Straight Lines: A Quick Overview
Straight lines are fundamental geometric figures that extend infinitely in both directions without curving. They are defined by their slope and intercept.
Key Concepts and Formulas
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Slope: The steepness of a line, measured as the ratio of vertical change (rise) to horizontal change (run) between two points on the line.
- Formula: Slope = (y₂ - y₁) / (x₂ - x₁)
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Intercept: The point where a line crosses the x-axis (x-intercept) or the y-axis (y-intercept).
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Equation of a Line:
- 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 (x₁, y₁) is a point on the line and m is the slope.
- General form: Ax + By + C = 0, where A, B, and C are constants.
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Parallel Lines: Two lines are parallel if their slopes are equal.
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Perpendicular Lines: Two lines are perpendicular if the product of their slopes is -1.
Common Applications
- Coordinate geometry: Locating points, finding distances, and determining equations of lines.
- Calculus: Finding tangents and normals to curves.
- Physics: Representing motion, forces, and relationships between variables.
- Engineering: Designing structures, analyzing systems, and solving problems.
Example: Find the equation of a line passing through the points (2, 3) and (5, 7).
- Solution:
- Calculate the slope: m = (7 - 3) / (5 - 2) = 4/3
- Use the point-slope form: y - 3 = (4/3)(x - 2)
- Simplify: y = (4/3)x - 2/3