High School Coordinate Geometry Study
Guide
Key Concepts and Definitions
Coordinate System: A system that uses ordered pairs (x, y) to locate points on a plane.
Distance Formula: The distance between points (x₁, y₁) and (x₂, y₂) is:
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Midpoint Formula: The midpoint between points (x₁, y₁) and (x₂, y₂) is:
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Slope Formula: The slope of a line passing through points (x₁, y₁) and (x₂, y₂) is:
m = (y₂ - y₁)/(x₂ - x₁)
Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept.
Point-Slope Form: y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line.
Standard Form: Ax + By + C = 0, where A, B, and C are constants.
Parallel Lines: Two lines are parallel if they have the same slope.
Perpendicular Lines: Two lines are perpendicular if the product of their slopes is -1.
Coordinate Proofs Techniques
1. Proving Points are Collinear
Three or more points are collinear if they all lie on the same straight line. To prove this:
● Calculate the slopes between pairs of points
● If all slopes are equal, the points are collinear
2. Proving a Quadrilateral is a Parallelogram
A quadrilateral with vertices A, B, C, and D is a parallelogram if:
Guide
Key Concepts and Definitions
Coordinate System: A system that uses ordered pairs (x, y) to locate points on a plane.
Distance Formula: The distance between points (x₁, y₁) and (x₂, y₂) is:
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Midpoint Formula: The midpoint between points (x₁, y₁) and (x₂, y₂) is:
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Slope Formula: The slope of a line passing through points (x₁, y₁) and (x₂, y₂) is:
m = (y₂ - y₁)/(x₂ - x₁)
Slope-Intercept Form: y = mx + b, where m is the slope and b is the y-intercept.
Point-Slope Form: y - y₁ = m(x - x₁), where m is the slope and (x₁, y₁) is a point on the line.
Standard Form: Ax + By + C = 0, where A, B, and C are constants.
Parallel Lines: Two lines are parallel if they have the same slope.
Perpendicular Lines: Two lines are perpendicular if the product of their slopes is -1.
Coordinate Proofs Techniques
1. Proving Points are Collinear
Three or more points are collinear if they all lie on the same straight line. To prove this:
● Calculate the slopes between pairs of points
● If all slopes are equal, the points are collinear
2. Proving a Quadrilateral is a Parallelogram
A quadrilateral with vertices A, B, C, and D is a parallelogram if:
