Perpendicular Line Equation Calculator

What is a Perpendicular Line Through a Point?

A perpendicular line is a line that intersects another line at a 90-degree angle. When we say "through a point," we mean that this perpendicular line passes through a specific point while maintaining that 90-degree angle with the original line. It's like drawing two lines that form a perfect "L" shape, with one line going through an exact point you choose!

How to Calculate a Perpendicular Line Through a Point

To find a perpendicular line through a point, we use the fact that perpendicular lines have slopes that are negative reciprocals of each other. If the original line has slope \(m\), the perpendicular line will have slope \(-\frac{1}{m}\). We then use this slope and our point to create the new line equation.

Formula

Given a line in the form: \(Ax + By + C = 0\)

The perpendicular line through point \((x_0, y_0)\) will be:

\[ Bx - Ay + D = 0 \]

Where:

• \(A\) and \(B\) are from the original line equation
• \(D = -(Bx_0 - Ay_0)\) makes the line pass through our point
• \(x\) and \(y\) are coordinates of any point on the new line

Calculation Steps

1. Start with original line: \(Ax + By + C = 0\)
2. Identify your point: \((x_0, y_0)\)
3. Swap and negate coefficients: \(B\) becomes new \(A\), \(-A\) becomes new \(B\)
4. Calculate \(D = -(Bx_0 - Ay_0)\)
5. Write final equation: \(Bx - Ay + D = 0\)

Interactive Example

Let's find a line perpendicular to \(2x + y - 4 = 0\) passing through point (2, 3).

Step-by-step solution:

1. Original line: \(2x + y - 4 = 0\) (blue line)
2. Point: (2, 3) (red dot)
3. Perpendicular slope: \(-\frac{2}{1} = -2\)
4. New coefficients: \(A = 1\), \(B = -2\)
5. Calculate D: \(D = -(1(2) - 2(3)) = 4\)
6. Final equation: \(x - 2y + 4 = 0\) (green dashed line)

The graph shows how the perpendicular line (green) intersects the original line (blue) at a perfect 90-degree angle while passing through our point (2, 3). The angle marker confirms this perpendicular relationship!