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 = mx−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 whereCisthehypotenuse
- Distance Formula for lines: √(x1−x2² + y1−y2²)
- 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.
