Orienting Yourself: The Use of Coordinates

Learning Objectives

• Understand the structure and components of the Cartesian Coordinate System, including the x-axis, y-axis, and the origin.
• Identify and describe the four Quadrants in the Coordinate Plane based on the signs of x and y coordinates.
• Calculate the distance between two points using the Distance Formula: $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
• Analyze the effects of Reflection in the Coordinate Plane on the coordinates of points.
• Determine the Midpoint of a line segment using the formula: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$.
• Evaluate the Collinearity of Points using coordinate-based methods.
• Plot and analyze Geometric Shapes in the Coordinate Plane using coordinates and geometric reasoning.
• Apply coordinate geometry to model real-life scenarios in Coordinate Geometry Applications.
• Recognize the coordinates of Points on Axes and Origin and their implications.
• Use Ordered Pair and Coordinate Notation to accurately describe locations in the plane.
• Calculate the Distance Between Axis-Parallel Points using absolute differences in coordinates.

Chapter Notes

Cartesian Coordinate System

• The Cartesian Coordinate System is a two-dimensional coordinate system that uses two perpendicular axes: the horizontal x-axis and the vertical y-axis.
• The intersection of these axes is the origin, denoted as $O(0,0)$.
• Points are located using ordered pairs $(x, y)$, where $x$ is the x-coordinate and $y$ is the y-coordinate.

Quadrants in Coordinate Plane

• The Cartesian plane is divided into four quadrants:
  • Quadrant I: $(+, +)$
  • Quadrant II: $(-, +)$
  • Quadrant III: $(-, -)$
  • Quadrant IV: $(+, -)$
• The signs of the coordinates determine the quadrant in which a point lies.

Distance Formula

• The distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated using the formula: $\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
• This formula is derived from the Baudhāyana–Pythagoras theorem.

Reflection in Coordinate Plane

• Reflecting a point across the x-axis changes the sign of the y-coordinate.
• Reflecting a point across the y-axis changes the sign of the x-coordinate.
• Reflection preserves distances and shapes.

Midpoint Formula

• The midpoint of a segment joining $(x_1, y_1)$ and $(x_2, y_2)$ is calculated as: $\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

Collinearity of Points

• Points are collinear if they lie on the same straight line.
• Collinearity can be checked using coordinate-based reasoning or distance-based verification.

Geometric Shapes in Coordinate Plane

• Coordinate geometry allows for the plotting and analysis of shapes like triangles, rectangles, squares, and circles.
• Analysis involves using coordinates, side lengths, perpendicular/parallel sides, and the distance formula.

Coordinate Geometry Applications

• Applications include modeling real-life layouts such as rooms, streets, maps, and computer screens.
• This involves using coordinates, scale, distances, and positions.

Points on Axes and Origin

• A point on the x-axis has coordinates $(x, 0)$.
• A point on the y-axis has coordinates $(0, y)$.
• The origin has coordinates $(0, 0)$.

Ordered Pair and Coordinate Notation

• Coordinates are written in the order $(x, y)$, where $x$ is the x-coordinate and $y$ is the y-coordinate.
• In general, $(x, y) \neq (y, x)$ unless $x = y$.

Distance Between Axis-Parallel Points

• If two points have the same y-coordinate, their distance is $|x_2 - x_1|$.
• If two points have the same x-coordinate, their distance is $|y_2 - y_1|$.