hey there everyone welcome back to mash up math my name is Anthony and thank you for joining me on this mini lesson just go ahead and check it [Music] out hey there everyone thanks again for joining me on this lesson where we are going to explore the relationship between circles and tangent lines so let's start with the question what is a tangent line to a Circle cool so let's take a look at a circle with a center at Point p and let's also imagine another point on the circle and we'll call that point t a tangent line to a circle is a line that intersects The Circle at exactly one point in this case our Point T now if we construct a line that connect that Center Point p with that point t on the outside of the circle we would be constructing a radius and we can call that line r so PT is a radius of circle p and if we look at the angle created by the radius and the tangent line we should see that this appears to be a right angle measuring 90° and this is the case with tangent lines in circles and we can say that a tangent line is perpendicular to a circle's radius so if we add another point on the tangent line Point J we can conclude that angle ptj has a measure of 90° and this perpendicular relationship between a circle's radius and a tangent line applies to every Circle in any occasion [Music] so the key Concepts to understand here is that a tangent line intersects a circle at exactly one point and that a tangent line and the radius it intersects are perpendicular cool so now let's go ahead and apply our understanding of tangent lines and circles to a practice problem in this case we have line segment UT and line segment UV both being tangent to Circle o and we're looking to find the value of x shown on the diagram so notice that lines UT and UV are line segments however they are still tangent to Circle o which means that they will still hold the same properties of tangent lines even though they are not infinite lines and we should also note that we know that angle o has a value of 121 degrees and that angle U is the value X that we are looking to find now we know that UT and UV are tangent lines to Circle o and we know that a tangent line is perpendicular to the radius it intersects so we can conclude that angles T and V are right angles and have measures of [Music] 90° we should also notice that we are dealing with a quadrilateral here in this diagram and we know that the sum of the interior angles in any quadrilateral is always equal to 360° so if we take these four angle measures and add them together their sum will equal 360 and we can use this fact to find the value of x 121 + 90 + 90 plus whatever X is is going to equal 360 and now we can just use some basic algebra to isolate the value of x and find out its value in this case adding the three angles that we do know gives a value of 301 subtracting 301 from 360 results in 59 which is the value of x and we can conclude that angle U is 59° [Music] so now that you understand the properties of tangent lines and circles and have got some more experience applying those relationships to practice problems like this one you can continue to apply your skills and understanding to Future problems so thank you again for stopping by everyone and I'll catch you soon cool all right so that's it for this lesson thank you again for stopping by I hope you found it helpful and if you did please click that link below and subscribe to our YouTube channel leave a thumbs up and a comment on this video we could really use your support and also don't forget to sign up for our Weekly Newsletter when you join our mailing list you get a free eBook download as well as weekly resources tips insights and some cool contests and giveaways as well so don't miss out on that there's a link on this page you can click to join the mailing list and that's all you have to do so thank you so much again for stopping by and for all your support and I will see you all next time bye