Perpendicular: what it means, with perpendicular lines and slopes

Perpendicular means at a right angle (90°) to something. Perpendicular lines are two lines that cross or meet at a right angle, like the edges at the corner of a page. The symbol is ⊥, so AB ⊥ CD means line AB is perpendicular to line CD. On a graph, two lines are perpendicular when their slopes multiply to make −1.

What does perpendicular mean?

Two lines, line segments or rays are perpendicular if the angle between them is exactly 90°. Diagrams show this with a small square in the corner instead of an arc. That square is a promise: you may assume the angle is 90° without measuring it.

Perpendicular lines are everywhere: the x and y axes on a graph, the sides of a square or rectangle, the two shorter sides of a right-angled triangle (right triangle in the US), a wall and the floor, or two streets at a crossroads. When two perpendicular lines cross, they make four right angles, not just one.

You will also meet the word on its own. The perpendicular height of a triangle is measured at 90° to the base, and it is the height in the area formula, area = ½ × base × height. A perpendicular from a point to a line is the line segment that meets the line at 90°, and it is the shortest way from the point to the line.

Perpendicular, parallel and intersecting lines

Perpendicular lines are a special kind of intersecting lines: every pair of perpendicular lines intersects, but most intersecting lines are not perpendicular.

Slopes of perpendicular lines

The slope (gradient in the UK) of a line says how steep it is: the rise divided by the run between any two points. For points (x₁, y₁) and (x₂, y₂):

Everything else on this page builds on finding a slope, so try one first. Press Mark to check your answer in the course.

Here is the rule. Two lines with slopes m₁ and m₂ are perpendicular when:

In words, the perpendicular slope is the negative reciprocal: flip the fraction upside down and change the sign. One of the two slopes is always positive and the other negative, because one line goes uphill and the other downhill.

Horizontal and vertical lines

A horizontal line, such as y = 4, has slope 0. A vertical line, such as x = −2, has an undefined slope, because the run is zero. The rule m₁ × m₂ = −1 can't be used on them, but they are still perpendicular: every horizontal line is perpendicular to every vertical line.

Why the slopes multiply to −1

A line with slope m goes 1 across and m up. Turn that step through 90° anticlockwise (counterclockwise in the US) and it becomes m to the left and 1 up, so the new slope is 1 ÷ (−m) = −1/m. Multiply the two slopes and you get m × (−1/m) = −1. The US Common Core standards ask high school students to prove this slope rule and use it, and AQA's GCSE Maths specification asks students to use y = mx + c to identify perpendicular lines.

Check yourself before the examples.

How to tell if two lines are perpendicular

1. Get each equation into the form y = mx + b (y = mx + c in the UK), so you can read off the slope m.
2. Multiply the two slopes.
3. If the answer is −1, the lines are perpendicular. If the slopes are equal, they are parallel. Anything else, they just intersect.

Example 1: two lines in slope-intercept form

Are y = 3x + 2 and y = −⅓x + 5 perpendicular? The slopes are 3 and −1/3, and 3 × (−1/3) = −1. Yes, they are perpendicular. The y-intercepts make no difference: they only decide where the lines cross.

Example 2: a line in standard form

Find the slope of a line perpendicular to 2x + 5y = 10. First make y the subject:

The slope is −2/5. Flip it and change the sign: the perpendicular slope is 5/2. A shortcut: a line Ax + By = C has slope −A/B, and any line Bx − Ay = D is perpendicular to it.

How to find the equation of a perpendicular line

Questions usually give you a line and a point, and ask for the line through the point that is perpendicular to the first one.

1. Find the slope of the given line.
2. Take the negative reciprocal to get the perpendicular slope.
3. Put that slope and the point into y − y₁ = m(x − x₁), the point-slope form.
4. Rearrange into the form the question asks for.

Example 3: a perpendicular line through a point

Find the equation of the line perpendicular to y = 4x − 1 that passes through (8, 3).

The given slope is 4, so the perpendicular slope is −1/4. Using point-slope form:

The answer is y = −¼x + 5. Check: when x = 8, y = −2 + 5 = 3, so the line does pass through (8, 3). Plot both lines in the linear graph calculator to see the right angle.

Example 4: is the triangle right-angled?

A triangle has corners A(1, 1), B(4, 3) and C(2, 6). Does it have a right angle?

Find the slope of each side. AB is (3 − 1) ÷ (4 − 1) = 2/3. BC is (6 − 3) ÷ (2 − 4) = −3/2. AC is (6 − 1) ÷ (2 − 1) = 5. Now 2/3 × (−3/2) = −1, so sides AB and BC meet at 90°, making B the right angle.

The perpendicular bisector

The perpendicular bisector of a line segment cuts it in half at a right angle. Every point on it is the same distance from both ends of the segment, which is why it comes up in loci questions.

Constructing it with a ruler and compasses

1. Open the compasses to more than half the length of the segment AB.
2. Put the point on A and draw an arc above and below the segment.
3. Without changing the width, put the point on B and draw two more arcs that cross the first ones.
4. Draw a straight line through the two crossing points. Leave your arcs showing: they are your working.

In England, GCSE Maths students must know this construction and how to construct a perpendicular to a line from or at a given point, and that the perpendicular distance from a point to a line is the shortest distance to it.

Example 5: the equation of a perpendicular bisector

Find the perpendicular bisector of A(2, 1) and B(6, 9).

• Midpoint: ((2 + 6) ÷ 2, (1 + 9) ÷ 2) = (4, 5).
• Slope of AB: (9 − 1) ÷ (6 − 2) = 2, so the perpendicular slope is −1/2.
• Equation: y − 5 = −½(x − 4), so y = −½x + 7.

The perpendicular bisector is y = −½x + 7. The midpoint and distance calculator checks the midpoint step.

Perpendicular distance from a point to a line

When a question asks how far a point is from a line, it means the perpendicular distance. To find it, draw the perpendicular from the point, find where it meets the line, and measure. Take the point (5, 2) and the line y = 2x. The perpendicular has slope −1/2 and passes through (5, 2), so it is y = −½x + 4.5. It meets y = 2x where 2x = −½x + 4.5, which gives x = 1.8 and y = 3.6. The distance from (5, 2) to (1.8, 3.6) is:

There is also a formula. For the line ax + by + c = 0 and the point (x₀, y₀), the distance is |ax₀ + by₀ + c| ÷ √(a² + b²). Here the line is 2x − y = 0, so the distance is |10 − 2| ÷ √5 = 8/√5 ≈ 3.58, the same answer.

Common mistakes

• Only flipping, or only changing the sign. The perpendicular slope to 2 is −1/2, not 1/2 and not −2. Do both.
• Reading the slope from the wrong form. In 2x + 5y = 10 the slope is not 2. Rearrange to y = mx + b first.
• Thinking lines that look square are perpendicular. On a graph with different scales on the axes, a right angle can look wrong and a wrong angle can look right. Check the slopes.
• Using the rule on vertical lines. A vertical line has no slope to multiply. Horizontal and vertical lines are perpendicular without it.
• Measuring a distance on a slant. The shortest route from a point to a line meets the line at 90°, so measure along that.

To practise this with other topics in a short timed paper, try today's Geometry, Algebra 2 & SAT Math or Pre-Algebra & Algebra 1 paper. In the UK, the same practice opens as GCSE Maths Higher.

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