This chapter introduces you to the Cartesian coordinate system, a fundamental tool in mathematics for describing the positions of points in a plane. You will learn how to use coordinates to plot points, understand the structure of the coordinate plane, and explore the properties of points in different quadrants. The chapter also touches on the historical development of coordinate geometry, highlighting its roots in ancient civilizations and its evolution through the works of mathematicians like Brahmagupta and René Descartes. By the end of this chapter, you will be able to visualize and manipulate points in a 2D space, setting the stage for more advanced studies in geometry and algebra.
What you'll learn
1Understand the concept of the Cartesian coordinate system and its components
2Learn to plot points using ordered pairs (coordinates) on the coordinate plane
3Identify the positions of points on the axes and in the four quadrants
4Calculate the distance between two points using the Baudhāyana–Pythagoras theorem
5Explore symmetry and reflection of points in the coordinate plane
Chapter at a glance
01Introduction to the Cartesian Plane
02Coordinates of a Point (Ordered Pairs)
03Coordinates on the Axes and in the Four Quadrants
04Symmetry on the Coordinate Plane
Detailed chapter notes
01
Introduction to the Cartesian Plane
The Cartesian plane, also known as the coordinate plane or xy-plane, is a two-dimensional space defined by two perpendicular lines called the x-axis (horizontal) and the y-axis (vertical). These axes intersect at a point called the origin, which has coordinates (0, 0). The Cartesian plane allows us to describe the exact location of any point using a pair of numbers called coordinates. The first number represents the distance from the y-axis (x-coordinate), and the second number represents the distance from the x-axis (y-coordinate). This system was formalized by René Descartes, building on the work of earlier mathematicians and civilizations.
The Cartesian plane is defined by the x-axis and y-axis
The origin is the point (0, 0) where the axes intersect
Coordinates (x, y) describe the position of a point in the plane
02
Coordinates of a Point (Ordered Pairs)
A point in the Cartesian plane is represented by an ordered pair of numbers (x, y), where x is the x-coordinate and y is the y-coordinate. The order of the numbers is important: the first number always refers to the x-coordinate, and the second to the y-coordinate. For example, the point (3, 4) is different from the point (4, 3). Points on the x-axis have a y-coordinate of 0, and points on the y-axis have an x-coordinate of 0. The origin is represented as (0, 0).
An ordered pair (x, y) represents a point in the plane
The order of coordinates matters(x, y) is not the same as (y, x)
Points on the x-axis have coordinates (x, 0)
Points on the y-axis have coordinates (0, y)
03
Plotting Points on the Coordinate Plane
To plot a point on the Cartesian plane, start at the origin and move horizontally along the x-axis to the x-coordinate of the point. Then, move vertically along the y-axis to the y-coordinate. The point where these movements intersect is the location of the point. For example, to plot the point (3, 4), move 3 units to the right of the origin along the x-axis and then 4 units up along the y-axis. This process helps in visualizing the position of points and understanding the layout of the coordinate plane.
Start at the origin to plot a point
Move horizontally to the x-coordinate and then vertically to the y-coordinate
The intersection of these movements is the point's location
04
Coordinates on the Axes and in the Four Quadrants
The Cartesian plane is divided into four quadrants by the x-axis and y-axis. Points in each quadrant have specific characteristics based on the signs of their coordinates. Quadrant I contains points with both x and y coordinates positive. Quadrant II has negative x-coordinates and positive y-coordinates. Quadrant III has both coordinates negative, and Quadrant IV has positive x-coordinates and negative y-coordinates. Points on the axes do not belong to any quadrant and are simply described by their coordinates on the respective axis.
Quadrant I(x, y) where x > 0 and y > 0
Quadrant II(x, y) where x < 0 and y > 0
Quadrant III(x, y) where x < 0 and y < 0
Quadrant IV(x, y) where x > 0 and y < 0
05
Symmetry on the Coordinate Plane
Symmetry on the coordinate plane involves reflecting points across the axes. Reflecting a point across the x-axis changes the sign of its y-coordinate, while reflecting it across the y-axis changes the sign of its x-coordinate. For example, the reflection of the point (3, 4) across the x-axis is (3, -4), and its reflection across the y-axis is (-3, 4). Understanding symmetry helps in visualizing and manipulating points in the plane, as well as in solving problems involving geometric transformations.
Reflection across the x-axis changes the sign of the y-coordinate
Reflection across the y-axis changes the sign of the x-coordinate
Symmetry helps in visualizing and manipulating points in the plane
06
Distance Between Two Points in the 2-D Plane
To find the distance between two points in the Cartesian plane, you can use the Baudhāyana–Pythagoras theorem. This theorem states that the distance between two points (x1, y1) and (x2, y2) is the square root of the sum of the squares of the differences in their x-coordinates and y-coordinates. Mathematically, this is expressed as distance = sqrt((x2 - x1)^2 + (y2 - y1)^2). This formula allows you to calculate the distance between any two points in the plane, regardless of their position or the quadrant they are in.
Distance formulasqrt((x2 - x1)^2 + (y2 - y1)^2)
The formula works for any two points in the plane
It is derived from the Baudhāyana–Pythagoras theorem
Want the complete chapter resources?Topic notes, quizzes and flashcards for Orienting Yourself: The Use of Coordinates.
Q1. Define the Cartesian plane and explain how the coordinates of a point are determined. Give the coordinates of the origin.
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Model answer
The Cartesian plane is a two-dimensional plane formed by two perpendicular lines, the horizontal x-axis and the vertical y-axis, intersecting at the origin. The coordinates of a point are an ordered pair (x, y), where x is the perpendicular distance from the y-axis (measured along the x-axis) and y is the perpendicular distance from the x-axis (measured along the y-axis). The origin has coordinates (0, 0).
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Q2. What are the coordinates of the origin? Explain the coordinates of a point on the x-axis and a point on the y-axis.
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Model answer
The origin O has coordinates (0, 0). A point on the x-axis has coordinates of the form (x, 0), where x is the distance from the y-axis along the x-axis. A point on the y-axis has coordinates of the form (0, y), where y is the distance from the x-axis along the y-axis.
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Q3. What are the coordinates of a point on the x-axis and a point on the y-axis? Give an example of each.
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Model answer
A point on the x-axis has coordinates of the form (x, 0), where x is the distance from the y-axis. For example, (4.5, 0) lies on the x-axis. A point on the y-axis has coordinates of the form (0, y), where y is the distance from the x-axis. For example, (0, -4.5) lies on the y-axis. The origin (0,0) is where both axes intersect.
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Q4. What are the coordinates of a point on the x-axis and a point on the y-axis? Give an example of each and state the quadrant in which the point (3, -4) lies.
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Model answer
A point on the x-axis has coordinates of the form (x, 0), where x is the distance from the origin along the x-axis. For example, (4, 0) lies on the x-axis. A point on the y-axis has coordinates of the form (0, y), such as (0, -3). The point (3, -4) has a positive x-coordinate and a negative y-coordinate, so it lies in Quadrant IV.
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Q5. If a point P(x, y) is reflected in the y-axis, what are the coordinates of its image? Explain with an example.
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When a point P(x, y) is reflected in the y-axis, the x-coordinate changes sign while the y-coordinate remains the same. So the image is P'(-x, y). For example, the reflection of A(3, 4) in the y-axis is A'(-3, 4), as shown in Fig. 1.9 of the chapter.
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The Cartesian plane is a two-dimensional space defined by the x-axis and y-axis, used to describe the positions of points using coordinates.
How do you plot a point on the coordinate plane?
To plot a point (x, y), start at the origin, move horizontally to the x-coordinate, and then vertically to the y-coordinate.
What are the characteristics of points in each quadrant?
Quadrant I: (x, y) where x > 0 and y > 0; Quadrant II: (x, y) where x < 0 and y > 0; Quadrant III: (x, y) where x < 0 and y < 0; Quadrant IV: (x, y) where x > 0 and y < 0.
How do you find the distance between two points in the plane?
Use the distance formula: sqrt((x2 - x1)^2 + (y2 - y1)^2), derived from the Baudhāyana–Pythagoras theorem.
What happens when you reflect a point across the x-axis or y-axis?
Reflecting across the x-axis changes the sign of the y-coordinate; reflecting across the y-axis changes the sign of the x-coordinate.