Perpendicular Lines – Definition with Examples

What Is Perpendicular?

Two distinct lines intersecting each other at 90°, or a right angle, are called perpendicular lines. 

Example:

Here, AB is perpendicular to XY because AB and XY intersect each other at 90°.

Non-Example:

The two lines are parallel and do not intersect each other. They can never be perpendicular to each other.	The two lines are intersecting each other at an acute angle. So, they are not perpendicular.	The lines are not intersecting each other at right angles.

Perpendicular Symbol

Perpendicular lines are represented by the symbol ‘⊥’. Suppose, m and n are two lines intersecting each other at 90°, then they are perpendicular to each other and are represented as m ⊥ n. The point of intersection of perpendicular lines is called the foot of the perpendicular.

Properties of Perpendicular Lines

• These lines always intersect at right angles.
• If two lines are perpendicular to the same line, they are parallel to each other and will never intersect. 

• Adjacent sides of a square and a rectangle are always perpendicular to each other.

• The sides of the right-angled triangle enclosing the right angle are perpendicular to each other.

Real Life Examples

We can observe many perpendicular lines in real life. 

Some examples are the sides of a set square, the arms of a clock at certain times of the day, the corners of the blackboard, a window, and the Red Cross symbol. 

Difference between Perpendicular Lines and Parallel Lines

Perpendicular Lines	Parallel Lines
Perpendicular lines are lines that intersect at a 90° angle.	Parallel lines are lines that are always the same distance apart. Parallel lines never intersect.
The symbol used to represent a perpendicular line is “⊥”.	The symbol used to represent a parallel line is “||”.

Example: The corner of the postcard

Example: Railway tracks

The corner of a ruler

The edges of a ruler

Construction of Perpendicular Lines

We can draw perpendicular lines for a given line in two ways.

• Using a protractor
• Using a compass

Drawing a perpendicular line using a protractor

Step 1: Let m be the given line and A the given point on it.

Step 2: Place the protractor on the line m such that its baseline coincides with m, and its center falls on A. 

Step 3: Mark a point B against the 90-degree mark on the protractor.

Step 4: Draw a line n passing through A and B.

Then line m ⊥ line n at A.

Drawing a perpendicular line at a given point on a line using a compass

Step 1: Place a compass tip on the given point A. Take an appropriate radius and draw an arc on either side of the given point.

Step 2: Stretch the compass wider. Place the compass tip on the new points and make a small arc above the given line. The two new arcs will intersect.

Step 3: Use the ruler to join the given point A and point B where the arcs intersect.

Solved Examples on Perpendicular

Example 1: State whether a given set of lines are perpendicular or parallel.

(a)	(b)
(c)	(d)

Solution:

1. Parallel lines
2. Perpendicular lines
3. Perpendicular lines
4. Parallel lines

Example 2: Write a relation between the line segments indicated by the arrow in each shape.

Solution: 

1. AB ⊥ AE
2.  PQ || SR
3. PQ || NM
4. AB ⊥ AD

Example 3: The diagonals of a rhombus are perpendicular to each other. How many right angles are formed at the intersection?

Solution:

Since the lines AC and BD intersect each other at 90 degrees, there are 4 right angles at the intersecting point. 

∠AED = ∠DEC = ∠CEB = ∠BEA = 90 degrees

Example 4: Name some two-dimensional shapes that have perpendicular lines.

Solution: