Quick answer

Enter m, r, x, y → get a, b, perpendicular equation, and intersection instantly.

Formula

  • Inputs: m, r, point (x, y)
  • Outputs: a, b, intersection (x, y)

Introduction

The Perpendicular Line Calculator lives on the home page directly under the hero section, so you can calculate before you scroll through formulas and FAQs.

It targets the common textbook setup: one line in slope-intercept form and one external point that defines the perpendicular direction.

If you want to understand the algebra behind the screen, read the step-by-step calculation guide first, then return here for button-by-button context.

Numbers from the tool match the longer cases in our worked examples collection, which is useful when you want paper practice with known answers.

What the calculator does

It computes a = -1/m, finds b, prints y = ax + b (or x = k for horizontal bases), and solves for the crossing point.

All processing happens locally in the browser. Nothing is uploaded to a server.

Results update as you type, which makes it easy to test what happens when you change only the point or only the slope.

The layout separates base line inputs, perpendicular output, and intersection so you can read each stage the way a teacher marks homework.

Formula and relationships

  • Inputs: m, r, point (x, y)
  • Outputs: a, b, intersection (x, y)

The logic follows the same algebra as the formula article; only arithmetic is automated.

When m is zero, the tool shows a vertical perpendicular instead of forcing a slope that would divide by zero.

Intersection coordinates use the same solve step you would apply by hand: equate the two expressions for y.

Step-by-step guide

  1. Enter m and r. Define y = mx + r for the line you are crossing at a right angle.
  2. Enter the point. Type the x and y values the perpendicular must pass through.
  3. Read slope a and intercept b. These fields populate when all four inputs are valid numbers.
  4. Read the intersection. Coordinates show where the two lines meet in the plane.
  5. Compare with hand work. Redo one result manually each study session to stay exam-ready.

Worked examples

Try m = 2, r = 1, point (3, 4). You should see a = -0.5, b = 5.5, and intersection (1.8, 4.6).

Change only the point and watch how b and the intersection shift while a depends solely on m.

Horizontal base m = 0 with r = 4 and point (2, 7) should return x = 2 as the perpendicular line form.