# Representation of Points in a Plane

### Representation of Points in a Plane:

#### Cartesian Coordinate System:

• Concept: A system that defines the position of points in a plane using two perpendicular lines called axes.
• Axes: Two lines intersecting at a point called the origin, typically labeled as x-axis (horizontal) and y-axis (vertical).
• Coordinates: Points are represented as ordered pairs (x, y), where x represents the horizontal distance and y represents the vertical distance.

#### Plotting Points:

• Ordered Pairs: A point's coordinates determine its location in the plane. The first coordinate (x) indicates the position along the x-axis, and the second coordinate (y) denotes the position along the y-axis.
• Origin: The point (0, 0) where both x and y are zero, located at the intersection of the axes.

• Divisions: The plane is divided into four quadrants, numbered I to IV, counterclockwise from the right side of the x-axis.

#### Distance and Midpoint:

• Distance Formula: Calculate the distance between two points using $d=\sqrt{\left({x}_{2}-{x}_{1}{\right)}^{2}+\left({y}_{2}-{y}_{1}{\right)}^{2}}$.
• Midpoint Formula: Find the point equidistant from two given points using $\left(\frac{{x}_{1}+{x}_{2}}{2},\frac{{y}_{1}+{y}_{2}}{2}\right)$.

### Understanding Coordinates:

#### x-Coordinate and y-Coordinate:

• x-Coordinate: Horizontal position from the origin along the x-axis; positive to the right and negative to the left.
• y-Coordinate: Vertical position from the origin along the y-axis; positive upwards and negative downwards.

#### Graphical Representations:

• Graphing Points: Plotting points on a plane by locating the respective x and y coordinates.
• Connecting Points: Joining plotted points to form shapes, lines, curves, or patterns.

### Applications:

#### Mathematics:

• Geometry: Defining shapes, angles, and distances.
• Algebra: Solving equations, graphing functions.

#### Physics:

• Kinematics: Describing the position of objects in motion.
• Vectors: Representing forces and velocities in a plane.

#### Engineering:

• CAD (Computer-Aided Design): Designing structures and objects using coordinate systems.
• Mapping: Representing geographical positions using coordinates.

#### Computer Science:

• Graphics: Creating images and animations by defining points in a 2D plane.
• Algorithms: Implementing algorithms that involve points and their coordinates.