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Circles and Tangents Examples

In mathematics, a tangent line to a plane curve at a given point is a straight line that "just touches" the curve at that point. More precisely, a tangent line is a line that has the same direction as the curve at the point where it touches the curve, and has a point of contact with the curve at that same point.

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Video Transcript

all right, we're here to do a few examples, work a few examples using the to circle tangent theories you talked about so you can use the exterior change in theory, um, Thio determine whether or not you have a right angle or, in other words, a true tangent. Like if a B is truly tangent to the circle, then we would have a Pythagorean relationship. That would be true. So keep in mind if this is 12 here. This also better be 12 because the radio of the same circle are congruent, which means we have a right triangle with side length 12, 16 and 20. So then we just got to ask ourselves, is 12 squared plus 16 squared? Is that equal to 20 squared? Well, you get 1 44 plus 16 squared, which is 2 56 equal to 400 sure enough, 1 44 and 2 56 is 400. So what that means is we do have a right triangle. That means we do have a right angle, and then ultimately we do have a tangent segment. If the Pythagorean theorem would have not held up in that segment, would have not been a tangent segment. We can work reverse on that and say, All right, let's assume we know it is tangent. Now let's find some missing values. Well, again, if we know this is 1.5 that I know this is 1.5. Well, now I know this. I have a triangle, has a site length at 1.5 a mystery side and then a high pot noose of 2.5. So the question is, Hey, 1.5 plus that mystery side squared has to equal 2.5 squared. Well, this is 2.25 plus X squared equals 6.251 other words, X squared equals 41 other words, X equals two. And it's an example of how you could use the same fear. Um, but in the reverse order, we also have now something along the lines of Let's say we wanted to find the perimeter of the shape Well, and here we now have common tangents emanating from the same points. So the fact that we have a third point here this is 13. I now know that this point here are this segment. Length is 13. Well, that means 25 minus 13. That tells us that this length is 12 and you could see all the dominoes falling. That means this length is 12. Well, that means the difference between that and 21.8 is 9.8 here. Which means this section here is 9.8. And obviously this point, this section here is 4.6. So if we were to some, all these values together now you have the perimeter of that circumscribed. Quite your lateral. Okay, using that same the're, um if I have to tangents, let's say no, they're tangents. I'm given that these air tangents and I know they're emanating from the same point. Well, then, by default, I know that they're equal. Okay, which gets just three x equals one. Or, in other words, X equals one third. If I wanted to go further than that and say, Okay, then what does that make the length of each of those tangents? I could plug one third into each of those values, and I could go ahead and find those legs. There's a whole host of questions that could be asked based upon the relationship But essentially, the key is knowing that if you have to external tangents emanating from the same point, they're gonna be equal. You can set them kill each other. Or if you have the one tangent, uh, to the circle, then you know, the radius that connects the center to the point of pregnancy is going to be perpendicular to that tangent line and essentially a former right angle. So those were two quick examples of how to use tangents with circles. I'll see you next time.