Parallel and Intersecting Lines – Formula & Equation Sheet
Essential formulas and equations from Ganita Prakash, tailored for Class 7 in Mathematics.
This one-pager compiles key formulas and equations from the Parallel and Intersecting Lines chapter of Ganita Prakash. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Formulas
Slope (m) = (y₂ - y₁) / (x₂ - x₁)
m represents the slope of a line connecting two points (x₁, y₁) and (x₂, y₂). Slope indicates how steep a line is. It's useful in determining the angle of a line.
General Form of Equation: Ax + By + C = 0
A, B, and C are constants in the equation of a line. This form is used to represent linear equations and find intersections.
y = mx + b
In this equation, m is the slope and b is the y-intercept. This form is key for graphing lines and understanding their behavior.
Perpendicular Slope: m₁ * m₂ = -1
If two lines are perpendicular, the product of their slopes (m₁ and m₂) is -1. This helps identify relationships in parallel and intersecting lines.
Parallel Lines: m₁ = m₂
When two lines are parallel, their slopes (m₁ and m₂) are equal. Essential in determining the nature of two given lines.
Point-Slope Form: y - y₁ = m(x - x₁)
Where (x₁, y₁) is a point on the line and m is the slope. This form is useful for writing the equation of a line quickly.
Distance Between Two Points: d = √((x₂ - x₁)² + (y₂ - y₁)²)
d represents the distance between two points (x₁, y₁) and (x₂, y₂). It can be applied in geometric problems involving point location.
Intercepts: x-intercept = -C/A, y-intercept = -C/B
These formulas give the x and y intercepts from the standard form of the equation. Useful for graphing lines and understanding their intersections with axes.
Angle between two lines: tan(θ) = |(m₁ - m₂) / (1 + m₁*m₂)|
Here, θ is the angle formed by two lines with slopes m₁ and m₂. This is used in geometry to find the angle of intersection.
Collinearity Condition: (y₂ - y₁)/(x₂ - x₁) = (y₃ - y₁)/(x₃ - x₁)
This checks if three points (x₁, y₁), (x₂, y₂), (x₃, y₃) are collinear via their slope equality, which is useful in various geometric proofs.
Equations
y = 2x + 3
This equation represents a line with a slope of 2 and a y-intercept at 3. It's a specific case of a linear function and is useful in graphing.
y = -1/2x + 4
A linear equation where the slope is -1/2. It illustrates a decreasing line and can be graphed to show negative correlation.
2x + 3y = 6
This linear equation can be converted to slope-intercept form. It represents a straight line and is key in understanding systems of equations.
x = 4 (Vertical Line)
This equation represents a vertical line where all points have an x-coordinate of 4. Understanding vertical lines is essential in coordinate geometry.
y = -3 (Horizontal Line)
A horizontal line equation indicating all points have a y-coordinate of -3, demonstrating basic line equations.
x + y = 5
This is a simple linear equation. It can be graphed to find intercepts, providing a basis for visualizing relationships.
y = 3x - 1
Represents a line with a slope of 3 and a y-intercept at -1, helpful for visualizing steep rising lines.
4x - y = 8
This equation can be rearranged to find the slope and intercepts, essential for plotting the line effectively.
y - y₁ = m(x - x₁) with (x₁,y₁) = (1,2), m=2
Substituting values into point-slope form provides the equation of the line passing through point (1,2) with slope 2.
1 + m₁*m₂ = 0 when m₁ and m₂ are slopes of two lines
Indicates that the two lines are perpendicular if this condition holds, foundational for understanding line orientations.
