Remember the difference between a tangent and a secant? A line that is tangent to a circle intersects the circle at exactly one point while a secant intersects a circle at two points. See if you can distinguish between the two types of lines in the images below. Click each image to see what types of lines are intersecting.
Once you can identify what lines are intersecting and where the intersection point is, you can find the length of a segment formed when tangents and secants intersect. Watch the video below to learn how.
As you watch this video, use the study guide to follow along if you'd like. Click the button below to download the study guide.
In this video, I would like to illustrate the Intersecting Secants Length Theorem. It states that when two secant lines, AB and CD intersect outside a circle at point P, then the following relationship is true. The product of lengths PA and PB is equal to the product of lengths PC and PD. With the help of this diagram, you should notice that segment PA is the segment along the dotted line, and segment PB is the entire length of that upper line. The same goes for segment PC and PD.
As we see in this interactive diagram, the lengths of these segments can drastically differ, especially when we drag point P to different locations on the outside of the circle. But you will also notice that the product remains the same… Additionally, if we were to drag point C so that it intersects with point D, we would have created a tangent line here instead of a secant line. And we can see that a little more clearly in this picture. The segment PC and what would have been segment PD are the exact same length. Therefore, we can modify our original theorem to state that the length of PC squared is equal to PA times PB. In this screen shot, the value PC squared is the same as length PC times PD since C and D are two concurrent points.
Now let's turn our attention to a problem we might need to solve involving this theorem. The diagram shows three side lengths and our job is to find the side length of x. Notice that the lengths of the upper segment are of PB and BA, not PA this time. And also the length of PD is 7 units whereas the length of DC is x units. We can establish an algebraic relationship with these dimensions if we are careful. Let's start with the theorem first.
The product of PA and PB is equal to the product of PC and PD… By substitution we see that PB is equal to length 12 and PD is equal to the length 7, but the length of PA is equal to the sum of 12 and 8, and the length of PC is equal to the sum of x and 7... Now that we have established the algebraic relationship, we should be able to solve for x. Please pause the video now and solve for x. Resume playback in a moment to check your work and compare notes… You should have determined that x is approximately equal to 27.3 units.
Please make sure that you understand this theorem and its implications as shown in this video before moving forward. Watch the video again, practice the calculations, and investigate the interactive diagram until you are comfortable. Good Luck!
Question
How can you find the length of a missing segment when two secants intersect or when a secant and a tangent intersect?
