Have you ever wondered what secant lines are? Do you have certain doubts about the meaning of this term? In that case, you’ve come to the right place. In doncomos we are going to explain what they are and what is the meaning of secant lines. From now on you will no longer have problems understanding what this term means.
What are secant lines?
If you look at the term “secant lines” it is made up of two words: lines and secants. To understand this term it is necessary to clarify both concepts.
- Straight. Lines in geometry are composed of a succession of points that have no end. That is, they are infinite and have no beginning or end. Along with the plane and the point, lines are basic geometric concepts. For a line to be considered a straight line, it must meet the following premises:
- It must be created by a succession of points that never end. That is, they are infinite.
- All these points that make up a straight line must always move towards the same direction.
- Lines are one dimensional. That is, they only have one dimension. The dimension they have is the length.
- They are determined by means of two points.
- Drying. The term secant in geometry refers to the line or surface that intersects another line or surface.
Now that we know what these terms mean in geometry, it will be easier to understand what a secant line is. A secant line is one that intersects either a curve or a different line. To be considered secants they must have a point in common. The point in common between both straight lines or between the straight line and the curve is the place where they intersect or intersect. In the event that the secant line intersects a curve, it can do so at one or several points on the curve. It all depends on where you cut the curve.
If you look at the example below these lines, we have a secant line, the line r, which intersects the line s. These two lines intersect at a single point, which is point A.
If you look at this new example under the text lines, you will see that we have a secant line, the line r, which is responsible for cutting a curve, the curve c. Since it is a circle, the line and the curve intersect at two points.
If these lines do not have any point in common, and yet they are in the same plane, we will be dealing with parallel lines.
Parallel lines and intersecting lines can coexist in the same plane. In fact, within Euclidean geometry they are used regularly to be able to solve many practical exercises and even problems.
If there are two parallel lines that are cut by a secant, different angles will be created. Depending on the type of angle that is created once the secant cuts the parallel lines, the angles that are created will be of one type or another. The normal thing when parallel lines are cut by a secant line is that one of the following angles can arise or be born:
- Alternate interior angles. Alternate interior angles are considered to be those born between parallel lines that are cut by another straight line. These spawn on different sides and are the same.
- Angles opposite the vertex. The angles opposite the vertex are those that are equal, and as their name indicates, opposite by the vertex.
- Alternate exterior angles. Alternate exterior angles are similar to alternate interior angles. They are created when a line intersects two parallel lines. Instead of appearing on the inside of the parallel lines, they appear on the outside. These are born on different sides of the parallel and the line that intersects them. As with alternate interior angles, alternate exterior angles are equal.
- adjacent angles. These angles are characterized by having one of their sides in common in addition to the vertex. The other sides that make up these angles are opposite rays.
- Corresponding angles. Corresponding angles are created when two parallel lines are cut by a transversal line. The 8 angles that arise from this cut are considered corresponding.
- External and internal collateral angles. The external collateral angles are born, like the rest of the angles we have studied, when a secant line cuts two parallel lines. In total, 8 angles are created when this cut is made. Depending on the position that these angles occupy, they will be considered external collaterals or internal collaterals. When the angles are located inside the line, and always on the same side of the secant, we will be facing internal angles. However, if they are on the outside of the line, and on the same side as the secant, we will be facing exterior angles.
Classification of secant lines according to their shape
Intersecting lines can be classified into different types based on their shape. Specifically, depending on the shape, we can find the following types:
- Oblique lines. When a secant line intersects another line, there is a point in common between them. Thanks to this point we talk about the fact that it is about secant lines. But when cut they can be either oblique or perpendicular. In the event that they intersect at a point creating angles that are not equal, we will find ourselves before oblique secant lines.
- Perpendicular straight lines. Now, in the event that a secant line cuts another line and 4 right or 90º angles are created when cut, we will be facing perpendicular secant lines or perpendicular lines.
In the event that the lines are in charge of cutting a curve or circumference, we will find ourselves before the following types:
- Secant lines. If the curve or circumference is cut at two different points, we will be facing a secant line. Within geometry and mathematics this type of line is known as a secant line to a curve.
- Tangent lines. However, if the line that cuts the circumference or curve does so at a single point, we will be facing a tangent line and the point of intersection will be called the point of tangency.
Here is an exercise with intersecting lines
Let’s see if we have truly understood what secant lines are. To do this, we suggest you carry out the following exercise. You will see how easy it is for you to find the solution. However, if you see that you get stuck and don’t know how to continue, don’t worry. We have enabled a space with the solutions so that you can solve the problems. This way you can check if your results are correct or find out what was the solution to the problem. Do you want to start? In that case, let’s go for that problem!
- Exercise 1. María and Pablo are beginning to study data related to lines in their mathematics classes. At this moment they are reviewing the fun world of intersecting lines. Your math teacher wanted to challenge you to create a practical example to see if you really know what a secant line is. For this he has asked them to draw a picture of a city with roads in which some of the streets form intersecting lines.
Since María and Pablo have a little trouble understanding the difference between intersecting lines and other lines, their teacher has had an idea. He has decided to show them photos of everyday elements where we can find intersecting lines. The photos that María and Pablo’s teacher has shown them are the following.
Solution to the exercise with secant lines
- Exercise 1. María and Pablo’s teacher proposed a challenge to these two children. Draw a road where intersecting lines meet. To help the children to be able to do this exercise, he showed them several pictures of our world where intersecting lines are incorporated. But he mixed these lines with other types of lines. In total he offered them 4 photos. Have you managed to find out which were intersecting lines and which were not? If you want to know for sure which is the correct answer… keep reading!
Photo 1. Secant line
Photo 2. Parallel lines… and intersecting lines. If you look closely, the man’s body cuts the lines of the ground, creating intersecting lines with his figure.
Photo 3. Parallel lines… and intersecting lines. The lines that form the floor are parallel. However, the lines that form the railing intersect creating intersecting lines.
Photo 4. Parallel lines. If you notice the columns do not cut each other. And they are also equidistant. Therefore they are parallel lines.
We encourage you, like María and Pablo, to make a map of a city with streets that form intersecting lines. You can have fun and add all kinds of straight lines. In addition to secants you can also include parallel lines. You can also make different angles depending on the type of cut made by the intersecting lines. Your parents will be able to help you solve this problem and confirm with you if you have chosen the lines correctly.