Perpendicular Line Calculator
Find the equation of a line perpendicular to a given line passing through a specific point
Calculate Perpendicular Line
Original Line Definition
Point the Perpendicular Line Passes Through
Line Equations and Properties
Original Line
Equation:
y = 2x -2
Standard Form:
2x - y -2 = 0
Slope: 2.0000
Perpendicular Line
Equation:
y = -0.5x +6.5
Standard Form:
-0.5x - y +6.5 = 0
Slope: -0.5000
Passes through: (3, 5)
Intersection Point
(3.4000, 4.8000)
Point where the two lines meet at a 90° angle
Perpendicularity Verification
Slope product: 2.0000 × -0.5000 = -1.0000 ≈ -1
Step-by-Step Solution
1. Identify the slope of the original line
Original line: y = 2x -2
Slope (m) = 2.0000
2. Calculate the perpendicular slope
For perpendicular lines, the product of their slopes equals -1
Perpendicular slope = -1 / 2.0000 = -0.5000
3. Find the equation using the given point
Point: (3, 5)
Using y = mx + b: 5 = -0.5000 × 3 + b
b = 5 - -0.5000 × 3 = 6.5000
4. Write the equation of the perpendicular line
Perpendicular line: y = -0.5x +6.5
5. Find the intersection point
Solve the system of equations:
y = 2x -2
y = -0.5x +6.5
Intersection point: (3.4000, 4.8000)
Example Calculation
Find Perpendicular Line
Given line: y = 2x - 2
Point: (3, 5)
Step 1: Original slope = 2
Step 2: Perpendicular slope = -1/2 = -0.5
Step 3: 5 = -0.5(3) + b → b = 6.5
Perpendicular line: y = -0.5x + 6.5
Intersection: (3.4, 4.8)
Perpendicular Lines Properties
Negative Reciprocal Slopes
Product of slopes equals -1
90° Intersection
Meet at right angles
Always Intersect
Unlike parallel lines
Special Cases
Vertical ⊥ Horizontal
Calculator Tips
Slope of perpendicular line = -1/original slope
Vertical lines are perpendicular to horizontal lines
Use any point to find y-intercept
Verify: slope product should equal -1
Understanding Perpendicular Lines
What are Perpendicular Lines?
Perpendicular lines are straight lines that intersect at a right angle (90°). They have slopes that are negative reciprocals of each other, meaning their product equals -1.
Key Characteristics
- •Negative reciprocal slopes: If one slope is m, the other is -1/m
- •Right angle intersection: They meet at exactly 90°
- •Always intersect: Unlike parallel lines, they always meet
- •Slope product: m₁ × m₂ = -1 (when both slopes exist)
Mathematical Formulas
Perpendicular Slope
a = -1/m
Where m is the original line's slope
Y-Intercept Formula
b = y₀ + x₀/m
Using point (x₀, y₀) that the line passes through
Verification
m₁ × m₂ = -1
Product of slopes for perpendicular lines
Special Cases
Vertical ⊥ Horizontal
x = a perpendicular to y = b
Zero Slope
Horizontal line ⊥ Vertical line
Equal Absolute Values
Slopes like 2 and -1/2
Real-World Applications
Construction
Building corners, foundation layouts, structural supports
Engineering
Mechanical joints, electrical circuits, stress analysis
Navigation
GPS coordinates, mapping systems, surveying