Let’s not fool ourselves. Sometimes understanding the principles of geometry can be a real pain. However, they are knowledge that must be acquired and that will help us to better understand the things that surround us. You may not be an ax at math. Geometry may not be your forte. In this article we want to help you understand what perpendicular lines are. We will also work on its meaning. In this way you will no longer have doubts about this concept and you will know how to apply it perfectly. But to understand well what perpendicular lines are, you must first explain what a line is.
Straight
When we talk about lines we refer to a sequence of points that is infinite. Due to this characteristic, straight lines do not have a beginning or an end. But they always move in the same direction. The line is one of the fundamental geometric elements together with the point or the plane. The line is characterized by the following:
- Its infinite character composed of a succession of points.
- The fact that it always moves in the same direction.
- That they are composed of a single dimension and this is the length.
- Which are determined by two points.
What are perpendicular lines? Meaning
The term perpendicular lines comes from the Latin. Specifically the word perpendicularis. Within geometry it is understood that perpendicular lines are those that at some point, and within the same plane, intersect each other. When cut, they create 4 equal right angles. Since the 4 angles are 90º.
As with perpendicular lines, planes and half planes can also be perpendicular to each other if they intersect at 4 right angles.
As with parallel lines, perpendicular lines have (or do not have) different properties. Thus they can be:
- Symmetrical perpendicular lines. We say that they have a symmetric property when a line is perpendicular to another line. Thus, if line A is perpendicular to line B, line B is also perpendicular to line A. This property is called symmetric.
- Reflexive property in perpendicular lines. A line can never be perpendicular to itself. Why is it so? Because a straight line can never cut itself at an angle of 90º. In this case we will say that the line also has a reflexive property.
- Transitive property on perpendicular lines. As in the previous case, a perpendicular line can never be transitive. What does this mean? Let’s take an example to understand it. Consider a plane in which 3 lines meet. If line A is perpendicular to line B and line B is also perpendicular to line C, it is not true that line C is perpendicular to line A or vice versa. These lines cannot be transitive. They must intersect each other and form right angles to be considered perpendicular. Therefore, since lines A and C do not intersect, they cannot be perpendicular to each other.
- Uniqueness. A line that is perpendicular can only pass through one point in the plane.
Therefore, the properties that perpendicular lines share are those of symmetry and uniqueness. They can never have reflexive or transitive properties between one another.
When beginning to study perpendicular lines, care must be taken not to confuse them with parallel lines. While the first ones cross each other forming 4 90º angles. The second ones never intersect and advance infinitely equidistantly. If you think about it carefully, the symbol “+” would perfectly represent the perpendicular lines. On the other hand, the symbol “=” would represent parallel lines.
What is the condition that must be met for two lines to be perpendicular?
As we have explained, two lines are perpendicular when they intersect each other and form 4 90º angles. To confirm that two lines are perpendicular, one of these two conditions must be met:
- The slope of one line must be the reciprocal of the other. In addition, it must have the opposite sign. Namely:
- Another way of looking at this condition is that if you multiply both slopes the result must always be -1. Namely
Let’s see if we have understood what conditions must be met for two lines to be perpendicular. We suggest you solve the following exercise:
Here you have a table in which several slopes appear. We challenge you to find out which slopes correspond to lines that are perpendicular.
To solve this exercise we need to know which pairs are perpendicular to each other. To find out we can use the two formulas that we have seen a few lines above. We remind you here.
- First of all we are going to find out if l 1and l 2 are perpendicular to each other. For this we use formula 2. That by which if we multiply both slopes the result must be -1. By multiplying them we will verify that this is not the result. Therefore they are not perpendicular to each other.
- Now we proceed to find out if l 1and l 3 are perpendicular to each other. In this case we can see that when doing the count we find numbers that are reciprocal to each other because they are flipped. However, in addition to reciprocals, they must have the opposite sign to be perpendicular. But this second variant is not fulfilled because, being both positive, the result is positive. So although at first they may seem reciprocal, the truth is that they are not.
- Now let’s do the same operation between l 1and l 4. In this case the fractions are reciprocals. That is, they are inverted. One is ¾ and the other is -4/3. In addition to being reciprocal, one of them is positive and the other negative. Therefore, when we multiply both, part of the formula will be fulfilled. Since when we multiply + by – the result is negative. Namely, -. Once the signs have been multiplied, we continue with the numbers. And clearing the formula as a result we get -1. Therefore, l 1 and l 4 are perpendicular to each other.
- Now let’s do the same operation between l 1and l 5. In this case the lines l 1 and l 5 have a slope of ¾ and -8/6. Although initially it is not appreciated that they are reciprocal to each other, we can see with the naked eye that one of the fractions is positive and the other is negative. Therefore, when multiplying both to solve the formula, we will obtain a negative result. When we multiply both fractions following the formula m 1 xm 2 = -1 we find that the result is -1. Therefore the l 1 and the l 5are two straight lines perpendicular to each other. Another way to do it is to simplify the fraction -8/6. If we do it this way when simplifying it would be -4/3. Once done we realize that it is reciprocal. Lines l 1 and l 5 are also perpendicular to each other. Same as l 1 and l 4.
- We begin to do the same operations with the rest of the lines from l 2. Just as we have done the operations with l 1 and all the other lines, we will do the same with l 2. That is, we will check if the lines l 2 and l 3 , l 2 and l 4 and l 2 and l 5 are perpendicular to each other. To solve the unknown we will act in the same way as in steps 1 to 4. After carrying out the pertinent operations we will verify that both l 2 and l 4 as well as l 2 and l 5are lines perpendicular to each other. Since when clearing the formula m 1 xm 2 the result in both cases is -1. However, if at a glance we can see that the fractions of the slopes are reciprocals and one negative and one positive, we won’t need to solve the formula. We will know in advance that these are lines that are perpendicular to each other.
- We will do the same exercise starting from the line l 3. In this way we will find out if the line l 3 is perpendicular to any of the other lines. If we look at the fractions of the slopes we can see that in this case there is no straight line that is perpendicular to each other. The remaining fractions do not satisfy both rules. That is, that they are reciprocal and that one is negative and the other positive. Since both rules are not fulfilled, it is impossible that once we solve the formula we obtain -1 as a result. Therefore l 3 and l 4 are not perpendicular to each other. And the l 3 and the l 5
- And finally… l 4and l 5. In this case these two lines are not perpendicular since both are negative. Also, once the fraction of -8/6 is simplified, the result is -4/3. Same as l 4. Since both fractions are not reciprocal, the result cannot be 1. If we clear (-4/3) * (-4/3) we will obtain 16/9 as a result. In total we would obtain 1.77777. Therefore these two lines are not perpendicular to each other.
Surely you have been able to clear up all the unknowns without having to read the explanation that we have written for you. If this is the case, that means that you already know how to apply the formulas about the condition for two lines to be perpendicular. From now on it will be very easy for you to know how to proceed. And you will always know when two lines are perpendicular to each other as long as you know their slope.