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 `AB ⊥ CD`. 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 $PQ \: \| \: RS$, 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 `AB ⊥ CD` 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 $m_1$ and the slope of line `CD` as $m_2$.

Slope $m_2$ would be the negative reciprocal of slope $m_1$ and vice-versa. 

Mathematically, we can explain this as:

$m_2$ `=` $-\frac{1}{m_1}$ and $m_1$ `=` $- \frac{1}{m_2}$

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:

$m_1 \times m_2 = -1$

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:

$m_1 = m_2$

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

Example: A line $L_1$ is `⊥` to line $L_2$. The slope of line $L_1$ is $\frac{3}{4}$. What is the slope of line $L_2$?

Solution: 

Slope of line $L_1$ `=` $\frac{3}{4}$

The slope of line $L_2$ `=` negative reciprocal of the slope of line $L_1$.

Hence slope of line $L_2$ `=` $-\frac{4}{3}$

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`:

$m_1 = \frac{7 - 3}{2 - (-1)} = \frac{4}{3}$

For Line `2`:

$m_2 = \frac{5 - (-1)}{(-2) - 4} = \frac{6}{(-6)} = -1$

Next, check if the lines are perpendicular:

$m_1 \times m_2 = \left(\frac{4}{3}\right) \times (-1) = -\frac{4}{3}$

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 $2x - 5y = 10$ and passes through the point `(3, 4)`.

Solution:

Given the equation $2x - 5y = 10$, we rewrite it in slope-intercept form:

$y = \frac{2}{5}x - 2$

So, the slope of the line $2x - 5y = 10$ is $\frac{2}{5}$.

The slope of the line perpendicular to it will be the negative reciprocal, which is $-\frac{5}{2}$.

Substituting the slope value of $-\frac{5}{2}$ and the coordinates of points `(3, 4)` into the equation `(y = mx + b)`, we get

$4 = -\frac{5}{2} \times 3 + b$

$4 = -\frac{15}{2} + b$

$b = \frac{23}{2}$

Therefore, the required equation of the perpendicular line is:

$y = -\frac{5}{2}x + \frac{23}{2}$

Practice Problems

Q`1`. 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

Q`2`. Which of the following symbols represents perpendicular lines?

1. `\|`  
2. `∠` 
3. `⊥`  
4. `≠`  

Answer: c

Q`3`. 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

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

1. $y = -\frac{4}{3}x - 3$  
2. $y = -\frac{3}{4}x - \frac{5}{3}$  
3. $y = \frac{4}{3}x - \frac{11}{3}$  
4. $y = -\frac{4}{3}x - \frac{11}{3}$  

Answer: c

Frequently Asked Questions

Q`1`. 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.

Q`2`. 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`.

Q`3`. 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 $-\frac{1}{m}$.

Q`4`. 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.

Q`5`. 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.