Point of Tangency: Definition

Point of Tangency: Definition & Example

In this lesson, you will learn the definition of a point of tangency. You will also see a few different examples of where points of tangency can occur.

What Is A Point Of Tangency?

A tangent is an object, like a line, which touches a curve. The tangent only touches the curve at one point. That point is called the point of tangency. The tangent does not intersect (pass through) the curve. Let's look at two different examples of points of tangency that you may encounter in math.

Point Of Tangency On A Circle

In geometry, we talk about tangents to circles. To visualize what a line tangent to a circle looks like, imagine that you balance a ruler on a ball, as is pictured here:

The ruler is the tangent. The red point where the ruler touches the ball is the point of tangency. Let's simplify the diagram to something you might see in a geometry book. The ball becomes a circle, and the ruler becomes a line, like this:

A single circle can have more than one point of tangency if it has more than one line 'balancing' on it. For example, if you put a square around a circle, then each side of the square has a point of tangency on the circle.

Points A, B, C, and D in the diagram above are all points of tangency on the circle.

Lines or segments are not the only objects that can be tangent to a circle. You could have two circles that are tangent to one another. Think about two basketballs next to each other on a shelf.

The green point where the two basketballs bump up against each other is the point of tangency. The same situation in a simplified geometry diagram would look like this:

Point Of Tangency To A Curve

Points of tangency do not happen just on circles. We can also talk about points of tangency on curves. So, if you have a graph with curves, like a parabola, it can have points of tangency as well. Let's look at an example of that situation. Below, we have the graph of y = x^2. The x-axis intersects that parabola at a single point, labeled V. Point V is the point of tangency of the parabola and the x-axis.

You could choose any other point on the parabola as well, and draw a tangent line through that point. So, let's put another point of tangency on this parabola. We will call that point T.

As you draw tangent lines to the more straightened part of the parabola, it might become harder to see that the line only touches the parabola at a single point. Remember that lines and curves in mathematics do not have a thickness to them. However, we cannot draw lines and curves that thin in real life and be able to see them. Therefore, in our diagrams, sometimes it may appear that the tangent line touches at more than just one point.

Lesson Summary

Let's review. A tangent is an object that just barely bumps up against a circle or a curve and touches at one point. The point where the tangent touches the curve is the point of tangency. Lines or segments can create a point of tangency with a circle or a curve. Two circles can also have a common point of tangency if they touch, but do not intersect.