Coordinate Geometry: Cartesian System

Introduction to Coordinate Geometry

• Chapter 3, Coordinate Geometry, is reviewed in a single session
• The chapter includes an introduction to the Cartesian system and plotting points on a plane using coordinates
• An example scenario introduces the core concept of coordinate geometry

Example: Locating a House

• Locating Shyam's house requires more than just a street number due to multiple houses on a street
• Knowing only the house number is insufficient because there are multiple streets
• To precisely locate the house, provide the street number and the house number on that street

René Descartes and the Cartesian System

• René Descartes developed the Cartesian system to locate a point on a plane
• The Cartesian system employs two perpendicular number lines

Constructing the Cartesian Plane

• Use two number lines, positioned at a 90-degree angle from each other, to construct the plane
• Number lines have zero in the middle, positives to the right, and negatives to the left
• One line is labeled X X', and the other is labeled Y Y'
• Intervals are typically set at 1-centimeter distances, similar to a graph

Locating a Point on the Cartesian Plane

• To specify the location of a point on the plane, start at the origin
• Move horizontally along the X-axis towards the point and note the distance
• Next, move vertically along the Y-axis to reach the point and note the distance
• The point's location is specified by two numbers, with the X-axis distance stated first, followed by the Y-axis distance
• For example, (3, 2) represents an exact location, ensuring anyone using this system finds the same point

Terminology

• The plane is also called the Cartesian plane, the coordinate plane, or the XY-plane
• Coordinate axes include the X-axis (horizontal) and the Y-axis (vertical)
• For a point (2, 3):
  • The X-coordinate (2) is called the abscissa
  • The Y-coordinate (3) is called the ordinate
  • Together, these are the point's coordinates
• René Descartes developed the Cartesian coordinate system

Signs and Directions on the Axes

• The X-axis to the right of the origin is positive
• The X-axis to the left of the origin is negative
• The Y-axis above the origin is positive
• The Y-axis below the origin is negative

Quadrants

• Coordinate axes divide the plane into four quadrants
• Quadrant signs:
  • First quadrant: X and Y are positive
  • Second quadrant: X is negative, Y is positive
  • Third quadrant: X and Y are negative
  • Fourth quadrant: X is positive, Y is negative
• Signs help identify the quadrant where coordinates are located

Coordinates of Points on the X and Y Axes

• For a point on the X-axis, move a certain distance along the X-axis from the origin, without moving along the Y-axis
• Points on the X-axis have coordinates of the form (x, 0)
• For a point on the Y-axis, move a certain distance along the Y-axis from the origin, without moving along the X-axis
• Points on the Y-axis have coordinates of the form (0, y)

Exercise 3.1 Discussion

Question 1

• Question 1, part 1: What are the names of the horizontal and vertical lines used to determine the position of a point in the Cartesian plane?
  • Answer: The horizontal line is the X-axis, and the vertical line is the Y-axis
• Question 1, part 2: What is the name of each part of the plane formed by these two lines?
  • Answer: The parts are called quadrants
• Question 1, part 3: What is the name of the point where these two lines intersect?
  • Answer: The intersection point is called the origin

Question 2

• In "See the given graph and write the following" questions, requested coordinates are identified from a graph

Exercise 3.3 Discussion

Question 1

• Identify the quadrant or axis for the points (-2,4), (3,-1), (-1,0), (1,2), and (-3,-5) by locating them on the Cartesian plane

Plotting Points on a Graph

• Review of plotting points:
  • (-2, 4) is in the second quadrant
  • (3, -1) is in the fourth quadrant
  • (-1, 0) is on the x-axis, to the left of the origin
  • (1, 2) is in the first quadrant
  • (-3, -5) is in the third quadrant

Question 2

• To plot points from a table, select appropriate units of distance on the axes

Setting up the Graph

• Set up a rough diagram to strategize the X and Y axis values, followed by a scale
• Based on the diagram:
  • X ranges from -2 to 3
  • Y ranges from about -2 to 8
• Each point is plotted according to its coordinates from the questions

End

• Plotting points when given a chart, is considered a completed chapter