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Distance between a point & line

Thinking about the distance between a point and a line. Proof that a point on an angle bisector is equidistant to the sides of the angle and a point equidistant to the sides is on an angle bisector. Created by Sal Khan.
Video transcript
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. 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. 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. 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 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. Now with that out of the way, let's think a little bit about angle bisectors. So let me draw an angle here, so let me, draw an angle. 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. 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. 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. So lets take an arbitrary point that sits on this angle bisector so let's take this point right over here, Ill call that arbitrary point F, or actually I havnt used E yet, so this is going to be and arbitrary point on our angle bisector, and now let's look at the distance between E and BC and the distance between E and BA we already say that, -we already said that the distance between point and a line is if you drop a perpendicular from a point to that line, which you can always do so lets draw a perpendicular right over here, this is one distance, and then this is the other distance this is, -this orange line right here is the distance between E and BC and this orange line is the distance between E and BA and what I want to prove is that these distances are equal well the first thing to realize, is that we have two right triangles over here, they both share th- they both have the same angle, they're not sharing it but angle ABE is congruent to angle CBE and we know that because DB bisects it so this angle is equal to that angle they're both right triangles so the actually have 2 angles in common, which actually means they actually have 3 angles in common and constrains what this other angle would be and they also have this side in common, -they have 3 angles are congruent to each other, they are not necessarily in common, but it does have this side in common, BE is the hypotenuse of both of these right triangles, and so, you can invoke using this angle, that angle and this side, this angle,that angle and this side you could say that these 2 triangles are going to be congruent to each other so we could say that triangle, and let me put some points over here, so let's see let's call this F and let's call this G, we can say that triangle EBF is congruent to triangle EBG and we can use AAS, by Angle Angle Side congruency or you could say hey if two angles, -if corresponding angles are the same then the third angle is also going to be the same so this angle right over here could also be the same and you could use ASA, but either way these two things are going to be congruent but if these two things are congruent, then the corresponding sides are going to be congruent so then, then length of EF, segment EF, is going to be congruent to segment EF is going to be congruent to segment EG which is the same thing as the length of EF is equal to the length of EG, these are really, these are really equivalent statements right over there, so the length of EF, the length of EF is equal to the length of EG and the length of those two segments are the distances between the point and those 2 respective sides. We've just proven the first case, if a point lies on an angle bisector, it is equidistant from the 2 sides of the angle, now let's go the other way around, let's say that I have, so let me draw another angle here, so let me draw another angle over here and let's call this A, B, and C and let's pick some arbitrary point E, let's point some arbitrary point E right over here and let's say we start off with the assumption that E is equidistant to BC and BA and what we want to do is prove that E must be on the angle bisector, so here, if you're on the angle bisector you're equidistant, over here we're going to show if you're equidistant you're on the angle bisector, so if it's equidistant to BC and BA, then this perpendicular right over here, this perpendicular right over here, is going to be congruent to this perpendicular right over there, and let me give these points' labels, so let call this point D, and let's call this point right over F, and let's just draw segment BE here, let's just draw segment BE right over here, so once again, we have 2 right triangles, we already know that 2 of the legs are congruent to each other, they both share the hypotenuse this hypotenuse is equal to its self, we know from the pythagorean theorm if you know 2 sides of a triangle, it determines the third side, so and we know 2 sides of both of these, so the third sides must be the same, so this side must be equal to that side so you could invoke SSS, Side Side Side, to show these 2 triangles are congruent or you actually didn't need to -have to good there you could have used a special case, the RSH case, where if you have a right triangle, so if you have the right triangle you have one set of sides that are congruent, and you have the hypotenuse that is congruent then you're also okay, you could use RSH to prove congruency as well, and so either way we know that triangle EBD, triangle EBD, is congruent to triangle EBF congruent to triangle EBF, we used Side Side Side here but you could have used RSH, let me write that RSH, which is -we know that Angle Side Side can be used for any general triangle but it can be, -RSH is essentially Angle Side Side for right triangles, if you have 2 sides of a right triangle and- in common, --if 2 sides of a right triangle are congruent then the 2 triangles are definitely congruent but once you know that two triangles are congruent, then they're corresponding angles have to be congruent, and angle EBD, angle EBD corresponds to angle EBF so we know that angle EBD must be congruent to angle EBF so EBD must be congruent to EBF, well if EBD is congruent to EBF then that means that segment EB must bisect, must bisect angle CBF, or actually CBA even, it could be called CEF angle CBA. So we're done! If over here, we show that if something sits on a bisector is equidistant from the sides of the angle, and here we showed if it's equidistant from the sides of the angle it sits on the angle bisector, or, or -it could even be the end point of an angle bisector but clearly, it sits on it.