Midpoints and Line Equations
This page covers essential geometric concepts including midpoints, equations of lines, parallel and perpendicular lines, and distance formulas.
The midpoint formula in geometry is introduced as a way to find the point that bisects a line segment.
Definition: A midpoint is a point that bisects a segment.
Various forms of line equations are presented:
- Point-Slope Form (PSF): y - y₁ = m(x - x₁)
- Slope-Intercept Form (SIF): y = mx + b
- Standard Form (SF): Ax + By = C
- General Form (GF): Ax + By + C = 0
The concept of parallel lines is explained:
Highlight: If lines are parallel, they have the same slope.
An example is provided: 2x + 2y = 4 and 2x + 10 = y are parallel lines.
Perpendicular lines are also discussed:
Highlight: When lines are perpendicular, they have reciprocal slopes and form 90° angles when intersected.
The Pythagorean theorem and distance formula are introduced:
- Pythagorean Theorem: A² + B² = C² (where C is the hypotenuse)
- Distance Formula for lines: √((x₁ - x₂)² + (y₁ - y₂)²)
- Distance Formula for point-to-line: |Ax + By + C| / √(A² + B²)
Example: Points (-2, -4) and (-4, x) form a 90° angle with a distance of 5 units between them.
The concept of collinearity is briefly mentioned, referring to points that can connect to form one line.
Lastly, the page touches on transformations and reflections:
Example: (x + h, y + k) represents a transformation where h is the transformation for x and k is the transformation for y.
(-x, y) represents a reflection over the y-axis, while (x, -y) represents a reflection over the x-axis.
These concepts form the foundation for more advanced topics in geometry and are crucial for solving various geometric problems.
