SM2 13.6-6: Using Secant and Tangent Lines to Solve for x, Simple Quadratic

Added: 2026-05-16

[00:00:00]Hello and welcome. So we have a circle with  a secant tangent line uh intersection there.

[00:00:05]So a secant line just means that you have  two intersection points that the line is hitting through the circle. A tangent line  just means that you're hitting it once.

[00:00:14]Okay, so that's all that definition  means. So we have a couple lengths here: given that this first length is 11, second length  is 12, the tangent line length is n. Now how this relationship works is it essentially is part  times the whole equals part times the whole.

[00:00:36]Okay. Now specifically if we want to be a little  more specific with that part it's going to be the outside portion and that's going to be of the same  line, right? So for example if we're looking at here we have two portions of this line. We have  until we hit the first intersection and we have between the points between the first and second  intersection of your secant line. So what this would look like is the outside, right? Outside of  the circle portion which is going to be 11 times the whole portion. Now that's not just saying the  whole portion inside the circle. We're saying the whole line for that thing which means that line  is 11 plus 12. It's the entire segment. It's not just one or the other. It's the whole shebang  which means that's 11 plus 12 on the inside.

[00:01:25]Okay. Now it looks a little more interesting  when it comes to tangent lines. Tangent lines you don't have two different segments to look  at. You only have one segment. So the part, the first part is the outside portion. That's  just n. But the whole portion is also n. So it's essentially, whenever you have a tangent line you  just multiply the same thing twice. All right. So first off 11 and then 11 plus 12 is 23. So we have  11 times 23 equals n times n which is n squared.

[00:01:59]All right. So 11 times 23 is 253 equals  n squared. Now if we want to solve for n we got to get rid of that square. How do we  do it? Square root. Take square root of both sides. And when we do remember you have a plus  and minus but think about it this way. If you have the square root of 253, right? Can that  be negative if we were to plug it back in?

[00:02:26]No, right? You're talking about solving for a  length. Lengths cannot be negative which means that you're not using that negative square  root. You're only using the positive one.

[00:02:34]If it simplifies you'd want to simplify it but we  already know that 253 is from 11 and 23. Those are both prime numbers. That's not simplifying. Which  means that your final answer is the square root of 253. All right. Be careful it's asking for a  simplified exact answer. Don't give a decimal. You want the square root. Simplify it if you can. And  that's all there is to it. Thanks for watching.