🔷Honors Geometry Unit 13 – Coordinate Geometry

Key Concepts and Definitions

• Coordinate geometry involves the study of geometric shapes and properties using a coordinate system
• The coordinate plane consists of two perpendicular number lines called the x-axis (horizontal) and y-axis (vertical) that intersect at the origin (0, 0)
  • The x-coordinate represents the horizontal distance from the origin
  • The y-coordinate represents the vertical distance from the origin
• Points on the coordinate plane are represented by ordered pairs (x, y)
• Distance is the length of the line segment connecting two points on the coordinate plane
• Midpoint is the point that divides a line segment into two equal parts
• Slope measures the steepness and direction of a line
  • Positive slope indicates an increasing line from left to right
  • Negative slope indicates a decreasing line from left to right
• Rate of change describes how one variable changes with respect to another variable
• Parallel lines have the same slope and never intersect
• Perpendicular lines have slopes that are negative reciprocals of each other and intersect at a 90-degree angle

The Coordinate Plane

• The coordinate plane is a two-dimensional surface formed by the intersection of a horizontal number line (x-axis) and a vertical number line (y-axis)
• The x-axis and y-axis intersect at a point called the origin, which has the coordinates (0, 0)
• The x-axis represents the horizontal direction, with positive values to the right of the origin and negative values to the left
• The y-axis represents the vertical direction, with positive values above the origin and negative values below
• The coordinate plane is divided into four quadrants:
  • Quadrant I: (+, +)
  • Quadrant II: (-, +)
  • Quadrant III: (-, -)
  • Quadrant IV: (+, -)
• Points on the coordinate plane are represented by ordered pairs (x, y), where x is the x-coordinate and y is the y-coordinate
• The x-coordinate indicates the horizontal distance from the origin, while the y-coordinate indicates the vertical distance from the origin

Distance Formula

• The distance formula calculates the length of the line segment connecting two points on the coordinate plane
• For points $(x_1, y_1)$ and $(x_2, y_2)$, the distance formula is:
  • $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
• The distance formula is derived from the Pythagorean theorem
• To find the distance between two points, substitute their coordinates into the formula and simplify
• The distance is always a non-negative value
• The distance between a point and itself is always 0
• The distance formula can be used to find the length of any line segment on the coordinate plane, not just horizontal or vertical segments

Midpoint Formula

• The midpoint formula finds the coordinates of the point that divides a line segment into two equal parts
• For a line segment with endpoints $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint formula is:
  • $(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$
• To find the midpoint, add the x-coordinates of the endpoints and divide by 2, then add the y-coordinates of the endpoints and divide by 2
• The midpoint is always on the line segment connecting the two endpoints
• The distance from the midpoint to either endpoint is equal
• The midpoint formula can be used to find the center of a circle or the center of a rectangle

Slope and Rate of Change

• Slope is a measure of the steepness and direction of a line
• The slope of a line is calculated by finding the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line
• The slope formula for a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is:
  • $m = \frac{y_2 - y_1}{x_2 - x_1}$
• A positive slope indicates an increasing line from left to right, while a negative slope indicates a decreasing line from left to right
• A slope of 0 represents a horizontal line, and an undefined slope represents a vertical line
• The rate of change is a measure of how one variable changes with respect to another variable
• In a linear relationship, the rate of change is constant and equal to the slope of the line
• The slope and rate of change can be used to interpret real-world relationships, such as the cost per item or the speed of an object

Equations of Lines

• The equation of a line is a linear equation that describes all the points on the line
• The slope-intercept form of a line is $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept
  • The y-intercept is the point where the line crosses the y-axis
• The point-slope form of a line is $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is a point on the line and $m$ is the slope
• To write the equation of a line, you need the slope and either a point on the line or the y-intercept
• Parallel lines have the same slope but different y-intercepts
• Perpendicular lines have slopes that are negative reciprocals of each other
• The x-intercept is the point where the line crosses the x-axis and can be found by setting y = 0 in the equation of the line

Parallel and Perpendicular Lines

• Parallel lines are lines that lie in the same plane and never intersect
• Parallel lines have the same slope but different y-intercepts
• The slopes of parallel lines are equal
• Perpendicular lines are lines that intersect at a 90-degree angle
• The slopes of perpendicular lines are negative reciprocals of each other
  • If the slope of one line is $m$, the slope of the perpendicular line is $-\frac{1}{m}$
• To find the equation of a line parallel to a given line, use the same slope and a different y-intercept
• To find the equation of a line perpendicular to a given line, use the negative reciprocal of the slope and a point on the line
• Parallel and perpendicular lines are used in various applications, such as in architecture, engineering, and computer graphics

Applications and Problem Solving

• Coordinate geometry has numerous real-world applications in fields such as physics, engineering, computer graphics, and navigation
• Distance formula applications:
  • Calculating the shortest path between two locations on a map
  • Determining the range of a wireless network or radio transmission
• Midpoint formula applications:
  • Finding the center of a circle or sphere
  • Locating the balance point of a physical object
• Slope and rate of change applications:
  • Analyzing the steepness of a hill or mountain
  • Calculating the speed or acceleration of an object
  • Determining the rate of change in population growth or decay
• Equations of lines applications:
  • Modeling the relationship between two variables (e.g., cost vs. quantity)
  • Predicting future values based on past data
• Parallel and perpendicular lines applications:
  • Designing structures with parallel or perpendicular support beams
  • Creating perpendicular streets in city planning
  • Developing computer graphics and video game environments
• Problem-solving strategies in coordinate geometry often involve:
  • Identifying the given information and the desired outcome
  • Sketching the problem on the coordinate plane
  • Applying the appropriate formulas or concepts
  • Simplifying and solving equations
  • Interpreting the results in the context of the problem