Perpendicular Lines: Definition, Examples, Slopes, Equations and Theorems

Perpendicular lines are lines that intersect at exactly 90°. The symbol for perpendicular is ⊥. In coordinate geometry, two non-vertical lines are perpendicular when their slopes are negative reciprocals, which means their slopes multiply to −1.

Perpendicular lines are one of the most important relationships in geometry. They help students recognize right angles, interpret diagrams, solve coordinate problems, construct shapes accurately, and prove geometric theorems. You can find perpendicular lines in squares, rectangles, coordinate axes, triangle altitudes, tangent circles, building corners, and grid-based designs.

Perpendicular Lines at a Glance

Concept	Rule or meaning
Perpendicular lines	Lines that meet at a right angle
Perpendicular angle	90°
Symbol	⊥
Non-vertical slope rule	m1 × m2 = −1
Horizontal line slope	0
Vertical line slope	Undefined
Horizontal and vertical lines	Perpendicular when they intersect
Perpendicular bisector	Divides a segment equally at 90°
Common examples	Coordinate axes, rectangle corners and square diagonals

What Are Perpendicular Lines?

Perpendicularity describes two lines, line segments, or rays that intersect to form a right angle of 90°.

If line AB and line CD meet at a right angle, write:

AB ⊥ CD

The point where they meet is called the point of intersection.

In geometry diagrams, a small square at a corner is the conventional notation showing that the marked angle is a right angle, which measures 90°. This mark gives direct evidence that the sides or lines forming the angle are perpendicular.

The symbol ⊥ describes the perpendicular relationship. Depending on the diagram, it may refer to complete lines, rays, or line segments.

Important: All perpendicular lines intersect, but not every pair of intersecting lines is perpendicular. The intersection must form an exact 90° angle.

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Perpendicular Lines vs Intersecting Lines

Intersecting lines meet at a point. Their angle can be small, wide, acute, obtuse, or right. Perpendicular lines are a special type of intersecting line because they meet at exactly 90°.

Type of lines	Do they meet?	Angle condition
Intersecting lines	Yes	Can form any angle
Perpendicular lines	Yes	Must form 90°
Parallel lines	No	Do not meet in the same plane

For example, two lines that cross at 45° are intersecting but not perpendicular. Lines that cross at 90° are intersecting and perpendicular.

This distinction is useful in exams. A diagram may look like it contains a right angle, but you should not assume perpendicularity unless the diagram includes a right-angle marker, a stated measurement, coordinate evidence, or a known shape property.

Perpendicular Lines Examples in Geometry

Perpendicular lines appear in many familiar figures and coordinate systems.

Coordinate Axes

The x-axis is horizontal, while the y-axis is vertical. They meet at the origin, (0, 0), forming a right angle.

Therefore:

x-axis ⊥ y-axis

This is one of the clearest examples of perpendicular lines in geometry and coordinate graphs.

Adjacent Sides of a Rectangle

A rectangle has four right angles. Each pair of adjacent sides is perpendicular.

If ABCD is a rectangle:

AB ⊥ BC

BC ⊥ CD

CD ⊥ DA

DA ⊥ AB

The opposite sides of a rectangle are parallel, not perpendicular.

Adjacent Sides and Diagonals of a Square

A square has four equal sides and four right angles. Therefore, all neighbouring sides are perpendicular.

A square’s diagonals are also perpendicular. They cross at the centre of the square and form four right angles.

Triangle Altitudes

An altitude of a triangle is a line segment drawn from one vertex perpendicular to the opposite side or the line containing that side.

In a right triangle, the two sides that form the right angle are perpendicular. In an acute or obtuse triangle, altitudes are useful for finding an area and locating the orthocenter.

Radius and Tangent of a Circle

A tangent touches a circle at exactly one point. The radius drawn to that point of contact is perpendicular to the tangent.

If radius OP meets tangent line l at point P:

OP ⊥ l

This relationship is frequently used in circle theorems and geometry proofs.

How to Identify Perpendicular Lines

The best method depends on the information in the question. Do not use a more complicated method when a simpler proof is available.

Given in the question	Best method
Right-angle marker	Identify the marked 90° angle
Angle measurement	Check whether the angle equals 90°
Two line equations	Compare the slopes
Two coordinate pairs	Calculate each slope
Horizontal and vertical lines	Use their orientation
Named shape	Apply its known properties
Segment and midpoint	Check perpendicular-bisector conditions

Use a Right-Angle Marker

A small square in a diagram proves that the marked angle is 90°. The two lines, rays, or segments forming that angle are perpendicular.

Measure the Angle

When the diagram has no right-angle mark, use a protractor. Place the centre of the protractor on the vertex and align the baseline with one side. If the other side passes through the 90° mark, the angle is a right angle.

Use Known Shape Properties

Geometry shapes have established properties that can prove perpendicularity.

• Adjacent sides of a rectangle are perpendicular.
• Adjacent sides of a square are perpendicular.
• Diagonals of a square are perpendicular.
• Diagonals of a rhombus are perpendicular.
• A right triangle has one pair of perpendicular sides.
• A radius and tangent are perpendicular at the point of contact.

Important: Not every quadrilateral has perpendicular diagonals. Squares and rhombuses have perpendicular diagonals, but a general parallelogram and most rectangles do not.

Perpendicular Lines and Slope

Slope measures how steep a line is and whether it rises or falls from left to right. In coordinate geometry, slope is one of the most reliable ways to prove that two non-vertical lines are perpendicular.

For points (x1, y1) and (x2, y2), use the slope formula:

m = (y2 − y1) ÷ (x2 − x1)

For two non-vertical perpendicular lines:

m1 × m2 = −1

This means their slopes are negative reciprocals.

What Is a Negative Reciprocal?

To find the perpendicular slope of a line:

1. Write the original slope as a fraction.
2. Reverse the numerator and denominator.
3. Change the sign.

Original slope	Perpendicular slope
2	−1/2
−5	1/5
3/4	−4/3
−2/7	7/2
1/6	−6
−3	1/3

For example, the negative reciprocal of 2/5 is −5/2.

The negative reciprocal of −4 is 1/4. Write −4 as −4/1, reverse the fraction to get 1/4, then change the negative sign to positive.

Important Limitation of the Slope Rule

The rule m1 × m2 = −1 works only when both lines are non-vertical.

A vertical line has an undefined slope, so it cannot be used in the slope multiplication rule. A horizontal line has slope 0. However, a horizontal line and a vertical line are perpendicular whenever they intersect.

For example:

• y = 4 is a horizontal line.
• x = −2 is a vertical line.

They meet at (−2, 4), so they are perpendicular.

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Example: Test Slopes for Perpendicularity

Consider the two equations:

y = 2x + 5

y = −1/2x + 1

The slope of the first line is 2. The slope of the second line is −1/2.

Now multiply the slopes:

2 × −1/2 = −1

Because the product is −1, the lines are perpendicular.

How to Prove Two Lines Are Perpendicular

A proof of perpendicularity should match the information in the question. You may use an angle, slopes, coordinates, or shape properties.

Prove It Using an Angle

If an angle between two lines measures 90°, the lines are perpendicular.

For example, if angle ABC equals 90°:

AB ⊥ BC

This is the most direct method in a labelled geometry diagram.

Prove It Using Slopes

For two non-vertical lines, find both slopes. If one slope is the negative reciprocal of the other, the lines are perpendicular.

For example, slopes 4 and −1/4 prove perpendicularity because:

4 × −1/4 = −1

Prove It Using Coordinates

When each line is defined by two points, calculate the slope of each line.

Suppose line AB passes through (1, 2) and (5, 8):

mAB = (8 − 2) ÷ (5 − 1)

mAB = 6 ÷ 4

mAB = 3/2

Suppose line CD passes through (2, 5) and (5, 3):

mCD = (3 − 5) ÷ (5 − 2)

mCD = −2 ÷ 3

mCD = −2/3

Now compare the slopes:

3/2 × −2/3 = −1

Therefore, AB is perpendicular to CD.

Prove It Using Shape Properties

A named geometric shape can give enough information to establish perpendicularity.

For example, a rectangle has four right angles by definition. Therefore, each side is perpendicular to its adjacent sides. A rhombus has diagonals that meet at 90°, while a square has both perpendicular sides and perpendicular diagonals.

How to Find a Perpendicular Line Equation

To find the equation of a perpendicular line, you normally need:

• The equation or slope of the original line.
• A point through which the new perpendicular line passes.

The basic process is:

1. Find the slope of the given line.
2. Find its negative reciprocal.
3. Use the new slope and the given point in point-slope form.

Point-slope form is:

y − y1 = m(x − x1)

Here, (x1, y1) is a point on the required line and m is its slope.

Example: Find a Perpendicular Equation Through a Point

Find the equation of a line perpendicular to:

y = 3x − 2

and passing through (2, 4).

The original slope is 3.

The perpendicular slope is −1/3.

Use point-slope form:

y − 4 = −1/3(x − 2)

This equation is already correct.

To write it in slope-intercept form:

y − 4 = −1/3x + 2/3

y = −1/3x + 14/3

So the perpendicular line equation is:

y = −1/3x + 14/3

Example: Find a Perpendicular Equation From Standard Form

Find the equation of a line perpendicular to:

2x + 5y − 10 = 0

and passing through (5, 2).

First, rearrange the original equation into slope-intercept form:

5y = −2x + 10

y = −2/5x + 2

The original slope is −2/5.

The perpendicular slope is 5/2.

Now use point-slope form:

y − 2 = 5/2(x − 5)

This is the equation of the required perpendicular line.

Perpendicular Bisector Explained

A perpendicular bisector is a line, ray, or segment that meets another segment at 90° and divides it into two equal parts.

For line l to be the perpendicular bisector of segment AB, it must meet both conditions:

AM = MB

l ⊥ AB

Here, M is the midpoint of segment AB.

A line can be perpendicular to a segment without being its perpendicular bisector. It becomes a perpendicular bisector only when it crosses the segment exactly at its midpoint.

Perpendicular Bisector Theorem

Any point on the perpendicular bisector of a segment is equally distant from the segment’s endpoints.

If point P lies on the perpendicular bisector of segment AB:

PA = PB

The converse is also true. If a point is equally distant from A and B, then it lies on the perpendicular bisector of segment AB.

This theorem is useful in constructions, triangle geometry, coordinate proofs, and questions involving equal distances.

Perpendicular Lines in Common Shapes

Shape	Are adjacent sides perpendicular?	Are diagonals perpendicular?
Rectangle	Yes	Generally no
Square	Yes	Yes
Rhombus	Not generally	Yes
Parallelogram	Not generally	Not generally
Right triangle	One pair of sides	Not applicable
Kite	Not generally	One diagonal is perpendicular to the other

A square is both a rectangle and a rhombus. That is why it has four right angles, equal sides, and diagonals that meet at 90°.

A rectangle has perpendicular adjacent sides, but its diagonals are generally not perpendicular. A rhombus has perpendicular diagonals, but its adjacent sides are not necessarily perpendicular.

How to Draw Perpendicular Lines

You can construct perpendicular lines using graph paper, a ruler and set square, or a compass and straightedge.

Drawing Perpendicular Lines on Graph Paper

1. Find the slope of the original line.
2. Find the negative reciprocal of that slope.
3. Start at the required point.
4. Follow the rise-and-run pattern of the new slope.
5. Mark a second point.
6. Draw a straight line through the two points.

For example, if the original slope is 2/3, the perpendicular slope is −3/2. From the starting point, move 2 units right and 3 units down to locate another point on the perpendicular line.

Drawing With a Set Square

Place one edge of the set square along the original line. Draw along the edge that forms the 90° corner. The new line will be perpendicular to the original line.

This method is useful in practical geometry, technical drawing, and construction exercises.

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Constructing a Perpendicular With a Compass

To construct a perpendicular line at a point on a given line:

1. Place the compass point on the given point.
2. Draw an arc that crosses the line on both sides.
3. Keep the compass width unchanged.
4. Draw an arc from each crossing point above or below the original line.
5. Mark the point where the two arcs meet.
6. Draw a line through the original point and the new intersection point.

The result is a line perpendicular to the original line.

Common Exam Questions on Perpendicular Lines

Basic Definition Question

Question: What are perpendicular lines?

Answer: Perpendicular lines are lines that intersect at exactly 90°.

Identify a Right Angle

Question: Two lines meet at 75°. Are they perpendicular?

Answer: No. Perpendicular lines must form an exact 90° angle.

Find a Negative Reciprocal

Question: What is the perpendicular slope of 5/8?

Answer: −8/5.

Test Two Slopes

Question: Are lines with slopes −4 and 1/4 perpendicular?

Answer: Yes, because:

−4 × 1/4 = −1

Identify Horizontal and Vertical Lines

Question: Are x = 7 and y = −2 perpendicular?

Answer: Yes. The line x = 7 is vertical, while y = −2 is horizontal. They form a right angle where they intersect.

Find a Perpendicular Equation

Question: Find the equation of a line perpendicular to y = −2x + 1 and passing through (3, 4).

The original slope is −2.

The perpendicular slope is 1/2.

Use point-slope form:

y − 4 = 1/2(x − 3)

This is a correct equation of the required line.

Common Mistakes to Avoid

• Assuming every pair of intersecting lines is perpendicular.
• Changing the sign of a slope without taking its reciprocal.
• Reversing a slope but forgetting to change its sign.
• Applying the slope multiplication rule when one line is vertical.
• Assuming every quadrilateral has perpendicular diagonals.
• Confusing a perpendicular line with a perpendicular bisector.
• Calling a line a perpendicular bisector without confirming that it passes through the midpoint.
• Confusing the perpendicular symbol ⊥ with the parallel symbol ∥.

The most useful question to ask is: What evidence proves that the angle is exactly 90°?

Frequently Asked Questions