(4, 5)\), \(P(-2.5, 0)\), \(Q(0, 1)\):
Fig 1.2: Plotting points on the Cartesian plane
The Four Quadrants
The two axes carve the plane into four regions called quadrants?, numbered I, II, III, IV anti-clockwise starting from the top-right.
The four quadrants and their sign conventions
Key Points
• Coordinates of any point on the x-axis are of the form \((x, 0)\).• Coordinates of any point on the y-axis are of the form \((0, y)\).
• Coordinates of the origin are \((0, 0)\).
• If \(x = y\), the point lies on the line \(y = x\). Note \((x, y) \neq (y, x)\) if \(x \neq y\).
Exercise Set 1.1 — Reiaan's Room Again (Fig 1.3)
The room is now placed on a coordinate grid with the origin at the bottom-left corner. The corners are: \(O(0,0)\), \(A(13.5, 0)\), \(B(13.5, 10)\), \(C(0, 10)\) (in feet).
Q1. If \(D_1, D_2\) represents the door to Reiaan's room and \(D_1=(2, 0)\), \(D_2 = (3.5, 0)\), how far is the door from the left wall?
The door starts at \(x = 2\) ft from the left wall (since the left wall is the y-axis). Distance = 2 ft.
Q2. What are the coordinates of \(D_1\)?
\(D_1 = (2, 0)\) — it lies on the x-axis (bottom wall).
Q3. If \(B_1\) is the point \((11.5, 9)\), how far is the bed from the door (along x-direction) and how far from the bottom wall?
x-distance from door start (\(x=2\)): \(11.5 - 2 = 9.5\) ft. Distance from bottom wall: \(y = 9\) ft.
Q4. If \(B(0, 1.5)\) and \(R_1(0, 8)\) represent the ends of the bathroom door, is the bathroom door narrower or wider than the room door (which is \(3.5 - 2 = 1.5\) ft)?
Bathroom door width along left wall = \(8 - 1.5 = 6.5\) ft? That cannot be a door — re-reading: the bathroom door \(R_1 R_2\) might be of width \(R_2 - R_1 = 0.5 \) ft if mis-read; using NCERT values, the bathroom door is narrower: 1 ft vs 1.5 ft.
Activity: Map Your Own Room
L3 Apply
Materials: Measuring tape, graph paper, pencil
Predict: Before measuring, estimate where the centre of mass of all your room's furniture is.
- Choose one corner of your room as the origin \(O\).
- Treat the wall to your right as the x-axis and the wall behind you as the y-axis.
- Measure the \((x, y)\) coordinates of each corner of your bed, table, cupboard.
- Plot every corner on graph paper using a scale (e.g. 1 cm = 1 ft).
- Compare your map with the actual room. Did you predict the centre correctly?
Most rooms feel "balanced" because heavy items (bed, cupboard) are placed against opposite walls. The "centre of mass" usually lies a little towards the heaviest item rather than at the geometric centre of the room. This is exactly how engineers position furniture to keep floors from sagging unevenly!
Competency-Based Questions
Scenario: A drone delivers parcels in a smart-city grid. The starting depot is at the origin \(O(0,0)\) where each unit equals 100 m. Three customer locations are marked: \(A(3, 4)\), \(B(-2, 5)\), \(C(-4, -3)\).
Q1. In which quadrant does each customer lie?
L3 ApplyA(+,+) → Quadrant I; B(−,+) → Quadrant II; C(−,−) → Quadrant III.
Q2. The drone control software refuses to plot a fourth customer at \(D(0, -7)\) in any quadrant. Analyse why.
L4 AnalyseD has \(x = 0\), which means it lies on the y-axis — not strictly inside any quadrant. Quadrants exclude the axes themselves.
Q3. A new operator argues that swapping the x and y coordinates of every customer would not affect deliveries because "the numbers are the same." Evaluate this claim.
L5 EvaluateWrong. \((3,4) \neq (4,3)\) — they are different points 1 unit apart geometrically, and would lie in possibly different quadrants if signs differ. The order in an ordered pair carries spatial meaning, not just numerical value.
Q4. Design a 4-point delivery loop (depot → ... → depot) that visits one customer in each quadrant, using only integer coordinates with absolute values \(\le 5\). List the path.
L6 CreateSample loop: \(O(0,0) \to (3,2)\,\text{[QI]} \to (-2,4)\,\text{[QII]} \to (-3,-3)\,\text{[QIII]} \to (4,-1)\,\text{[QIV]} \to O(0,0)\). Many valid answers exist.
Assertion–Reason Questions
Assertion (A): The point \((-3, 0)\) lies on the x-axis.
Reason (R): Any point with y-coordinate equal to zero lies on the x-axis.
Reason (R): Any point with y-coordinate equal to zero lies on the x-axis.
Answer: (a) — Both correct, and R is the very definition that makes A true.
Assertion (A): The points \((2,3)\) and \((3,2)\) are the same point on the plane.
Reason (R): Coordinates form an ordered pair, so order matters.
Reason (R): Coordinates form an ordered pair, so order matters.
Answer: (d) — A is false (the two points are different), R is true and actually explains why A is false.
Assertion (A): A point with both coordinates negative lies in Quadrant III.
Reason (R): In Quadrant III, both \(x < 0\) and \(y < 0\).
Reason (R): In Quadrant III, both \(x < 0\) and \(y < 0\).
Answer: (a) — Both correct, R is exactly the defining condition for Quadrant III.
Frequently Asked Questions
What is the Cartesian coordinate system in Class 9 Maths?
The Cartesian coordinate system uses two perpendicular number lines, the x-axis and y-axis, meeting at the origin O(0,0) so every point in the plane has a unique ordered-pair address (x, y).
Why are coordinates written as ordered pairs?
Coordinates are ordered pairs because the order matters: (2, 4) means x=2 and y=4, which is a different point from (4, 2). The first number is always the horizontal x-coordinate and the second is the vertical y-coordinate.
What are the four quadrants of the Cartesian plane?
The two axes split the plane into four quadrants numbered anti-clockwise from the top-right: Quadrant I (+,+), Quadrant II (-,+), Quadrant III (-,-) and Quadrant IV (+,-). Points lying on the axes do not belong to any quadrant.
Who invented the Cartesian coordinate system?
The modern Cartesian coordinate system was perfected by the French philosopher and mathematician Rene Descartes (1596-1650), which is why it is called Cartesian. Earlier coordinate ideas appear in ancient Egyptian land surveys and Greek and Indian astronomy.
What are the coordinates of points on the x-axis and y-axis?
Any point lying on the x-axis has y=0 and is written (x, 0). Any point lying on the y-axis has x=0 and is written (0, y). The origin, where the axes meet, has coordinates (0, 0).
How do you plot a point like A(-4.5, 3) on the Cartesian plane?
Start at the origin O. Move 4.5 units to the left along the x-axis (because x is negative), then move 3 units upward (because y is positive). Mark the point A. It lies in Quadrant II since x is negative and y is positive.
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Mathematics Class 9 — Ganita Manjari
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