Perpendicular Lines

  • Introduction
  • Perpendicular Symbol
  • Properties of Perpendicular Lines
  • Perpendicular Lines and Parallel Lines
  • Difference Between Perpendicular and Parallel Lines
  • Drawing Perpendicular lines 
  • Slope of Perpendicular Lines
  • Solved Examples
  • Practice Problems
  • Frequently Asked Questions

 

Introduction

When two lines intersect each other at a right angle, then the lines are said to be perpendicular to each other. Perpendicular lines are like the corners of a square or a rectangle: when two lines meet, they form a perfect right angle. This makes them distinct from lines that meet at other angles, like those forming a parallelogram.

Perpendicular lines form a 90-degree angle where they meet. You can think of them as one line standing vertically while the other lying horizontally, creating a right angle where they cross. A perfect example would be the x and y axes of a cartesian plane. Perpendicular lines are often used in various mathematical and real-world applications to describe relationships between different objects or directions.

 

Perpendicular Symbol

When two lines are perpendicular to each other, we represent this relationship using the perpendicular symbol "". For instance, if line AB is perpendicular to line CD, we express it as ABCD. This symbol indicates that the two lines AB and CD meet at a right angle.

When representing two perpendicular lines through a graph, we draw a square arc at the point of intersection to signify that the lines meet at 90°.

 

Properties of Perpendicular Lines

While all perpendicular lines intersect, not all intersecting lines are perpendicular. Remember, two main properties that characterize perpendicular lines:

  • Perpendicular lines always cross each other.
  • Wherever two perpendicular lines meet, they make a perfect 90-degree angle.

 

Perpendicular Lines and Parallel Lines

Parallel Lines: Two straight lines are parallel if they are always the same distance apart and never intersect, even if extended indefinitely. For instance, the opposite sides of your ruler or the lines on a zebra crossing are parallel. We denote parallel lines using the symbol "", expressed as PQRS, meaning line PQ is parallel to line RS.

Perpendicular Lines: When two lines intersect to form a right angle, they are perpendicular. In the given figure, AB is perpendicular to CD. Perpendicular lines resemble the shape of the letter "L". They are commonly observed in structures like door frames or window corners. 

 

Difference Between Perpendicular and Parallel Lines

 

Drawing Perpendicular Lines 

  • Start with Two Points: Begin by marking two points A and B on your paper or surface. These points will serve as the endpoints for one of the two perpendicular lines.
  • Draw the First Line: Using a ruler or straight edge, draw a straight line connecting the two points you marked. This line AB will act as a reference for one of the perpendicular lines.
  • Find the Midpoint: Locate the midpoint of the line you just drew. You can do this by measuring the length of the line and marking its midpoint. Label the midpoint as E.
  • Draw a Perpendicular Line: From the midpoint you found, use a protractor to mark a 90° angle and then use your ruler to draw a line CD that intersects the first line at point E making a right angle. 
  • Draw a Square Arc: Draw a square arc at the point of intersection E, to signify that line AB meets line CD at 90°.
  • Label or Use Symbols: You can label the lines as perpendicular using the symbol "" at their intersection point and say ABCD meaning AB is perpendicular to CD.

 

Slope of Perpendicular Lines

When two lines, let's say AB and CD, are perpendicular to each other, their slopes have a special relationship. When two lines are perpendicular, the slope of one line is the negative reciprocal of the slope of the other line.

Let's denote the slope of line AB as m1 and the slope of line CD as m2.

Slope m2 would be the negative reciprocal of slope m1 and vice-versa. 

Mathematically, we can explain this as:

m2 = 1m1 and m1 = 1m2

From the above relationship, we can infer that two lines are perpendicular to each other if and only if the product of their slopes equals minus one (1). In other words, if you multiply the slope of one line by the slope of the other, you get -1.

So, mathematically, we express this relationship as:

m1×m2=1

This formula helps us determine the slope of a line if we know the slope of the line perpendicular to it. 

Also, it comes in handy to know that when two lines are parallel, their slopes are equal to each other:

m1=m2

In the figure above, line AB has a slope of -2 and line CD has a slope of 12. When we multiply their slope values we get -212=-1, which verifies that AB is perpendicular to CD (ABCD).

 

Example: A line L1 is to line L2. The slope of line L1 is 34. What is the slope of line L2?

Solution: 

Slope of line L1 = 34

The slope of line L2 = negative reciprocal of the slope of line L1.

Hence slope of line L2 = 43

 

Solved Examples

Example 1. In the given figure, lines PQ and RS intersect at point T. The lines meet at 90° and STU=40°. Determine the value of x.

 

Solution:

Given that PTS=90° and STU=40°, we can express this as an equation: x+40°=90°.

Solving for x, we find: x=50°.

 

Example 2. Line 1 passes through the points (-1,3) and (2,7).  Line 2 passes through the points (4,-1) and (-2,5). Are these lines perpendicular?

Solution:

First. let’s calculate the slope of each line using the slope formula:

For Line 1:

m1=732(1)=43

For Line 2:

m2=5(1)(2)4=6(6)=1

Next, check if the lines are perpendicular:

m1×m2=(43)×(1)=43

Since the product of the slopes is not -1, the lines are not perpendicular.

 

Example 3. Find the equation for the line that is perpendicular to 2x5y=10 and passes through the point (3,4).

Solution:

Given the equation 2x5y=10, we rewrite it in slope-intercept form:

y=25x2

So, the slope of the line 2x5y=10 is 25.

The slope of the line perpendicular to it will be the negative reciprocal, which is 52.

Substituting the slope value of 52 and the coordinates of points (3,4) into the equation (y=mx+b), we get

4=52×3+b

4=152+b

b=232

Therefore, the required equation of the perpendicular line is:

y=52x+232

 

Practice Problems

Q1. Which of the following scenarios represents perpendicular lines?

  1. Two roads intersecting at a right angle  
  2. Two parallel railway tracks  
  3. The curves of a circle
  4. Two non-intersecting lines in a 3D space  

Answer: a

 

Q2. Which of the following symbols represents perpendicular lines?

  1.   
  2.  
  3.   
  4.   

Answer: c

 

Q3. In the diagram below, lines AB and CD intersect at point E. If the lines intersect at 90° and BEF=30°, find the value of x.

  1. 0°  
  2. 45°  
  3. 60°  
  4. 90°  

Answer: c

 

Q4. Find the equation of a line that is perpendicular to 3x+4y=12 and passes through the point (2,-1)

  1. y=43x3  
  2. y=34x53  
  3. y=43x113  
  4. y=43x113  

Answer: c

 

Frequently Asked Questions

Q1. What are perpendicular lines?

Answer: Perpendicular lines are lines that intersect at a right angle (90 degrees). This means that the lines form an "L" shape where they meet.

 

Q2. How do you know if two lines are perpendicular?

Answer: Two lines are perpendicular if the angle formed between them measures 90 degrees. Alternatively, the product of their slopes is -1.

 

Q3. What is the relationship between the slopes of perpendicular lines?

Answer: The slopes of perpendicular lines are negative reciprocals of each other. In other words, if the slope of one line is m, the slope of the perpendicular line will be 1m.

 

Q4. Can perpendicular lines be parallel?

Answer: No, perpendicular lines cannot be parallel. Perpendicular lines intersect at a right angle, while parallel lines never intersect and maintain a constant distance from each other.

 

Q5. What are some real-life examples of perpendicular lines?

Answer: Examples of perpendicular lines in real life include the corners of a rectangular door frame, the edges of a bookshelf, the crossroads of streets, the meeting points of walls in buildings, and many more.