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Orienting Yourself: The Use of Coordinates

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Chapter Notes

Overview
This chapter introduces the Cartesian coordinate system, a framework that uses two perpendicular axes (x-axis and y-axis) to locate any point in a two-dimensional plane using an ordered pair of numbers (x, y). It traces the historical development of coordinate geometry from the Sindhu-Sarasvatī Civilisation and Indian mathematicians like Baudhāyana, Brahmagupta, and Āryabhaṭa to Descartes. The one big idea a student must walk away with is that every point in a plane can be uniquely identified by its distances from two fixed perpendicular lines, and this system bridges algebra and geometry.
Key Concepts (12)
Coordinate axes
Two perpendicular lines, one horizontal (x-axis) and one vertical (y-axis), used to locate points in a two-dimensional plane.
e.g. In a classroom, the floor tiles form a grid; the lines along the length and width of the room are like the coordinate axes.
Origin
The point of intersection of the x-axis and y-axis, denoted by O, with coordinates (0, 0).
e.g. The centre of a city where the main North-South and East-West roads cross is like the origin.
x-axis
The horizontal line in the coordinate plane; distances to the right of the origin are positive and to the left are negative.
e.g. On a graph, the line running left to right through the origin is the x-axis.
y-axis
The vertical line in the coordinate plane; distances above the origin are positive and below are negative.
e.g. On a graph, the line running up and down through the origin is the y-axis.
Quadrants
The four parts into which the coordinate axes divide the plane, numbered I, II, III, and IV in anticlockwise order starting from the top-right.
e.g. The top-right part of a graph paper is Quadrant I.
Coordinates of a point
An ordered pair (x, y) where x is the perpendicular distance of the point from the y-axis (x-coordinate) and y is the perpendicular distance from the x-axis (y-coordinate).
e.g. Point P(3, 4) means 3 units to the right of the y-axis and 4 units above the x-axis.
Cartesian plane
The plane in which the coordinate axes are situated, also called the coordinate plane or xy-plane.
e.g. The entire sheet of graph paper with the axes drawn is the Cartesian plane.
x-coordinate
The distance of a point from the y-axis, measured along the x-axis; also called the abscissa.
e.g. In point (5, -2), the x-coordinate is 5.
y-coordinate
The distance of a point from the x-axis, measured along the y-axis; also called the ordinate.
e.g. In point (5, -2), the y-coordinate is -2.
Baudhāyana–Pythagoras Theorem
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.
e.g. For a triangle with legs 3 and 4, the hypotenuse is 5 units.
Origin as zero
The origin (0, 0) represents the zero point on both axes, and negative axes represent values less than zero, formalised by Brahmagupta.
e.g. Without zero and negative numbers, the four-quadrant Cartesian plane would be impossible.
Reflection in axes
When a point is reflected across the y-axis or x-axis, its coordinates change sign accordingly, but distances are preserved.
e.g. Point A(3, 4) reflected in the y-axis becomes A'(-3, 4).
Formulas & Equations (6)
Distance = √[(x₂ - x₁)² + (y₂ - y₁)²]
Reads as: Distance equals the square root of (x two minus x one) squared plus (y two minus y one) squared
Variables: (x₁, y₁) and (x₂, y₂) are the coordinates of the two points; all distances are in the same unit (e.g., cm, feet)
Use when: When finding the straight-line distance between any two points in the Cartesian plane, especially when the line is not parallel to an axis
⚠ Students forget to square the differences or add before taking the square root, or they subtract in the wrong order (e.g., x₁ - x₂ instead of x₂ - x₁) — the order does not matter because squaring removes the sign.
Distance between points (x₁, y) and (x₂, y) = |x₂ - x₁|
Reads as: Distance equals the absolute value of x two minus x one
Variables: x₁ and x₂ are the x-coordinates of two points sharing the same y-coordinate
Use when: When two points lie on a horizontal line (same y-coordinate), e.g., finding the width of a door or table
⚠ Students forget to take the absolute value and write a negative distance.
Distance between points (x, y₁) and (x, y₂) = |y₂ - y₁|
Reads as: Distance equals the absolute value of y two minus y one
Variables: y₁ and y₂ are the y-coordinates of two points sharing the same x-coordinate
Use when: When two points lie on a vertical line (same x-coordinate), e.g., finding the height of a room or table
⚠ Students confuse this with the horizontal distance formula.
Coordinates of a point on x-axis: (x, 0)
Reads as: The coordinates of a point on the x-axis are x comma zero
Variables: x is any real number; y is always 0
Use when: Identifying or plotting points that lie exactly on the x-axis
⚠ Students write (0, x) instead of (x, 0).
Coordinates of a point on y-axis: (0, y)
Reads as: The coordinates of a point on the y-axis are zero comma y
Variables: y is any real number; x is always 0
Use when: Identifying or plotting points that lie exactly on the y-axis
⚠ Students write (y, 0) instead of (0, y).
Coordinates of origin: (0, 0)
Reads as: The coordinates of the origin are zero comma zero
Variables: Both x and y are zero
Use when: Always the reference point for all coordinate measurements
⚠ Students sometimes write (0) or forget it is an ordered pair.
How It Works
Locating a point using coordinates
  1. Step 1: Identify the x-coordinate — the perpendicular distance from the y-axis.
  2. Step 2: Move that many units along the x-axis from the origin (right if positive, left if negative).
  3. Step 3: Identify the y-coordinate — the perpendicular distance from the x-axis.
  4. Step 4: From the point on the x-axis, move vertically that many units (up if positive, down if negative).
  5. Step 5: Mark the final point and label it with its ordered pair (x, y).
💡 Always move along the x-axis first, then the y-axis. The order (x, y) is fixed — swapping them gives a different point.
Finding distance between two points using Baudhāyana–Pythagoras Theorem
  1. Step 1: Identify the coordinates of the two points: (x₁, y₁) and (x₂, y₂).
  2. Step 2: Find the horizontal shift: x₂ - x₁ (the difference in x-coordinates).
  3. Step 3: Find the vertical shift: y₂ - y₁ (the difference in y-coordinates).
  4. Step 4: Square both differences.
  5. Step 5: Add the squares.
  6. Step 6: Take the square root of the sum to get the distance.
💡 The formula works even if the differences are negative because squaring removes the sign. Do not forget the square root at the end.
Determining the quadrant of a point
  1. Step 1: Look at the sign of the x-coordinate (positive or negative).
  2. Step 2: Look at the sign of the y-coordinate (positive or negative).
  3. Step 3: If both are positive → Quadrant I.
  4. Step 4: If x is negative, y is positive → Quadrant II.
  5. Step 5: If both are negative → Quadrant III.
  6. Step 6: If x is positive, y is negative → Quadrant IV.
💡 Points on the axes (where x = 0 or y = 0) do not lie in any quadrant. The origin is the intersection of axes.
Reflecting a point across the y-axis
  1. Step 1: Take the original point (x, y).
  2. Step 2: Change the sign of the x-coordinate only.
  3. Step 3: The reflected point is (-x, y).
  4. Step 4: The y-coordinate remains unchanged.
💡 Reflection across the y-axis changes only the x-coordinate sign. Reflection across the x-axis changes only the y-coordinate sign.
Exam Traps — Don't Make These Mistakes
⚠ Confusing the order of coordinates
✗ They write (y, x) instead of (x, y), e.g., for point (3, 4) they write (4, 3).
→ In an ordered pair, the first number is always the x-coordinate (distance from y-axis) and the second is the y-coordinate (distance from x-axis). Swapping them gives a completely different location.
✓ Always write the x-coordinate first, then the y-coordinate: (x, y).
⚠ Forgetting the absolute value for distances on axes
✗ They write distance = x₂ - x₁ without absolute value, getting a negative number.
→ Distance is always positive. The formula |x₂ - x₁| ensures a non-negative result regardless of subtraction order.
✓ Write distance = |x₂ - x₁| or |y₂ - y₁| as applicable.
⚠ Misidentifying quadrant signs
✗ They say Quadrant II has (+, -) or Quadrant IV has (-, +).
→ Quadrant II is (-, +) and Quadrant IV is (+, -). The signs are specific to each quadrant.
✓ Quadrant I: (+, +); Quadrant II: (-, +); Quadrant III: (-, -); Quadrant IV: (+, -).
⚠ Thinking (x, y) = (y, x) always
✗ They claim (3, 5) is the same as (5, 3).
→ (x, y) = (y, x) only if x = y. If x ≠ y, the two ordered pairs represent different points.
✓ State: (x, y) = (y, x) if and only if x = y.
⚠ Omitting the square root in distance formula
✗ They write distance = (x₂ - x₁)² + (y₂ - y₁)² without the square root.
→ The Baudhāyana–Pythagoras Theorem gives the square of the hypotenuse; you must take the square root to get the actual distance.
✓ Always write distance = √[(x₂ - x₁)² + (y₂ - y₁)²].
⚠ Confusing x-axis and y-axis points
✗ They say a point on the x-axis has coordinates (0, y).
→ A point on the x-axis has y = 0, so its coordinates are (x, 0). A point on the y-axis has x = 0, so its coordinates are (0, y).
✓ Point on x-axis: (x, 0). Point on y-axis: (0, y).
Compare & Contrast
Definition and measurement
x-coordinate
y-coordinate
x-coordinate is the perpendicular distance from the y-axis (measured horizontally); y-coordinate is the perpendicular distance from the x-axis (measured vertically).
Coordinate form
Point on x-axis
Point on y-axis
A point on the x-axis has coordinates (x, 0) — y is always zero. A point on the y-axis has coordinates (0, y) — x is always zero.
Signs of coordinates
Quadrant I
Quadrant III
Quadrant I: both x and y are positive (+, +). Quadrant III: both x and y are negative (-, -).
Signs of coordinates
Quadrant II
Quadrant IV
Quadrant II: x is negative, y is positive (-, +). Quadrant IV: x is positive, y is negative (+, -).
Exam Focus
Marks pattern: 2 MCQs (1 mark each) on quadrant identification and coordinate writing, 1 Short Answer (2-3 marks) on distance formula, 1 Long Answer (4-5 marks) on plotting points and finding distances or verifying collinearity
Most asked: Writing coordinates of a point from a graph · Identifying the quadrant of a given point · Finding distance between two points using the distance formula · Properties of points on axes (x, 0) and (0, y) · Condition for (x, y) = (y, x)
Don't skip: Distance formula application — finding distance between two points or verifying if points are collinear; Plotting points and identifying shapes (square, triangle) with area calculation
One-Line Revision (read aloud before the exam)
  1. The coordinate axes are two perpendicular lines — the horizontal x-axis and the vertical y-axis — intersecting at the origin (0, 0).
  2. The x-coordinate is the perpendicular distance of a point from the y-axis; the y-coordinate is the perpendicular distance from the x-axis.
  3. Coordinates are always written as an ordered pair (x, y) — x first, then y.
  4. A point on the x-axis has coordinates (x, 0); a point on the y-axis has coordinates (0, y).
  5. The origin has coordinates (0, 0).
  6. Quadrant I: (+, +); Quadrant II: (-, +); Quadrant III: (-, -); Quadrant IV: (+, -).
  7. The distance between points (x₁, y₁) and (x₂, y₂) is √[(x₂ - x₁)² + (y₂ - y₁)²].
  8. For points on a horizontal line (same y), distance = |x₂ - x₁|.
  9. For points on a vertical line (same x), distance = |y₂ - y₁|.
  10. (x, y) = (y, x) only if x = y; otherwise they are different points.
  11. Reflection across the y-axis changes the sign of x only; reflection across the x-axis changes the sign of y only.
  12. The Cartesian plane is divided into four quadrants by the axes.
  13. Without zero and negative numbers (Brahmagupta's contribution), the four-quadrant system would be impossible.
  14. The Baudhāyana–Pythagoras Theorem is used to derive the distance formula.
  15. When plotting, always move along the x-axis first, then the y-axis.

MCQs — 10 Questions with Answers

Q1. What are the coordinates of the origin in a coordinate system?

  • A. (0, 0) ✓
  • B. (1, 1)
  • C. (0, 1)
  • D. (1, 0)

Answer: A — The origin is defined as the point where the x-axis and y-axis intersect, which is always at (0, 0).

Q2. A point P lies on the x-axis at a distance of 5 units to the left of the origin. What are the coordinates of P?

  • A. (5, 0)
  • B. (−5, 0) ✓
  • C. (0, 5)
  • D. (0, −5)

Answer: B — Points to the left of the origin have negative x-coordinates; since P is on the x-axis, its y-coordinate is zero, giving (−5, 0).

Q3. Which of the following points lies on the y-axis?

  • A. (3, 5)
  • B. (0, −7) ✓
  • C. (4, 0)
  • D. (2, 2)

Answer: B — Points on the y-axis have x-coordinate equal to zero; only (0, −7) satisfies this condition.

Q4. If a point Q has coordinates (−3.5, 2.5), which of the following statements is correct?

  • A. Q is 3.5 units above the origin and 2.5 units to the right
  • B. Q is 3.5 units to the left of the origin and 2.5 units above ✓
  • C. Q is 3.5 units below the origin and 2.5 units to the left
  • D. Q is 3.5 units to the right of the origin and 2.5 units below

Answer: B — The x-coordinate (−3.5) indicates 3.5 units to the left; the y-coordinate (2.5) indicates 2.5 units above the origin.

Q5. Ramesh marks a point on a floor plan using coordinates (0, 6). Which of the following best describes this point's location?

  • A. 6 units to the right of the origin on the floor
  • B. 6 units above the origin, directly on a vertical line through the origin ✓
  • C. 6 units below the origin on the x-axis
  • D. At the origin itself

Answer: B — When x-coordinate is zero, the point lies on the y-axis; the positive y-coordinate (6) means 6 units above the origin.

Q6. Which of the following is NOT correct about the coordinate system?

  • A. The x-axis is horizontal and the y-axis is vertical
  • B. A point (x, y) has its first coordinate as distance from the y-axis
  • C. The origin has coordinates (0, 1) ✓
  • D. Distances to the right of the origin are positive on the x-axis

Answer: C — The origin is always at (0, 0), not (0, 1); this is the fundamental definition of the coordinate system.

Q7. A door in a room extends from point D₁(5, 0) to point R₁(11.5, 0). How wide is the door?

  • A. 5 units
  • B. 6.5 units ✓
  • C. 11.5 units
  • D. 16.5 units

Answer: B — Both points are on the x-axis; the door width is the difference in x-coordinates: 11.5 − 5 = 6.5 units.

Q8. Two bathroom door ends are at B₁(0, 1.5) and B₂(0, 4). How tall is this bathroom door opening?

  • A. 1.5 units
  • B. 2.5 units ✓
  • C. 4 units
  • D. 5.5 units

Answer: B — Both points are on the y-axis; the door height is the difference in y-coordinates: 4 − 1.5 = 2.5 units.

Q9. In the Sindhu-Sarasvatī Civilization, merchants used a grid system to locate shops. This ancient practice is most directly related to which modern concept?

  • A. The invention of the calculator
  • B. The development of coordinate geometry and location mapping ✓
  • C. The creation of the printing press
  • D. The discovery of electricity

Answer: B — The grid system with North-South and East-West lines used by ancient merchants is the foundational principle of coordinate systems for locating positions.

Q10. Brahmagupta's formalization of zero and negative numbers was crucial to coordinate geometry because they allow:

  • A. Only positive coordinates to exist
  • B. The origin to be at any point other than (0, 0)
  • C. Points to exist in all four quadrants, including those in negative directions from the origin ✓
  • D. The x-axis and y-axis to be non-perpendicular

Answer: C — Without zero (origin) and negative numbers (left and down directions), the four-quadrant Cartesian system with points in all directions would be impossible.

Flashcards

What is the origin in a coordinate system?

The origin is the point where the x-axis and y-axis intersect, with coordinates (0, 0).

What does the x-coordinate of a point tell you?

The x-coordinate tells the distance of the point from the y-axis, measured horizontally (positive to the right, negative to the left).

What does the y-coordinate of a point tell you?

The y-coordinate tells the distance of the point from the x-axis, measured vertically (positive upward, negative downward).

What are the coordinates of a point on the x-axis?

Any point on the x-axis has coordinates of the form (x, 0) where y-coordinate is always zero.

What are the coordinates of a point on the y-axis?

Any point on the y-axis has coordinates of the form (0, y) where x-coordinate is always zero.

How are positive and negative directions marked on the coordinate plane?

Distances to the right or upward from the origin are positive; distances to the left or downward are negative.

Why is the coordinate system called 2-D?

It is called 2-D because two perpendicular lines (x-axis and y-axis) are used to locate any point in a two-dimensional plane.

Who formalized the modern Cartesian coordinate system?

René Descartes formalized in 1637 CE that any point in a two-dimensional plane can be defined by two numbers representing distances from two perpendicular axes.

What did Brahmagupta contribute to the development of coordinates?

Brahmagupta formalized the notion and use of zero and negative numbers as algebraic entities, making the four-quadrant Cartesian plane possible.

How is the position of a point written using coordinates?

The position of a point is written as P(x, y) or P = (x, y), where the first number is the x-coordinate and the second is the y-coordinate.

Important Board Questions

Define the coordinate system and explain why the origin is represented as (0, 0). [2 marks]

State that a coordinate system uses two perpendicular axes (x and y) to locate points; the origin is their intersection point where both distances are zero.

A room has corners at O(0, 0), A(12, 0), B(12, 8), and C(0, 8). If a door is located from D(3, 0) to R(6, 0), calculate the width of the door and explain why both coordinates have y = 0. [3 marks]

Door width = difference in x-coordinates: 6 − 3 = 3 units; y = 0 means the door lies on the x-axis, so it is a horizontal distance measurement only.

Explain how the ancient Indian city planning in the Sindhu-Sarasvatī Civilization using a North-South and East-West grid system relates to the modern Cartesian coordinate system. Also, describe Brahmagupta's contribution that made the four-quadrant system possible. [5 marks]

Connect the ancient grid (perpendicular lines for location) to modern axes; explain that Brahmagupta's formalization of zero (origin) and negative numbers (left/down directions) enabled points in all four quadrants, making Descartes' two-number system complete.

Assertion-Reason Questions for Orienting Yourself: The Use of Coordinates

Q1. Assertion (A):

A point lying on the positive y-axis has coordinates (0, y) where y > 0.

Reason (R):

Points on the y-axis have an x-coordinate of 0, and positive values are marked upward from the origin.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The reason correctly explains why the assertion is true: the y-axis is vertical with x = 0 for all points on it, and positive y-values extend upward from the origin.

Q2. Assertion (A):

The origin divides the coordinate plane into four equal quadrants.

Reason (R):

The x-axis and y-axis are perpendicular lines that intersect at right angles at the origin.

Answer: B — Both A and R are true, but R is NOT the correct explanation of A — Both statements are true, but the reason does not fully explain the assertion. While perpendicular axes do create regions, the quadrants may not be 'equal' in all contexts (they depend on the scale and positioning). The two statements are logically separate.

Q3. Assertion (A):

In the ancient Sindhu-Sarasvatī Civilisation, the coordinate system was used for navigation and locating buildings within cities.

Reason (R):

Brahmagupta developed the formal mathematical concept of the four-quadrant Cartesian plane which was essential for navigation.

Answer: C — A is true but R is false — The assertion is true—historical evidence shows grid-based navigation in the Sindhu-Sarasvatī Civilisation. However, the reason is false—Brahmagupta (c. 628 CE) came much later and formalized zero and negative numbers, not the original coordinate system for navigation.

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Case-Based Questions (CBQ) for Orienting Yourself: The Use of Coordinates

Case Study:

Shalini designed a school campus map using a coordinate grid where the main entrance gate is at the origin O(0, 0). The administration building is located at A(4, 3), the library is at L(−2, 5), the science laboratory is at S(3, −4), and the sports field is at F(−5, −3). Each unit on the grid represents 10 metres of actual distance on the campus. Students need to navigate between these locations efficiently.

Q1. Which building is located in the second quadrant of the campus map?

  • A. Administration building
  • B. Library ✓
  • C. Science laboratory
  • D. Sports field

Answer: B

Q2. What is the actual distance of the science laboratory from the main entrance gate?

  • A. 50 metres
  • B. 70 metres ✓
  • C. 40 metres
  • D. 100 metres

Answer: B

Q3. If a student travels from the administration building A(4, 3) to the library L(−2, 5), how far does the student move in the x-direction (left-right)?

  • A. 2 units or 20 metres
  • B. 8 units or 80 metres
  • C. 6 units or 60 metres ✓
  • D. 1 unit or 10 metres

Answer: C

Q4. Which two buildings are located in the fourth quadrant of the campus?

  • A. Administration building and Science laboratory ✓
  • B. Library and Sports field
  • C. Science laboratory and Administration building
  • D. Only the Science laboratory

Answer: A

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