In this video, I would like to introduce two theorems involving lines tangent to a circle. As you might recall a tangent line touches a circle only at one point, the point of tangency.

Take a look at this diagram. A line travels through two points on the circle. This line is not considered a tangent line now since it touches the circle more than once. But when I move this point closer to the other, the line becomes more like a tangent line. In fact when these two points coincide, the line only touches the circle once at one point, this point is at the intersection of the radius and the point of tangency. What do you notice about the angle formed by these two lines? What is the angle measure when the two points coincide? With this angle measure displayed, it becomes easy to see that when these two points coincide when they get closer and closer to each other, until they actually line up with one another, the angle measure approaches ninety degrees. In fact, the first tangent line theorem states just that… If a line is tangent to a circle it is perpendicular to the radius drawn to the point of tangency and perpendicular means 90 degrees.

In this next example, we have two points of tangency, and two line segments that coincide with the tangent lines. Pay close attention to the lengths of each of these segments when I drag the points of tangency to different locations… Also watch what happens when I drag the third point to resize the circle… Did you notice that the lengths of each of these segments remain equal at all times? Tangent segments to a circle from the same external point are congruent. This is the second Tangent Line Theorem…

Hopefully these illustrations clarified the two tangent theorems. We will make use of them later in the course so it is important to have a good understanding of them before you move forward. Good luck!