Midpoint Formula and Segment Length

This page focuses on finding the midpoint of a line segment and calculating segment length in the coordinate plane.

The midpoint formula is introduced: M = $(x₁ + x₂)/2, (y₁ + y₂)/2$

For segment length, the distance formula is given: AB = √$(x₂ - x₁)² + (y₂ - y₁)²$

Several examples demonstrate the application of these formulas.

The page then transitions to equations of lines, presenting two main methods:

1. Using two points on the line $point-point form$
2. Using a point and the slope $point-slope form$

Vocabulary: The punto medio (midpoint) is the point that divides a line segment into two equal parts.

Example: For a segment with endpoints A(2,1) and B(-3,5), the midpoint M is calculated as (-1/2, 3).

Definition: The length of a line segment is the distance between its endpoints, calculated using the distance formula derived from the Pythagorean theorem.

Highlight: The midpoint and length formulas are fundamental tools in coordinate geometry, used in many advanced problems and proofs.

Slope and Distance to a Line

This page delves deeper into finding the slope (coefficiente angolare) of a line and introduces the concept of distance from a point to a line.

The slope formula is revisited: m = $y₂ - y₁$ / $x₂ - x₁$

Examples are provided for calculating slope given two points.

The point-slope form of a line equation is emphasized: y - y₁ = m$x - x₁$

The page concludes with the formula for the distance from a point to a line: d = |ax₀ + by₀ + c| / √$a² + b²$

Where (x₀, y₀) is the point and ax + by + c = 0 is the equation of the line in implicit form.

Definition: The coefficiente angolare (slope) of a line measures its steepness and direction, represented by m in the equation y = mx + q.

Example: For a line passing through A(2,-3) and B(-1,4), the slope is calculated as m = (4 - (-3)) / (-1 - 2) = 7 / -3 = -7/3.

Highlight: Understanding slope is crucial for analyzing the relationship between lines, including parallel and perpendicular lines.

Vocabulary: The distance from a point to a line is the length of the perpendicular segment from the point to the line.

Converting Between Implicit and Explicit Forms

This page explains how to convert a linear equation from forma implicita to forma esplicita. The process involves isolating the y-variable on one side of the equation.

The general steps are:

1. Start with the implicit form ax + by + c = 0
2. Subtract ax and c from both sides
3. Factor out b from the y term
4. Divide both sides by b to isolate y

An example is given: 2x + 3y - 6 = 0 is converted to y = -2/3x + 2

The page also covers:

• Graphing lines using a point-slope approach
• Checking if a point lies on a line by substituting coordinates
• Special cases of vertical and horizontal lines

Definition: The forma implicita of a line is ax + by + c = 0, while the forma esplicita is y = mx + q, where m is the slope and q is the y-intercept.

Example: To check if point P(-2,1) lies on y = -3x + 5, substitute the x-coordinate: 1 ?= -3(-2) + 5. Since 1 = 1, the point does lie on the line.

Highlight: Understanding how to convert between implicit and explicit forms is crucial for solving many geometry problems involving lines.