# Perpendicular Lines Lesson

Perpendicular lines are two lines that intersect at a 90-degree angle (right angle). This means that the slopes of perpendicular lines are negative reciprocals of each other. An example of perpendicular lines in the real world would be the intersection of the floor and a wall, or two walls, because two walls intersect at 90° angles, as do the floor and wall. Therefore, these form perpendicular lines. A line is perpendicular to another line if the two lines intersect at a 90° angle, or right angle.

The figure above illustrates two perpendicular lines, the little square at the intersection of the two lines indicates that the two lines form a right angle (90 degrees), which means the two lines are perpendicular.

Perpendicular lines are not just any two lines that intersect, the lines must intersect at a 90° angle in order to be perpendicular lines. In math, we use notation for a right angle in the diagram to show that two lines intersect at a 90° angle. We also use notation to show that two lines are perpendicular when describing two lines. For example, if I wanted to say that line AB and line CD were perpendicular, I would use the mathematical notation,

The two main characteristics of **perpendicular lines **are:

The lines must intersect at a 90 angle (right angle).

The slopes of the two lines must be negative reciprocals of each other. In other words, the product of the slopes must equal -1. (m _{1} x m _{2} = -1).

For example, if the slope of a line is 2/3, then the slope of the line perpendicular to that line is – 3/2.

Additionally, you can see that the product of these two slopes will result in -1.

The symbol ⊥ is used to denote perpendicular lines. If l_{1} is perpendicular to l_{2}, it would be written as l_{1} ⊥ l_{2}.

Using a Protractor

We can draw perpendicular lines using a protractor or a compass. Let’s take a look at the first method, drawing perpendicular lines using a protractor.

**Step One:** Use the straight edge of the protractor to draw a straight line, and label a point A on the line.

**Step Two:** Place the protractor on the line from step one, and center it on Point A. Then place point B on the 90 ° mark of the protractor.

**Step Three:** Pick up the protractor, then use the straight edge of the protractor to connect Points A and B to form the perpendicular line.

**Step Four:** Remove the protractor and label the angle where the two lines intersect as a right angle. Now, AB is perpendicular to the given line.

Using a Compass

That is how we draw perpendicular lines with a protractor. Now we will look at how to draw perpendicular lines using a compass.

**Step One: **Use a straightedge to draw a line, and label a point C near the middle of the line.

**Step Two: **Adjust the compass to your desired width. Then place the point of the compass on point C and construct a semi circle that cuts the line in two places. Label those intersections points J and K.

**Step Three:** Without adjusting the width of the compass, place the point of the compass on point J, and draw an arc through the semi circle. Then repeat the same thing from point K, resulting in two small arcs cutting the semi circle. Label those marks point L and point M.

**Step Four:** Again, without adjusting the width of the compass, place the point of the compass on point L and then point M, and draw two intersecting arcs above the semi circle. Label this point D.

**Step Five:** Use a straight edge to connect point C and D, which forms a perpendicular line to the beginning line. Line JK is perpendicular to line CD.

**Examples**

**Example 1:** If the slope of a line is 3/4, find the slope of the line perpendicular to that line.

**Solution:** Since the slopes of perpendicular lines are negative reciprocals of each, we know that the slope of the perpendicular line must be negative, and the reciprocal 3/4 of 4/3 is . Therefore, the slope of the perpendicular line is – 4/3.

We can verify this by multiplying together 3/4 and – 4/3 , since the product is -1, we know that – 4/3 is the slope of the line perpendicular to a line with slope of 3/4.

**Example 2:** Determine if the following statement is sometimes, always, or never true.

If two lines intersect, then they are perpendicular.

**Solution:** Since we know that perpendicular lines are two lines that intersect at a 90 ° angle, the answer to this is sometimes. Not all intersecting lines will intersect at a right angle, and therefore, this is not always true, but it can be true if the intersecting lines do meet at a right angle.

The answer is sometimes true.

**Example 3:** If lines RS and PQ are perpendicular, find the measure of angle PAR.

**Solution:** Since lines RS and PQ are perpendicular lines, we know that they intersect at a right angle, which means that the measure of angle PAR is 90°.

**FAQs on Perpendicular Lines**

**1) What are perpendicular lines?**

Perpendicular lines are two lines that intersect at a right angle, forming a 90-degree angle between them. This means that if you extend the lines infinitely in both directions, they will never meet. The symbol ⊥ is often used to indicate perpendicularity.

**2) How can I determine if two lines are perpendicular?**

Two lines are perpendicular if the product of their slopes is -1. If the slopes m_{1} and m_{2} of two lines satisfy the equation m_{1} x m_{2} = -1, then the lines are perpendicular. Another way is to check if the angles formed by the lines are 90 degrees.

**3) Can any two lines be perpendicular?**

No, not all pairs of lines can be perpendicular. For two lines to be perpendicular, they must intersect at a right angle. If lines are parallel or have different slopes, they cannot be perpendicular.

**4) What is the relationship between slopes and perpendicular lines?**

The product of the slopes of two perpendicular lines is always -1. If the slope of one line is m, the slope of a line perpendicular to it will be -1/m. The slopes of perpendicular lines are negative reciprocals of each other. For example, if the slope of one line is ⅕ to slope of the perpendicular line is -5.

**5) How can I find the equation of a line perpendicular to another line?**

If you have the equation of a line in the form y = mx + b, where m is the slope, the equation of a line perpendicular to it can be found by taking the negative reciprocal of the original slope. So, if the original slope is m, the perpendicular slope is -1/m.

**6) Do perpendicular lines have the same length?**

Not necessarily. The length of a line is not determined by its orientation or its relationship to other lines. Perpendicular lines only describe the geometric relationship between two lines forming a right angle; it doesn’t dictate their lengths.

**7) Can perpendicular lines exist in three-dimensional space?**

Yes,perpendicular lines can exist in three-dimensional space. In this case, instead of forming a 90-degree angle in a flat plane, the lines would form right angles in three-dimensional space.

**8) What is the significance of perpendicular lines in geometry and mathematics?**

Perpendicular lines play a crucial role in geometry and mathematics. They are fundamental for concepts such as coordinate geometry, trigonometry, and vector algebra. Understanding perpendicularity helps solve problems involving angles, slopes, and geometric relationships in various mathematical applications.