Formula Sheet: Straight Lines
This chapter explores the properties and equations of straight lines in coordinate geometry, emphasizing their significance in mathematics and real-life applications.
Straight Lines – Formula & Equation Sheet
Essential formulas and equations from Mathematics, tailored for Class 11 in Mathematics.
This one-pager compiles key formulas and equations from the Straight Lines chapter of Mathematics. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formula sheet
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
Distance between two points: d = √((x₂ - x₁)² + (y₂ - y₁)²)
d is the distance, (x₁, y₁) and (x₂, y₂) are the coordinates of the points. This formula calculates the distance between two points in a Cartesian plane.
Midpoint: M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
M is the midpoint of the line segment joining points (x₁, y₁) and (x₂, y₂). It provides the average coordinates of the two endpoints.
Slope (m): m = (y₂ - y₁)/(x₂ - x₁)
m is the slope of the line through points (x₁, y₁) and (x₂, y₂). It represents the steepness and direction of the line.
Equation of a line (Point-Slope Form): y - y₀ = m(x - x₀)
Where (x₀, y₀) is a point on the line and m is the slope. It combines a point's coordinates with the slope to define a line.
Slope-Intercept Form: y = mx + c
m is the slope and c is the y-intercept. This form is useful for quickly graphing lines based on their slope and y-intercept.
Two-Point Form: (y - y₁) = (y₂ - y₁)/(x₂ - x₁) (x - x₁)
This form gives the equation of the line through two points (x₁, y₁) and (x₂, y₂). It is useful for deriving a line's equation from its endpoints.
Condition for parallel lines: m₁ = m₂
If two lines have equal slopes (m₁ and m₂), they are parallel and will never intersect.
Condition for perpendicular lines: m₁ * m₂ = -1
If the product of the slopes of two lines equals -1, these lines are perpendicular and intersect at right angles.
Distance from point (x₁, y₁) to line Ax + By + C = 0: d = |Ax₁ + By₁ + C| / √(A² + B²)
This formula gives the perpendicular distance from a point to a line represented in general form. It is useful in geometric proofs.
Area of triangle formed by points (x₁, y₁), (x₂, y₂), (x₃, y₃): Area = 0.5 * |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|
This formula helps calculate the area of a triangle given its vertices in a coordinate plane, facilitating geometric analysis.
Equations
Distance PQ: d(PQ) = √[(x₂ - x₁)² + (y₂ - y₁)²]
Calculates the distance between points P and Q in Cartesian coordinates.
Line through P(x₁, y₁) with slope m: y - y₁ = m(x - x₁)
Defines a straight line using a point and slope, commonly used in coordinate geometry.
y-intercept form: y = mx + c
This expresses the equation of a line where c is the y-intercept, showing where the line intersects the y-axis.
Slope of horizontal line: m = 0
Indicates that a horizontal line has zero slope as there is no rise over run.
Slope of vertical line: Undefined
A vertical line's slope cannot be defined as it involves division by zero (change in x = 0).
Equation for a line with y-intercept: y = c (when m = 0)
Describes a horizontal line where y-value remains constant, at the y-intercept c.
Equation for a vertical line through point (a, 0): x = a
Describes a line parallel to the y-axis, consistent for all y-values when x is fixed.
Collinearity Condition: A1(B2 - B3) + A2(B3 - B1) + A3(B1 - B2) = 0
This determinant condition checks if three points are collinear based on their coordinates.
Perpendicular distance: d = |Ax₁ + By₁ + C| / √(A² + B²)
Describes how far a point is from a line, providing a geometric measure important in proofs.
Area of triangle: Area = 1/2 * Base * Height
General concept for calculating the area of a triangle using base and height measurements.
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