Perpendicular Line

Perpendicular Line

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Definition of Perpendicular

A perpendicular is a line that makes an angle of \(\mathbf{90^{\circ}}\) with another line.

\(90^{\circ}\) is also called a right angle.

Example of a perpendicular line:

Definition of perpendicular: Angle between two lines is 90 degrees

Here, the blue line and the green line are perpendicular to each other.

Examples of lines that are not perpendicular:

Examples of lines that are not perpendicular - The angle between two lines is NOT 90 degrees.

Here, in each example, the angle between the two lines is NOT \(90^{\circ}\)

Hence, they are NOT perpendicular.


Perpendicular Lines

Perpendicular lines in math are two lines where the angle between them is \(90^{\circ}\)

The symbol used in between the two lines to say that they are perpendicular is \( \mathbf{\perp}\)

If two lines \(\overline{AB}\) and \(\overline{CD}\) are perpendicular to each other, we represent it using the notation \( \mathbf{ \overline{AB}  \perp \overline{CD}}\).

Perpendicular lines in math - the angle between two perpendicular lines is 90 degrees.

Perpendicular shapes in geometry refer to shapes that include perpendicular lines.

Some perpendicular shapes in geometry are:

Square has a perpendicular shape. A right-angled triangle has two sides that are perpendicular to each other.

Rectangle has a perpendicular shape.

Here are some real-life examples of perpendicular lines.

Real-life examples of a perpendicular include a clock showing a right angle between the hands.Real-life examples of a perpendicular include a protractor showing 90 degrees.

Real-life examples of a perpendicular include a white board with right angles at its corners.

You can understand more about perpendicular lines using the following simulation.

Here, you can drag the line \(OA\) by holding it at \(A\) and you can see whether \(OA \perp OB\).

 
tips and tricks
Tips and Tricks

If there is an "L" shape in a figure, the corresponding angle at the vertex is a right angle.

Perpendicular lines form the shape of the alphabet L in uppercase.

How to Draw Perpendicular Lines?

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

  1. Using protractor
  2. Using compass

Drawing a Perpendicular Using a Protractor

To draw a perpendicular line at a point \(P\) on a line,

Step 1: Place the baseline of the protractor along the line such that its centre is at \(P\).

Step 2: Mark a point or a dot \(B\) at \(90^{\circ}\) of the protractor.

Drawing a perpendicular using a protractor: a dot is placed at 90 degrees of a protractor.

Step 3: Remove the protractor and join \(P\) and \(B\). 

This forms a perpendicular line for the given line.

Drawing a perpendicular using a protractor: P and B are joined. A perpendicular line is formed.

Drawing a Perpendicular Using a Compass

We can draw a perpendicular line without using a protractor but by using a compass instead.

Let's see how.

To draw a perpendicular line at a point \(P\) on a line,

Step 1: Open the compass to a desired radius.

Step 2: Place the pointer of the compass at \(P\) and draw a semi-circle which cuts the line at \(A\) and \(B\).

Steps for drawing a perpendicular using a compass: The compass is placed at P and a semi-circle is drawn.

Step 3: Without disturbing the radius of the compass, draw two arcs which cut the semi-circle at \(C\) and \(D\) by placing the pointer of the compass at \(A\) and at \(B\).

Steps for drawing a perpendicular using a compass: Draw two arcs C and D cutting the semi-circle using the compass and keeping the same radius.

Step 4: Without disturbing the radius of the compass, draw two intersecting arcs which intersect at \(Q\) by placing the pointer of the compass at \(C\) and at \(D\).

Steps for drawing a perpendicular using a compass: Draw two intersecting arcs from C and D using the compass. Mark this as Q.

Step 5: Join \(P\) and \(Q\) which forms a perpendicular line of the given line.

Steps for drawing a perpendicular using a compass: Join P and Q which forms a perpendicular.


Properties of Perpendicular Lines

There are two important properties of perpendicular lines in math.

  • The angle between any two perpendicular lines is always equal to \(90^{\circ}\)
  • If two lines are perpendicular to the same line, the two lines are parallel.
    Properties of perpendicular lines: Two parallel lines m and n with a perpendicular transversal

Solved Examples

Example 1

 

 

Are the hands of the following clock perpendicular to each other?

Perpendicular solved examples: A clock showing the time 3 o'clock

Solution:

Place the centre of the protractor at the intersecting point of the hands of the clock.

The baseline of the protractor should be placed along one of the hands of the clock.

Next, measure the angle.

Perpendicular solved examples: A clock showing the time 3 o'clock. A protractor is placed to find the angle between the hands. The angle between them is shown to be 90 degrees.

Since the angle between the hands of the clock is \(90^{\circ}\), they are perpendicular.

\(\therefore\)  The hands are perpendicular
Example 2

 

 

In the following figure, \(AB \perp CD\).

If \(\angle BOC=90^{\circ}\), then find the value of \(x\).

Perpendicular solved examples: finding an unknown angle x.

Solution:

Since \(\angle BOC=90^{\circ}\),

\[\begin{align} x+63&=90\\x &=27 \end{align}\]

\[x=27\]
 
Challenge your math skills
Challenging Questions

Find \(x\) in the following:

Perpendicular challenging question: finding an unknown angle x.

Practice Questions

Here are a few problems for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 

Maths Olympiad Sample Papers

IMO (International Maths Olympiad) is a competitive exam in Mathematics conducted annually for school students. It encourages children to develop their math solving skills from a competition perspective.

You can download the FREE grade-wise sample papers from below:

To know more about the Maths Olympiad you can click here


Frequently Asked Questions (FAQs)

1. Do perpendicular lines have to be 90 degrees?

Yes, the angle between two perpendicular lines must be \(90^{\circ}\)

2. What is the difference between parallel and perpendicular?

The angle between two parallel lines is \(0^{\circ}\)

The angle between two perpendicular lines is \(90^{\circ}\)

3. Does perpendicular mean straight?

No, the perpendicular lines form an "L" shape.

You can learn more about perpendicular lines under the "Definition of Perpendicular" section of this page.

4. What is the perpendicular symbol?

The symbol used in between the two lines to say that they are perpendicular is \( \mathbf{\perp}\)

 
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