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Perpendicualr Bisectors

In mathematics, the perpendicular bisector of a segment is a line that passes through the midpoint of the segment and is perpendicular to it. The perpendicular bisector of a ray is a line that passes through the midpoint of the ray and is perpendicular to it.

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Christine G.

Cairn University

Kristen K.

University of Michigan - Ann Arbor

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University of Nottingham

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Boston College

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Video Transcript

Hey guys, we're here today to talk about perpendicular by sectors. This won't be a super duper long lesson, but, um, I've had videos have posted, referring to medians and perpendicular bias sectors and angle by sectors and altitudes. And this is a couple remaining interesting results that come from being a perpetrator bicycle. It's first of all, go back and talk about a few things. What does it mean to be a by sector? If you have, let's say a segment A B, and you have something that crosses it in such a way that cuts it into equal pieces. Let's say it crosses. Appoint a C such that a C equals B. C. Okay, this line l we're gonna call l A by sector, not a perpendicular by sector. At this point, it's a by sector because it bisects. It cuts in half segment a B into two congruent segments. So, as you can guess, a perpendicular by sector is going to be a segment or array or a line that takes a segment. In this case, let's say segment X y, and not only does it bisected, but it bisects it in such a way that it intersects at a 90 degree angle, so you have to equal segments. Obviously, we'll call this point Z and we'll call this line mm and or segment M whatever you'd like. Um, and it basically cuts segment X y and two congruent or equal pieces, and also, as it does so intersex at a 90 degree angle. That is what a perpendicular by sector is Now. There's two very special things that happen with the perpendicular by sector. Okay, and here they are. One is if you have a segment I mean called the segment X y again and you have perpendicular by sector to that segment, we again we can call this line hell any point on the perpendicular by sector. And by the way, I wouldn't have to tell you this is a perpendicular by sector. If I gave you the tick marks and the right angle measurement, you would know that that is indeed a perpendicular by sector because you have to equal segments and you also the 19 to your ankle. What is true but a perpendicular by sectors at any point on a perpendicular by sector will be equally distant to the end point of the segment that it by six. Don't you think about why that iss If I were to connect these points then what you have is essentially a side angle side situation a side angle side congruence which will prove triangle one congruent to triangle to and therefore the corresponding pieces of a triangle will be congruent. And therefore, if I were to call, let me go ahead and get rid of this hour to call um, this point right here Let's say I call it Z then, by proving triangle one and two congruent, I now know that xz has to be equal toe y Z because of corresponding parts of congruent triangles are congruent. That is how I can say to you how I can prove to you at any point on the perpendicular by sector will be equi distant. Okay, that's the That's the vocabulary award to the end points of the bisected line or line segment. The vice versa is also true if you have a line segment which we can call BC and you have a segment that goes through BC, and we know that any given point on that line is equi distant to the end points of B. C. So we're basically saying is we know that this point that's on this line or Ray or segment If we know that these two are equal meaning we know that this point we'll call it Point D is equidistant from B and D. Then we know that d lies on the perpendicular by sector of BC. So if I know that BD is equal to CD, then what I know is that not only does we'll call this point e d bisect BC, but it's also a perpendicular to And here's how we know this if I know that beady and seedier equal well, I also know that d is equal to itself. I also know that triangle B D. C is ice oscillates because you told me B d and C D or equal Well, that means remember an isosceles triangle. If you have an isosceles triangle, the opposite angles, they're also gonna be equal. Okay, so here we have a situation where we have a pair of sides, another pair of sides and a pair of angles. Now, at this point, what I've described to you is essentially it would be very careful a side side angle situation. So going forward, if we were going to, you know, thoroughly approve this what we call the converse of the perpendicular bisect with your, um would have to do a conditional statement. We have to say, Let's either assume is the mid point or do this thing called indirect proof and say Let's say is not the midpoint and we would pick a point off to the side and then we basically go ahead, draw some more triangles, draw some more segments in by contradiction. We prove that the point we picked has to indeed be point e. So for time's sake and for the scope and range of what we're doing here, I'm just going to reiterate the converse of the perpendicular by sector Theorem says, You've got a segment, you have a line that crosses it. If I know a point on that line happens to be equally distant from the end point of the segment that I know that the line where we're talking about will not only bisect that line X y, it will also be perpendicular to it. So you essentially have two things going on. If you have a perpendicular by sector. Any point on the perpendicular by sector will be equidistant to the end points. Conversely, if you have any line being crossed by a second line and we know that a point on that line is equal to the endpoints, it is a perpendicular by sector, so it can go both ways. It Z is a by conditional statement, So some quick examples would be. Here's a triangle here is a little nicer. Here is a perpendicular by sector and let's say I know this is four units. Then I automatically know this is for you. It's because this point has to be equally distant from this point at this point knowing that we have a perpendicular by sector. Now, on the other hand, if you have a triangle or just a segment and you the line that crosses that segment and you know here's the important part of the statement at any point on this line is equidistant to the end points. So, for example, those distances are equal, those are equal, those are equal etcetera. If you know that every point along this line is equidistant to these points net line is for sure the perpendicular by sector of the second. It's very important to note that every point has to be equidistant because otherwise we could have something like this or have pardon me alliance segment. I have a point here which happens to be equidistant to here and equidistant to hear. But that lion could do something like this where these air clearly not congruent. So it has to be that every point along that line is equidistant to the end points. Alright, I'll go back with some examples.