Perpendicular Lines – Definition, Symbol, Properties, Examples

What are Perpendicular Lines?

Two distinct lines intersecting each other at $90^{\circ}$ or at a right angle are called perpendicular lines. 

Example:

Here, AB is perpendicular to XY because AB and XY intersect each other at $90^{\circ}$.

Non-Example:

The two lines are parallel and do not intersect each other.	The two lines are intersecting each other at an acute angle. So, they are not perpendicular.	The lines are not intersecting at right angles.

Definition of Perpendicular Lines

In geometry, perpendicular lines are defined as two lines that meet or intersect each other at right angles $(90^{\circ})$. The term ‘perpendicular’ originated from the Latin word ‘perpendicularis,’ meaning a plumb line. 

If two lines AB and CD are perpendicular, then we can write them as AB $\bot$ CD. The symbol $\bot$ is used to indicate the lines are perpendicular.

Perpendicular Lines Around Us

We can observe many perpendicular lines in real life. 

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

Here are a few more examples!

Perpendicular Symbol

Perpendicular lines are represented by the symbol, ‘$\bot$’. Suppose, m and n are two lines intersecting each other at 90 degrees, then they are perpendicular to each other and are represented as m $\bot$ 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.  For example k and l are two lines and both are perpendicular to the same line r, then lines k and l are parallel to each other 

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

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

Difference between perpendicular lines and parallel lines

Perpendicular lines	Parallel Lines
Perpendicular lines are lines that intersect at a 90 degrees 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 “$\bot$”.	The symbol used to represent a parallel line is “$\left|\right|$”.
Example: The corner of postcard The corner of the ruler.	Example:Railway tracks Edges of 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 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 base line coincides with m, and its centre 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 $bot$ line n at A.

Drawing A Perpendicular Line At A Given Point On A Line Using 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 the point B where the arcs intersect.

Conclusion

Give your children opportunities to observe perpendicular lines in objects or places around them, such as a tall tree on the ground, an electric pole on the pavement, railway intersection, corner of two adjacent walls, and high buildings.

Let’s solve a few examples for better understanding.

Solved Examples On Perpendicular Lines

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.

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

Solution: 

1. AB $\bot$ AE
2.  PQ $\left|\right|$ SR
3. PQ $\left|\right|$ NM
4. AB $\bot$ AD

Example 3: The diagonals of 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. 

$\angle \text{AED} = \angle \text{DEC} = \angle \text{CEB} = \angle \text{BEA} = 90$ degrees

Example 4: Name two dimensional shapes having perpendicular lines.

Solution:

Rectangle	Right angled triangle	Square

Example 5: If AB $\left|\right|$ CD and CD $\left|\right|$ EF. What can you say about AB and EF? 

Solution: 

Line AB is perpendicular to line CD. 

Also, line CD is perpendicular to line EF. 

When two lines are perpendicular to the same line, then the 2 lines are parallel to each other. 

Thus, lines AB and EF are parallel to each other. 

Example 6: Are these lines perpendicular?

Solution: 

No, these lines are not perpendicular because they do not intersect at right angles.

Example 7: Are these lines perpendicular?

Solution: Yes, these are perpendicular lines. They intersect at $90^{\circ}$.