DB bisects BISECTS angle ABC, ABC Fare enough, now, the whole reason why I started this video talking about distances between points and lines, is that I want to show you, that any point that is on an angle bisector, is actually going to be equidistant from the sides of the angle, and then were going to go the other way saying, any point that is equidistant from the sides of an angle is going on the angle bisector. And an angle bisector is essentially line, or a segment, or a ray that splits an angle into two equal angles and we've talked about this before, so for example if we want to bisect angle ABC, so this angle right over here, we want to split it in two, we are going to -I can draw a better version of that, we want to split it in two, we want to split it in two like, let me draw a little bit better, my drawing still doesn't look like, that looks decent, alright, so lets call this point right over here D, and again we could even say that, that's a ray, or we could call that a segment, or whatever, but the way to think about this, is if now angle DBC is equal to angle DBA, so this angle, DBC is equal to angle DBA, we can say that DB bisects angle ABC, so we can say that DB and now Im talking about segment DB we could have made it a ray if we had -keep going to the right, or a line. So lets call this point, let me do it in a different color, let's call that point A, lets call this point B, and let's call this point C right over here. So let me draw an angle here, so let me, draw an angle. Now with that out of the way, let's think a little bit about angle bisectors. So hopefully that at least gives you a decent sense why dropping the perpendicular will always give you the shortest distance between a point and a line, and that unique shortest distance is what we call the distance between a point and a line. The hypotenuse is always going to be the longest side a triangle D squared plus whatever length this squared, is going to be equal to this length squared. so pick another point on this line right over here, let's call this point E, and think about this -now this is an arbitrary point I could have drawn E here, I could have drawn E here, I could have drawn E anywhere, but regardless of where you draw E, if you draw a line segment between A and E, you see that we've form a right triangle from A to E, to the point we had the perpendicular, so let me call this point right here F, you're always going to draw a right triangle assuming that E is different than F and if you will immediately see that you see that D has to be shorter than this orange length, because this orange length is the hypotenuse. So say that that is some point, point A this is some line right over here, we'll call that line, BC, so when you're taking the distance between a point and another point, it's very obvious, you just draw a line to that other point, Well I already used B, you just draw a line to that other point and you find the length of that line, so distance seems very straight foreward between two points, but what about a point and a line because there are many points on this line maybe were going to find this distance, or maybe were going to find this distance, or this distance, and these are all going to be different lengths so how do we have one unique distance? And the way that we think about this, and we're going to do this in much more depth in future math courses especially when you start vectors and linear algebra and all the rest, is distance between a point and a line is really the shortest distance and that shortest distance is as if you were to drop a perpendicular from that point to the line, so this right over here, this right over here, is what we call the distance the distance between the point and the line, and this is perpendicular right over here, to recognize that this is indeed the shortest distance think about this relative -the distance between this point and any other point on this line. In this video I'm going to talk a little more about points on angle bisector but before that I want to at least make sure we understand what we mean when we talk about the distance between a point and a line.
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