NCERT Solutions for Class 9 Maths Exercise 3.1 Question 1
Welcome to the detailed solution for Class 9 Maths Chapter 3. This question introduces the fundamental idea of coordinate geometry, which is all about describing the position of a point. We’ll explore how a simple concept like the Cartesian plane can help us give a precise location to any object, like a table lamp. Let’s break it down!
Question 1: How will you describe the position of a table lamp on your study table to another person?
| Concept | Method |
|---|---|
| Treating the table as a Plane | Using two perpendicular edges of the table as reference lines or axes. |
| Measuring Position | Finding the perpendicular distances of the lamp from these two axes. |
| Representation | Expressing the position as an ordered pair, like &&(x, y)&&. |
Detailed Step-by-Step Solution 📝
To describe the position of the lamp to someone else, we need a clear and unambiguous method. Simply saying “it’s in the middle” isn’t precise. Here’s how we can use mathematics to give an exact location.
- Step 1: Consider the Table as a Plane
Imagine your study table’s flat top surface is a Cartesian plane. This plane extends infinitely in two dimensions, but we’re just focusing on the area of the table. - Step 2: Choose Your Axes
We need two fixed reference lines. The easiest choice is to pick two adjacent edges of the table that are perpendicular to each other. Let’s say we choose the bottom edge as the x-axis and the left edge as the y-axis. - Step 3: Identify the Origin
The point where these two edges meet (the bottom-left corner of the table) will be our origin. The coordinates of the origin are &&(0, 0)&&. - Step 4: Measure the Distances
Now, take a ruler and measure two distances:- Measure the perpendicular distance of the table lamp from the left edge (the y-axis). Let’s say this distance is &&20&& cm. This will be our x-coordinate.
- Measure the perpendicular distance of the table lamp from the bottom edge (the x-axis). Let’s say this distance is &&30&& cm. This will be our y-coordinate.
- Step 5: State the Position as an Ordered Pair
The position of the lamp can now be precisely described by the ordered pair of these distances. The position is written as &&(x, y)&&.
Therefore, the position of the table lamp is &&(20, 30)&&. You can tell another person that by considering the bottom-left corner as the origin, the lamp is located at point &&(20, 30)&&, with all measurements in centimeters.
Common Mistakes to Avoid 🚫
- Mixing up Coordinates: A very common error is to write the coordinates in the wrong order. Remember, it’s always &&(x, y)&&. The point &&(20, 30)&& is very different from &&(30, 20)&&.
- Not Specifying the Origin/Axes: If you just say the position is &&(20, 30)&& without telling the other person which corner you chose as the origin and which edges are the axes, the information is useless. Always define your reference frame first!
Real-World Application 🌍
This simple concept is the foundation of modern navigation! GPS systems use a similar principle with latitude and longitude as coordinates to pinpoint any location on Earth. It’s also used in computer graphics, engineering design (CAD), and even in games like Battleship to specify locations on a grid.
Frequently Asked Questions (FAQ)
Q: What is a Cartesian plane?
A: A Cartesian plane is a two-dimensional plane formed by the intersection of two perpendicular number lines, called the x-axis (horizontal) and the y-axis (vertical). It’s used to locate points with a pair of coordinates.
Q: Why do we need two axes to describe a position on a table?
A: A single line or axis can only describe a position in one dimension (like a point on a ruler). To locate a point on a two-dimensional flat surface like a table, you need two independent references, which are provided by the two perpendicular axes.
Q: What is an ordered pair?
A: An ordered pair is a set of two numbers, written as &&(x, y)&&, where the order is crucial. The first number (x-coordinate) represents the horizontal position, and the second number (y-coordinate) represents the vertical position.
Q: Can the coordinates &&(15, 25)&& and &&(25, 15)&& represent the same position?
A: No, they represent different positions. In an ordered pair, the order matters. The point &&(15, 25)&& is &&15&& units along the x-axis and &&25&& units along the y-axis, while &&(25, 15)&& is &&25&& units along the x-axis and &&15&& units along the y-axis.
Q: What is the origin in a coordinate system?
A: The origin is the point where the x-axis and the y-axis intersect. It has the coordinates &&(0, 0)&& and serves as the starting point for all measurements on the Cartesian plane.
Further Reading
- For more practice, refer to the official NCERT website for textbooks and resources.
- To understand the full context of this exercise, check out our summary of NCERT Solutions for Class 9 Maths Chapter 3 Coordinate Geometry.