Angle bisectors
Point Line Distance and Angle Bisectors 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
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- 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.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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