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Tangent Line

As we get further into the world of geometry and math, we will encounter tangent lines from time to time. It's important that we gain a full understanding of this concept, as tangent lines can help us solve a range of math problems. So what exactly is a tangent line, and why are they so important? Let's find out:

What is a tangent line?

A tangent line is a line that touches a curve at a single point. And that's really all there is to it. If we wanted to give a fancier definition, we would say that a line is "tangent" when its slope equals the slope of a curve. This second definition is easier to understand when we view it on a graph:

As we can see, the curved line and the tangent line share a common coordinate point at (-2, 0). In other words, their slopes are equal at this point.

But why do we call this a "tangent line?" Like many other mathematical terms, the word "tangent" comes from Latin. "Tangere" is a Latin infinitive verb that means "to touch." A tangent line is simply a "touching" line.

Interesting properties regarding tangent lines

Tangent lines may also touch the outer edges of any shape as opposed to simple curved lines. Take a look at the following graph:

What do we notice about this graph? Look at the "point of tangency," which is the coordinate at which the circle and the tangent line touch. What do you notice about the relationship between the radius and the tangent line? That's right -- they create a perfect, right angle. In other words, the tangent line to a circle is perpendicular to its radius.

Aside from this basic rule, one of the most interesting properties involving tangent lines is the "two tangents theorem."

This theorem states that if we draw two tangent segments to a circle from the same point external to the circle, then these two lines must be congruent.

Secant lines vs. tangent lines

You might also encounter secant lines when solving geometrical problems.

So what's the difference between secant lines and tangent lines?

A secant line "cuts through" a curved line or a circle rather than simply touching it. More specifically, it intersects the curved line at two points. In contrast, a tangent line touches the curved line but never actually "slices" it.

An easy way to remember the difference between tangent and secant lines is to consider the Latin roots of the words. "Secant" comes from the Latin verb "secare," which means "to cut." This is also where we get the word "section." A secant line cuts a "section" off the curved line, while a tangent line merely "touches" it.

Both secant and tangent lines are infinite, and they continue indefinitely outside of the circle.

Other lines in a circle

Aside from tangent and secant lines, there are a few other important lines to know when dealing with circles:

• Chord: A chord is a straight line drawn between any two points on a circle or curve. Unlike secant and tangent lines, chords are finite and only exist within the circle.
• Diameter: A diameter is a special type of chord that perfectly intersects the midpoint of a circle and divides the circle into two equal halves.
• Radius: The radius, as we all know, extends from the midpoint or center of the circle until it touches the perimeter.

Is the tangent ratio related to tangent lines?

There is indeed a relationship between tangent lines and the tangent ratio of an angle. To understand this connection, let's examine a diagram with a circle and tangent line.

Consider a circle centered at point O, with a radius OA that intersects the circle at point A. Draw a tangent line to the circle at point A, and extend this tangent line to meet the x-axis at point E. Now, let $\angle \mathrm{AOE}$ be $\left(\theta \right)$ , where O is the center of the circle.

With this setup, the distance between points A and E is equal to the tangent of the $\angle \mathrm{AOE}$ . This is because if we draw a right triangle with vertices at O, A, and E, with a right angle at A, the tangent of ∠θ would be equal to the ratio of the opposite side AE to the adjacent side OA, which is the radius of the circle.

So, the tangent ratio of an angle is related to the tangent lines, as the length of the segment AE along the tangent line is equal to the tangent of the central $\angle \theta$ . However, it is crucial to note that tangent lines and tangent functions are related concepts but are not the same thing.

A short history of tangent lines

Tangent lines were mentioned by Euclid about 2300 years ago. Archimedes also studied tangent lines when studying his Archimedean spirals. By the 1630s, Fermat had developed a new technique called adequality to calculate the tangents of a parabola. Combined with Descartes' innovative method of normals, the study of tangent lines help develop differential calculus by the 17th century.

Real-world applications of tangent line calculations

Since the real world is filled with circles and lines, there are many real-world applications and examples involving tangent lines. On a very basic level, a "tangent line" is created whenever our bicycles touch the road. Our circular bicycle wheels might touch the ground, but the road never "intersects" our wheels (unless something goes very wrong).

On a much more advanced level, NASA engineers use tangent-line equations to make sure space probes and other interstellar vehicles travel in the right direction after leaving orbit. When spacecraft travel around a planet or a moon, their path can be plotted in the same way as a tangent line. The goal is to have the spacecraft approach the planet and "touch" the outer limits of its orbit and gravity without making it crash. With these methods, engineers can "slingshot" spacecraft around planets -- using the power of gravity to supplement fuel-powered propulsion.

Infinity

Flashcards covering the Tangent Line

Calculus 1 Flashcards