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# Distance between a point & line

## Video transcript

in this video I'm going to talk a little bit about points on angle bisectors but before that I want to at least make sure we understand what we mean when we talk about the dis the distance between a point and a line so let's say that that is some point Point a and this is some line right over here we'll call that line line BC so you're taking in the distance between a point and another point it's very obvious you just you just draw a line to that other point well I already use B you just draw a line to that other point and you find the length of that line so distance seems very straightforward between two points but what about a point in line because there are many points on this line so maybe we're gonna find this distance or maybe we're gonna find this distance or maybe we're gonna find 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 when in in future math courses especially once you start vectors in linear algebra and all the rest is distance between a point and a line is really the shortest distance and that shortest distance is 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 and to recognize that this is indeed the shortest distance think about this relative to the distance between this point and and and and and and any other point on this line so pick another point on this line right over here let's call this point E and let's think about this distance now this is an arbitrary point I could have drawn a here I could have drawn a here I could have drawn eat anywhere but regardless of where you draw e if you draw a line segment between a and E you see that we form a right triangle from A to E to the point where 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 do that you'll immediately see that D has to be shorter than this orange length because this orange length is a hypotenuse the hypotenuse is always going to be the longest side of a triangle d squared plus whatever length this squared is going to be equal to this length squared so hopefully that 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 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 let's call this point do it in different colors let's call that point a let's call this point B and let's call this point C right over here and an angle bisector is essentially a line or segment or array that splits an angle into two equal angles and we've talked about this a little bit before so for example if we want to bisect angle a ABC so this angle right over here we want to split it in two we are going to split it in that I can actually draw a better version of that we want to split it in two we want to split it in two like let me draw it a little bit better the drawing is that still doesn't look like that looks decent all right and so let's call this point right over here and I know let's call this D and maybe we could even say that's an array 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 D ba so this angle DBC is equal to angle DBA we can say that D B bisects angle ABC so we could say that D B and now I'm talking about segment DB we could've made an array if we made it kept keep going to the right or a line DB bisects bisects angle a BC ABC fair 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 we're going to go the other way to show that any point that is equidistant from the sides of an angle is going to be on the angle bisector so let's take an arbitrary point that sits on the angle bisector so let's take this point right over here I'll call that arbitrary point F well actually I could use a I haven't used yet so this is going to be an 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 a point of the line is if you drop a perpendicular from a point to that line but you can always do so let's 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 this is the distance between and this orange line is the distance between E and B a and what I want to prove is is that these distances are equal well the first thing to realize is that we have two right triangles over here they both share they both have this same angle they're not sharing it but angle a I should say angle a B E is congruent to angle CBE and we know that because D B bisects it so this angle is equal to that angle they're both right triangles so they actually have two angles in common which actually means that they actually have three angles in common it constrains what this other angle would be and they also have this side in common and when I say they have three angles that are congruent to each other they're not necessarily in common but it does have this side in common B e 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 on this side you could say that these two 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 this call this F and let's call this G we can say that triangle e be F is congruent to triangle e be G and we could use it angle angle side by angle angle side congruence or you say hey if two angles if two corresponding angles are saying then that third angle is also going to be the same so this angle right over here could also be the same and you could use angle-side-angle but either way these two things are going to be congruent but these two things are congruent than 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 two respective size so we've just proved that we've just proven the first case if a point lies on an angle bisector it is equidistant from the two 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 right over here and let's call this a B and C and let's pick some arbitrary point e let's pick 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 II must be on the angle bisector so here if you're on the angle bisector you're equidistant over here we're gonna show if you're equidistant you're on the angle bisector so if it's equidistant to BC and B a 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's call this point D and let's call this point right over here f and let's just draw a segment to be e here let's just draw a segment let's just draw a segment to be e right over here so once again we have two right triangles we already know that two of the legs are congruent to each other they both share the hypotenuse this hypotenuse is equal to itself we know from the Pythagorean theorem if you know two sides of a right triangle it determines the third side so and we know two sides of both of these so their 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 that these two triangles are can or you actually didn't even have to go there you could have used a special case the rsh case where if you have a right triangle and 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 K okay you could use rsh to prove congruence as well and so either way we know that triangle EBD triangle II B D is congruent to triangle EB F congruent to triangle e B F we used side-side-side here but you could have used R SH let me write that RSH which is we know that angle side side can't be used for any general triangle but it can be our SH is essentially angle side side for right triangles if you have two sides of a right triangle and a Nang and and and if you have two sides of a right triangle in common then the they are definitely two sides of a right triangle are congruent then the two triangles are definitely congruent which is what essentially this is saying but once you know that two triangles are congruent then their corresponding angles have to be congruent an angle EBD angle abd corresponds to angle EB F so we know that angle e BD must be congruent to angle E bf so eb d must be congruent to EB F well if abd is congruent to EB F then that means that means that segment EB that means that segment EB must bisect must bisect angle C B F or I should say CBA even it could be called CB f angle C B a so we're done if over here we showed that if something sits on a 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 it or it could even be the endpoint of an angle bisector but clearly it sits on it