Tangent lines are part of geometry. It is a staple of this part of mathematics. You may have started studying this concept at school and you are somewhat lost. If that’s your case, don’t worry. In this article we are going to explain in the simplest way possible what tangent lines are. And what is its meaning. If you are interested in knowing more information don’t wait any longer to continue reading!
What are tangent lines? What is its meaning?
The most basic definition of a tangent line could be the following. A line is tangent to a curve when it touches it at a single point and does not intersect it. If you look at this definition, several terms are mentioned that are very important in order to understand what tangent lines are. The first of these terms is the word straight. Next up is the word curve.
What is a line?
When we talk about a line, what do we mean? A straight line, or rather a straight line, is made up of an infinite number of points that extend in the same direction. Extending in one direction, they also have a single dimension. If we brought a magnifying glass closer to the straight line that we see, we would see that it is made up of many tiny points that have no end.
Surely, when reading this, some of you will ask yourself “what is an infinite number of points?” Yes, an infinite number never ends! If it were not so, we would be talking about other elements of geometry. For example on one side
- Rays. When we talk about rays, we refer to those lines that have an origin point but lack a point with which they end their journey. That is to say, that its other end extends to infinity.
- Segments. When we talk about a segment we are referring to a straight line that is delimited by two points. One start and one end. Many people when they begin to study geometry confuse segments with lines. This happens because, if you think about it, we can’t visually represent a line. It never ends! When the teacher explains it, he creates a straight line on the blackboard that truly has an end and a beginning. That is, a segment. This is where the confusion comes from, because you have to use your imagination in the teacher’s explanation and think that these lines never end.
Now that we have a little better understanding of what a line is, let’s take a quick look at the general equation of a line.
The general equation of a line is given by the following expression: Ax + By + C = 0.
In this equation the values A, B and C belong to the set of real numbers. Do you know what real numbers are? The real numbers are divided into rational and irrational numbers. Rationals are fractions of one number by another. While the irrationals are all the other numbers. But let’s go back to the general equation of a line. While the values A, B and C belong to the set of real numbers, the value B is always different from 0.
Starting from that equation we can learn to clear the “and” with what we would have:
By =-A x – C –> y = – A / Bx – C / B
This other way of expressing the same equation allows us to clearly see that the slope of a line would be – A / B and the ordinate at the origin would be – C / B.
With this information we can draw or represent any line in the Cartesian plane. Cartesian planes are those planes that have two coordinates. The X coordinate and the Y coordinate. The X coordinate is usually drawn horizontally and the Y coordinate vertically. These coordinates form a 90º angle between them and are also known as axes. If two lines make a 90 degree angle, how are they in relation to each other? Perpendiculars.
What is a curve?
Another of the important concepts that we have mentioned is the curve. Do not forget that we have explained that a tangent line is one that has a single point in common with a curved line but does not intersect it. That is, it touches at a point.
Well, what is the curved line? The curved line is very similar to the straight line, since it is composed of a continuous line that is formed by an infinite number of points. But beware:
- It will have an infinite number of points when we talk about open curves, like the parabola.
- However, it can have a finite number of points (as was the case with segments or rays in straight lines) and not follow the same direction. In this case we speak of closed curves. Can you think of an example of a closed curve? Indeed! A good example of a closed curve is the circle. Can you think of another example of a closed curve? The ellipse! The projection of the Earth onto a plane would be an ellipse. Remember that the planet Earth is not round at all.
The equation of a curve is much more complicated than that of a straight line. Do not worry, we are going to explain it to you in a very simple way so that you can understand it. In general terms, we could say that to obtain an equation, what is done is to break down the curve into as many segments as necessary and define equations for each of them that comply with the following:
The equation between point A and B: (A<T<B ) would be:
x=f1(t), y=f2(t), z= f3(t)
Finally we are going to try to express a tangent mathematically in the simplest way possible.
If we imagine a circle of radius one, whose center is at the coordinate origin of a Cartesian system, and we draw a straight line that passes through the coordinate origin and intersects the circle at any point in the positive quadrant, we will obtain an angle Alpha between the x – axis and the line. And if we project to the x axis the point where the line intersects the circumference we will obtain a segment AB. Likewise, if we project the point where the circumference intersects the X coordinate axis onto the straight line that we have drawn, we will obtain the segment CD.
Applying trigonometry, which is based on the measurement of triangles, we know that the tangent is the ratio between the opposite side and the adjacent side, in this case:
So (alpha) = AB / OB = CD / OD.
Since the radius of the circle (OD) was 1:
Tan(alpha) = CD.
Let’s do an exercise to find out if we have understood what tangent lines are
Well, now that we have finished explaining the meaning of tangent lines, we want to propose an exercise to find out if you have understood what these lines are. Go for it!
This afternoon Paquito has gone with his parents to the circus. The circus is a bit far from his house, so they have gone there by train. It is one of the shows that Paquito likes the most. He loves having fun with the tightrope walkers, watching the clowns and seeing how big the elephants are. While they are sitting in the stands, Paquito observes everything around him. His father knows that he has begun to learn the principles of geometry at school. The world of straight lines is no longer something unknown to Paquito and he seems to enjoy it a lot. For this reason, his father has come up with an idea while they wait for the show to start. Look for different types of lines both inside and outside the circus! Would you like to join the challenge that his father has proposed to Paquito? In that case we are going to show you a series of images so that you can tell us the type of lines you see in them. Let’s see how much you know about lines.
- Image 1. In this image you can see the tracks of a train. If you look closely, these tracks are made up of rails. What relationship do the two lines that form the rails have? Can you tell what kind of lines they are?
- Image 2. In this image we see a tightrope walker juggling on the rope. In this case, the exercise he does is truly difficult. I don’t walk on his feet, but on a bicycle. The wheels of the bicycle touch the string. What kind of line is that string in relation to the wheels of the bike?
- Image 3. Here you can enjoy the planet Earth. We have marked on it the lines of the horizon and those of the poles. Which completely cross the Earth. How are these lines related to each other?
Solution
- Image 1. Do you like trains? Have you found out what kind of relationship the lines that make up the rails have? Indeed! These are parallel lines since they do not touch and have the same slope.
- Image 2. What a great balance this juggler has! It must be really hard riding a bicycle on a rope. What type of line does this rope form in relation to the wheels of the bike? It is a tangent line since it touches the circle formed by the wheels but does not cross it.
- Image 3. If we cut the Earth based on the horizon line and the two poles, what we find are perpendicular lines. These intersect each other forming 4 90º angles.