Introduction to Coordinate Geometry; Distance, Midpoint, and Slope Fundamentals; Forms of Linear Equations; Parallel and Perpendicular Lines; Applications and Problem Solving

Coordinate Geometry: The Line

Mathematics

Coordinate Geometry: The Line

Mastering lines and their equations on the Cartesian plane.

Agenda

• Introduction to Coordinate Geometry

• Distance, Midpoint, and Slope Fundamentals

• Forms of Linear Equations

• Parallel and Perpendicular Lines

• Applications and Problem Solving

Introduction to Coordinate Geometry: Mapping the Plane

Coordinate geometry uses ordered pairs (x, y) to precisely map points and geometric figures on a 2D plane, connecting algebraic equations with visual shapes. For example, the point (3, 2) is uniquely located by moving 3 units right and 2 units up from the origin.

Cartesian coordinates plot points on a 2D plane.

Calculating Distance: The Length of a Segment

The distance formula, d = √[(x₂ - x₁)² + (y₂ - y₁)²], is derived from the Pythagorean theorem. Use it to find the length of a segment. For points A(1,2) and B(4,6), the distance is 5 units.

Midpoint formula: Average of coordinates for line segment center.

Finding the Midpoint: The Center of a Segment

• Locates the exact center of any line segment.

• Formula: Average of x-coordinates, average of y-coordinates.

• M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

• Example: P(-2, 5) and Q(6, 1) → M(2, 3)

Understanding Slope (Gradient): Steepness and Direction

Slope (gradient), m = (y₂ - y₁)/(x₂ - x₁), measures a line's steepness and direction. It can be positive, negative, zero, or undefined, indicating how the line rises or falls. For (2,3) and (5,9), m = (9-3)/(5-2) = 6/3 = 2.

Graph of a linear function with positive slope.

Equation Forms: Slope-Intercept Form (y = mx + b)

The slope-intercept form is y = mx + b. Here, 'm' is the slope and 'b' is the y-intercept. It's ideal for graphing, like y = 2x + 3.

Linear equation y = 2x - 3 graph on Cartesian coordinate system.

Equation Forms: Point-Slope Form (y - y₁ = m(x - x₁))

• Formula: y - y₁ = m(x - x₁)

• m = slope; (x₁, y₁) = any point

• Example: slope 3, point (1, -2)

• Substitute: y - (-2) = 3(x - 1)

Equation Forms: General Form (Ax + By + C = 0)

The General Form is Ax + By + C = 0, where A, B, C are integers. This versatile form represents *any* straight line, including vertical lines (e.g., x - 5 = 0).

Graph of y = (7/2)x - (13/2) showing a line with positive slope.

What is Intercept Form?

Example: Quick Intercepts

The intercept form of a linear equation is `x/a + y/b = 1`. Here, 'a' represents the x-intercept and 'b' represents the y-intercept. This form efficiently identifies where a line crosses the coordinate axes.

Consider the equation `x/3 + y/4 = 1`. From this form, we immediately know the x-intercept is 3 and the y-intercept is 4. The line passes through (3, 0) and (0, 4).

Coordinate Geometry: The Line

Introduction to Coordinate Geometry; Distance, Midpoint, and Slope Fundamentals; Forms of Linear Equations; Parallel and Perpendicular Lines; Applications and Problem Solving

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