NCERT Solutions for Class 9 Maths Chapter 3 Coordinate Geometry Exercise 3.1

The Concept of Relative Positioning in 2D Space

To locate any point on a flat surface (a plane), we need two independent pieces of information. For example, if you want to tell someone where a point is on a blank sheet of paper, just saying 'it is near the top' or 'it is on the left' is not precise. Instead, we can establish two perpendicular reference lines: a horizontal line and a vertical line. The distance of the point from the vertical line gives its horizontal position, and its distance from the horizontal line gives its vertical position. This pair of distances acts as the unique 'address' of the point. In coordinate geometry, these reference lines are called axes, and the address is called an ordered pair $(x, y)$. Exercise 3.1 helps you build this exact visualization before we define formal coordinate terminology.

How to Describe the Position of an Object (The Study Lamp)

1. Choose Two Perpendicular Reference Edges — Consider the top of your study table as a plane. Identify two perpendicular edges: the shorter edge (left edge) and the longer edge (front edge near which you sit).
2. Measure the Perpendicular Distances — Measure the perpendicular distance of the study lamp from both chosen edges. Suppose the distance of the lamp from the left edge is $30\text{ cm}$ and the distance from the front edge is $25\text{ cm}$.
3. Write the Position as an Ordered Pair — Write down these measurements in a fixed order. If you state the distance from the left edge first, the position of the lamp can be written as $(30, 25)$. If you state the distance from the front edge first, it is written as $(25, 30)$.

NCERT Exercise 3.1 Textbook Solutions Worked Out

• Question 1: How will you describe the position of a table lamp on your study table to another person? Detailed Solution: 1. Let the table surface represent a plane sheet. 2. Choose two perpendicular edges of the table as your reference lines: say, the left edge and the bottom (front) edge. 3. Measure the perpendicular distance of the lamp from the left edge. Let this distance be $30\text{ cm}$. 4. Measure the perpendicular distance of the lamp from the front edge. Let this distance be $25\text{ cm}$. 5. Now, describe the position of the lamp to your friend as: 'The lamp is located $30\text{ cm}$ from the left edge and $25\text{ cm}$ from the front edge.' 6. Mathematically, you can represent this position as the ordered pair $(30, 25)$ relative to the bottom-left corner of the desk.
• Question 2: Street Plan Problem. A city has two main roads which cross each other at the center of the city... Draw a model of the city and answer: (i) How many cross-streets can be referred to as $(4, 3)$? (ii) How many cross-streets can be referred to as $(3, 4)$? Detailed Solution: 1. Let the two main roads represent the horizontal axis (East-West) and the vertical axis (North-South). They intersect at the origin $(0,0)$. 2. The streets are parallel to these main roads, spaced $200\text{ m}$ apart. Represent these streets using parallel grid lines ($1\text{ cm} = 200\text{ m}$). 3. Any cross-street (intersection) is identified uniquely by the horizontal street number and the vertical street number. 4. Part (i): The notation $(4, 3)$ means Street 4 running North-South and Street 3 running East-West. Since these two unique lines can only intersect at exactly one point, there is only one unique cross-street that can be referred to as $(4, 3)$. 5. Part (ii): The notation $(3, 4)$ represents Street 3 running North-South and Street 4 running East-West. Similarly, these two perpendicular streets intersect at exactly one point, so there is only one unique cross-street referred to as $(3, 4)$.

Crucial Exam Tip: Why Order Matters in Coordinates

In board exams, students often confuse $(x, y)$ with $(y, x)$. This is a critical mistake! The term ordered pair is called 'ordered' for a reason. The sequence of numbers conveys specific information. In Question 2 of Exercise 3.1, you can clearly see that the point $(4, 3)$ and $(3, 4)$ represent two completely different physical intersections on the street map. Always define which coordinate represents which direction/edge before writing down your coordinate pairs.

Practice Questions with Solutions

• Q: A student is sitting in a classroom. Describe their position if they are sitting in the 4th row (from the front) and the 3rd column (from the left side of the room). A: Step 1: Let the front wall and left wall of the classroom act as our perpendicular reference axes. Step 2: Let the horizontal coordinate represent the column number (from left), which is $3$. Step 3: Let the vertical coordinate represent the row number (from front), which is $4$. Step 4: Writing these together as an ordered pair (Column, Row) gives the unique position $(3, 4)$. Final answer: The unique location is represented as $(3, 4)$.
• Q: A rectangular chess board has coordinates on its edges. A pawn is located 2 units away from the left edge and 5 units away from the bottom edge. Express its position coordinates. A: Step 1: Establish the bottom-left corner of the chessboard as the origin $(0,0)$. Step 2: The horizontal distance (from the left edge) is $2$ units. Step 3: The vertical distance (from the bottom edge) is $5$ units. Step 4: Combine these as an ordered pair $(x, y)$, where $x$ is horizontal distance and $y$ is vertical distance. Final answer: The coordinates of the pawn's position are $(2, 5)$.
• Q: On a grid sheet, a point P is located such that it is 6 units to the right of the vertical axis and 8 units above the horizontal axis. What are its coordinates? A: Step 1: Identify the horizontal distance along the x-axis, which is $6$ units to the right. Step 2: Identify the vertical distance along the y-axis, which is $8$ units upward. Step 3: Arrange them in the standard coordinate format $(x, y)$. Final answer: The coordinates of point P are $(6, 8)$.
• Q: If a city street map defines a crossroad at coordinates $(a, b)$, under what mathematical condition would the crossroad $(a, b)$ be identical to the crossroad $(b, a)$? A: Step 1: For two coordinate pairs $(a, b)$ and $(b, a)$ to represent the exact same physical location, their corresponding coordinates must be equal. Step 2: This means the horizontal coordinate of the first point must equal the horizontal coordinate of the second point ($a = b$), and the vertical coordinates must also match ($b = a$). Final answer: The two crossroads will only be identical if $a = b$.

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