Equation of a Perpendicular Line: Step-by-Step Guide with Examples
Finding the equation of a perpendicular line is one of those skills that comes up in geometry, algebra, and standardized tests more often than students expect. Two lines are perpendicular when they meet at a 90° angle, and that geometric fact translates directly into an algebraic rule about their slopes. Once you know that rule — and how to apply it through point-slope form — writing the equation of a perpendicular line becomes a routine process. This guide walks through the theory, the steps, and multiple worked examples so you can handle any perpendicular line problem that comes your way.
Contents
- 01What Makes Two Lines Perpendicular?
- 02How to Find the Negative Reciprocal of a Slope
- 03How to Find the Equation of a Perpendicular Line: 5-Step Method
- 04Worked Example 1: Perpendicular to a Whole-Number Slope
- 05Worked Example 2: Perpendicular to a Line in Standard Form
- 06Worked Example 3: Perpendicular to a Negative Fractional Slope
- 07Special Cases: Perpendicular to Horizontal and Vertical Lines
- 08Common Mistakes to Avoid
- 09Practice Problems with Full Solutions
- 10Where Perpendicular Line Equations Are Used
- 11Frequently Asked Questions
What Makes Two Lines Perpendicular?
Two lines are perpendicular when they intersect at exactly 90°. You see this everywhere in real life — the corner of a page, a floor meeting a wall, a street crossing at a right angle. In coordinate geometry, perpendicularity has a precise algebraic meaning that lets you work with it using equations and slope values rather than a protractor. The key fact is this: if line 1 has slope m₁ and line 2 is perpendicular to it, then the slope of line 2 is the negative reciprocal of m₁. Written as a formula: m₂ = −1 ÷ m₁, or equivalently, m₁ × m₂ = −1. That product of −1 is the quick test for perpendicularity — multiply the two slopes together and if you get −1, the lines are perpendicular. This rule applies to every pair of perpendicular lines on the coordinate plane, except the special case of horizontal and vertical lines (which are perpendicular to each other but have slopes of 0 and undefined, respectively — covered at the end of this guide).
If line 1 has slope m₁ and line 2 is perpendicular to line 1, then m₁ × m₂ = −1. The slopes are negative reciprocals of each other.
How to Find the Negative Reciprocal of a Slope
The negative reciprocal is the foundation of every perpendicular line equation problem. Finding it takes two operations: flip the fraction (take the reciprocal) and change the sign (negate). You must do both — doing only one gives the wrong slope and a line that is not perpendicular.
1. Step 1 — Write the slope as a fraction
If the slope is a whole number, write it over 1. Slope = 3 becomes 3/1. Slope = −5 becomes −5/1. If it is already a fraction, like 2/7, leave it as is.
2. Step 2 — Flip the fraction (take the reciprocal)
Swap numerator and denominator. 3/1 becomes 1/3. −5/1 becomes −1/5. 2/7 becomes 7/2. −3/4 becomes −4/3.
3. Step 3 — Change the sign (negate)
If the reciprocal is positive, make it negative. If it is negative, make it positive. • 1/3 becomes −1/3 • −1/5 becomes +1/5 • 7/2 becomes −7/2 • −4/3 becomes +4/3
4. Step 4 — Verify with multiplication
Multiply original slope × perpendicular slope. The product must equal −1. • 3 × (−1/3) = −1 ✓ • −5 × (1/5) = −1 ✓ • 2/7 × (−7/2) = −14/14 = −1 ✓ • −3/4 × (4/3) = −12/12 = −1 ✓
Quick pattern: if a slope is a/b, the perpendicular slope is −b/a. Flip and negate in one step.
How to Find the Equation of a Perpendicular Line: 5-Step Method
To write the equation of a perpendicular line, you need two pieces of information: the slope of the original line (so you can calculate the perpendicular slope) and a specific point the new line must pass through. With those in hand, point-slope form does the work.
1. Step 1 — Find the slope of the original line
If the line is given as y = mx + b, the slope is m — read it directly. If the line is in standard form Ax + By = C, rearrange to slope-intercept form first: y = (−A/B)x + (C/B), giving slope m = −A/B.
2. Step 2 — Calculate the perpendicular slope
Take the slope from Step 1, flip the fraction, and negate the sign. This is the slope of the perpendicular line, m⊥. Verify: original slope × m⊥ should equal −1.
3. Step 3 — Plug into point-slope form
Use the formula y − y₁ = m⊥(x − x₁), where (x₁, y₁) is the given point the perpendicular line passes through and m⊥ is the perpendicular slope from Step 2.
4. Step 4 — Simplify to slope-intercept form
Distribute m⊥, then isolate y. Collect like terms to reach y = m⊥x + b. If the problem asks for standard form (Ax + By = C), move the x-term to the left and clear fractions by multiplying through by the denominator.
5. Step 5 — Check your answer
Substitute the given point into your equation — both sides should be equal. Then multiply the two slopes: original × perpendicular. The result must be −1. If either check fails, review Steps 2 or 3 first, as those are where most errors occur.
The equation of a perpendicular line always uses the negative reciprocal slope. No other slope produces a 90° intersection.
Worked Example 1: Perpendicular to a Whole-Number Slope
Problem: Find the equation of a perpendicular line to y = 2x + 5 that passes through the point (4, 1). This is the most straightforward type — the original slope is a whole number, so the perpendicular slope is a simple fraction.
1. Step 1 — Identify the original slope
The equation y = 2x + 5 is in slope-intercept form. The slope is m = 2.
2. Step 2 — Find the perpendicular slope
Write 2 as 2/1. Flip to 1/2. Negate: m⊥ = −1/2. Verify: 2 × (−1/2) = −1 ✓
3. Step 3 — Point-slope form with (4, 1)
y − 1 = −1/2 · (x − 4)
4. Step 4 — Simplify
y − 1 = −1/2 · x + 2 y = −1/2 · x + 2 + 1 y = −1/2 · x + 3
5. Step 5 — Verify
Check the point: y = −1/2 · (4) + 3 = −2 + 3 = 1 ✓ Check slopes: 2 × (−1/2) = −1 ✓ Final answer: y = −½x + 3
Answer: y = −½x + 3. This line passes through (4, 1) and meets y = 2x + 5 at a right angle.
Worked Example 2: Perpendicular to a Line in Standard Form
Problem: Find the equation of a perpendicular line to 3x − 4y = 12 that passes through (−3, 2). Standard form requires an extra conversion step before you can identify the slope. This is where students often make their first error — trying to guess the slope from the coefficients without converting properly.
1. Step 1 — Convert to slope-intercept form
3x − 4y = 12 Subtract 3x from both sides: −4y = −3x + 12 Divide every term by −4: y = (3/4)x − 3 The slope of the original line is m = 3/4.
2. Step 2 — Find the perpendicular slope
Slope is 3/4. Flip to 4/3. Negate: m⊥ = −4/3. Verify: (3/4) × (−4/3) = −12/12 = −1 ✓
3. Step 3 — Point-slope form with (−3, 2)
y − 2 = −4/3 · (x − (−3)) y − 2 = −4/3 · (x + 3)
4. Step 4 — Simplify
y − 2 = −4/3 · x − 4/3 · 3 y − 2 = −4/3 · x − 4 y = −4/3 · x − 4 + 2 y = −4/3 · x − 2
5. Step 5 — Verify
Check the point (−3, 2): y = −4/3 · (−3) − 2 = 4 − 2 = 2 ✓ Check slopes: (3/4) × (−4/3) = −1 ✓ Final answer: y = −⁴⁄₃x − 2
When a line is in standard form Ax + By = C, always convert to y = mx + b first. The slope is −A/B, not A or B alone.
Worked Example 3: Perpendicular to a Negative Fractional Slope
Problem: Find the equation of a perpendicular line to y = −2/3 · x + 1 that passes through (−4, 5). This example illustrates a useful pattern: when the original slope is negative, the perpendicular slope comes out positive. Two negatives cancel during the negation step.
1. Step 1 — Identify the original slope
The slope is m = −2/3 (read directly from slope-intercept form).
2. Step 2 — Find the perpendicular slope
Slope is −2/3. Flip the fraction: −3/2. Negate: −(−3/2) = +3/2. So m⊥ = 3/2. Verify: (−2/3) × (3/2) = −6/6 = −1 ✓ Note how the original negative slope becomes a positive perpendicular slope. This is not a mistake — it is expected when you negate a negative number.
3. Step 3 — Point-slope form with (−4, 5)
y − 5 = 3/2 · (x − (−4)) y − 5 = 3/2 · (x + 4)
4. Step 4 — Simplify
y − 5 = 3/2 · x + 3/2 · 4 y − 5 = 3/2 · x + 6 y = 3/2 · x + 11
5. Step 5 — Verify
Check the point (−4, 5): y = 3/2 · (−4) + 11 = −6 + 11 = 5 ✓ Check slopes: (−2/3) × (3/2) = −1 ✓ Final answer: y = ³⁄₂x + 11
Pattern: when the original slope is negative, the perpendicular slope is positive. When the original slope is positive, the perpendicular slope is negative. They always have opposite signs.
Special Cases: Perpendicular to Horizontal and Vertical Lines
Horizontal lines (y = k, slope = 0) and vertical lines (x = h, slope undefined) are perpendicular to each other. They do not fit the negative reciprocal formula because you cannot take the reciprocal of 0 or of an undefined value. Instead, remember these two rules directly: the perpendicular to a horizontal line is vertical, and the perpendicular to a vertical line is horizontal.
1. Perpendicular to a horizontal line y = 3 through the point (5, 7)
y = 3 is a horizontal line. Any line perpendicular to a horizontal line is vertical. The vertical line through (5, 7) is x = 5. All points on this line have x-coordinate 5, regardless of y. It includes (5, 7), (5, 0), (5, −10), etc.
2. Perpendicular to a vertical line x = −2 through the point (3, 6)
x = −2 is a vertical line. Any line perpendicular to a vertical line is horizontal. The horizontal line through (3, 6) is y = 6. All points on this line have y-coordinate 6, regardless of x.
Perpendicular to a horizontal line → vertical line (x = constant). Perpendicular to a vertical line → horizontal line (y = constant).
Common Mistakes to Avoid
Most errors in perpendicular line problems come from a handful of predictable sources. Recognizing these mistakes ahead of time is the most efficient way to avoid them on a test.
1. Mistake 1: Only negating, not flipping (or vice versa)
If the slope is 3, the perpendicular slope is NOT −3 (only negated, not flipped). It is also NOT 1/3 (only flipped, not negated). You must do both. The correct perpendicular slope is −1/3. Quick check: 3 × (−3) = −9 ≠ −1. 3 × (1/3) = 1 ≠ −1. Only 3 × (−1/3) = −1 ✓.
2. Mistake 2: Reading the slope from standard form without converting
In Ax + By = C, the slope is NOT A or the coefficient of x alone. For 3x − 4y = 12, the slope is found by converting: y = (3/4)x − 3, so m = 3/4. Skipping the conversion and reading m = 3 directly from the original equation produces a completely wrong perpendicular slope.
3. Mistake 3: Using the wrong point in point-slope form
The point you substitute into y − y₁ = m⊥(x − x₁) must be the specific point the new perpendicular line passes through — as stated in the problem. Do not accidentally use a point that lies on the original line instead.
4. Mistake 4: Fraction arithmetic errors when distributing
When m⊥ is a fraction like −4/3, multiplying by (x + 3) means −4/3 × 3 = −4 (not −4/3). Simplify each multiplication separately. Write out −4/3 × x and −4/3 × 3 as two distinct steps before combining.
5. Mistake 5: Skipping the verification step
Substituting the given point takes 20 seconds and catches the majority of errors. If the given point is (−3, 2) and your equation does not produce y = 2 when x = −3, something went wrong — revisit Steps 2 through 4 before writing a final answer.
Practice Problems with Full Solutions
Work through each problem on your own before reading the solution. Start with Problems 1 and 2 (whole-number slopes) before moving to the fraction and standard-form problems.
1. Problem 1
Find the equation of a perpendicular line to y = 4x − 7 that passes through (8, −3). Solution: m = 4, so m⊥ = −1/4 (flip 4/1 to 1/4, then negate) Point-slope: y − (−3) = −1/4 · (x − 8) y + 3 = −1/4 · x + 2 y = −1/4 · x − 1 Check point: −1/4 · (8) − 1 = −2 − 1 = −3 ✓ Check slopes: 4 × (−1/4) = −1 ✓ Answer: y = −¼x − 1
2. Problem 2
Find the equation of a perpendicular line to y = −3x + 2 that passes through (−6, 4). Solution: m = −3, so m⊥ = 1/3 (flip −3/1 to −1/3, then negate the negative to get +1/3) Point-slope: y − 4 = 1/3 · (x − (−6)) y − 4 = 1/3 · (x + 6) y − 4 = 1/3 · x + 2 y = 1/3 · x + 6 Check point: 1/3 · (−6) + 6 = −2 + 6 = 4 ✓ Check slopes: (−3) × (1/3) = −1 ✓ Answer: y = ⅓x + 6
3. Problem 3
Find the equation of a perpendicular line to 5x + 2y = 10 that passes through (0, −4). Solution: Convert to slope-intercept: 2y = −5x + 10 → y = −5/2 · x + 5. So m = −5/2. m⊥: flip −5/2 to −2/5, negate to +2/5 Point-slope with (0, −4): y − (−4) = 2/5 · (x − 0) y + 4 = 2/5 · x y = 2/5 · x − 4 Check point: 2/5 · (0) − 4 = −4 ✓ Check slopes: (−5/2) × (2/5) = −10/10 = −1 ✓ Answer: y = ²⁄₅x − 4
4. Problem 4 (Challenge)
Find the equation of a perpendicular line to 2x − 7y = 14 that passes through (2, −1). Write the answer in standard form. Solution: Convert: −7y = −2x + 14 → y = 2/7 · x − 2. So m = 2/7. m⊥ = −7/2 Point-slope with (2, −1): y − (−1) = −7/2 · (x − 2) y + 1 = −7/2 · x + 7 y = −7/2 · x + 6 Convert to standard form: multiply every term by 2 to clear fractions: 2y = −7x + 12 7x + 2y = 12 Check point: 7(2) + 2(−1) = 14 − 2 = 12 ✓ Answer: 7x + 2y = 12
After solving, always substitute the given point back into your equation. One 20-second check catches the majority of mistakes before they cost marks.
Where Perpendicular Line Equations Are Used
The equation of a perpendicular line is not just an isolated textbook skill — it appears in several places across geometry and algebra courses where you may not immediately recognize it. Shortest distance from a point to a line: The shortest path from a point P to a line L is along the perpendicular from P to L. To find that distance, you write the equation of a perpendicular line through P, find the intersection with L, and then compute the distance between P and the intersection point. Altitudes in triangles: An altitude of a triangle runs from a vertex perpendicular to the opposite side. Finding where an altitude meets a side requires writing the equation of a perpendicular line from the vertex to that side. Proving rectangles and right angles: If you need to show that two sides of a quadrilateral are perpendicular, compute their slopes and verify the product is −1. This proof technique relies directly on the perpendicular slope rule. Graphing reflections: When reflecting a point across a line, the perpendicular from the point to the line gives the direction of the reflection. The reflection point is equidistant from the line along that perpendicular.
Any problem that mentions 'shortest distance from a point to a line' or 'altitude of a triangle' is almost certainly asking you to find the equation of a perpendicular line.
Frequently Asked Questions
These are the questions students ask most often when first working with perpendicular line equations.
1. Q: How do I know which slope belongs to which line?
The original line is whatever line the problem gives you — read its slope from its equation. The perpendicular line is the one you are finding — its slope is the negative reciprocal of the original. Label them clearly: m_original and m⊥ so you do not mix them up.
2. Q: Can two perpendicular lines have the same y-intercept?
Yes. The y-intercept depends on where the line crosses the y-axis, which is determined by the given point — not by the slope alone. If the perpendicular line happens to pass through a point on the y-axis, the two lines will share a y-intercept. Their slopes will still be negative reciprocals.
3. Q: What is the difference between a parallel line equation and a perpendicular line equation?
For a parallel line, the slope stays the same — you just change the y-intercept to pass through the new point. For a perpendicular line, the slope changes to the negative reciprocal. In both cases, you use point-slope form with the given point; the only difference is which slope value you substitute.
4. Q: What if the problem asks for the perpendicular bisector?
A perpendicular bisector is a perpendicular line that also passes through the midpoint of a segment. Find the midpoint of the given segment using the midpoint formula: ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2). Then use that midpoint as your given point and follow the same 5 steps to find the equation of a perpendicular line.
5. Q: How do I convert the perpendicular line equation to standard form?
Once you have y = m⊥x + b, move the x-term to the left: −m⊥x + y = b. If m⊥ is a fraction like −4/3, multiply every term by the denominator (3) to clear fractions: 4x + 3y = 3b. Then check that the coefficient of x is positive — if not, multiply through by −1.
Related Articles
How to Find the Equation of a Line: 4 Methods with Worked Examples
Master all four methods for writing a line equation — slope-intercept, point-slope, two-point, and standard form — with full worked examples.
How to Graph a Linear Equation Step by Step
Learn how to plot any linear equation on the coordinate plane using slope and y-intercept — a key visual skill for understanding perpendicular lines.
Geometry Problems: Practice with Lines, Angles, and Shapes
Work through geometry problems involving lines, angles, and coordinate geometry, including problems that use perpendicular lines.
Related Math Solvers
Step-by-Step Solutions
Get detailed explanations for every step, not just the final answer.
Smart Scan Solver
Snap a photo of any math problem and get an instant step-by-step solution.
Concept Explainer
Understand the 'why' behind every formula with deep concept breakdowns.