Now, there's a couple of different cases of perpendicular lines. And let's talk about it. In one situation, you've got a line and a point on the line. And what you're looking to do is you're looking to construct a perpendicular line that goes through that point. Another line that's perpendicular to the original and goes through that point. The second case that we've got today concerns what happens when we have a line and a point not on the line key right in the first case, we're going to have a line at a point on it. The second case will have a line and a point not on the line. And what we're trying to do in that situation, hopefully. Is, oh, I lost my line. It moved all the way up here. But what I'm trying to do in that situation. Is construct a perpendicular line that goes through that point perpendicular to this one. All right. This one probably sounds a little bit familiar to one that we did recently where we had a line, a point not on the line, and we wanted to construct a parallel line. Here, we'll be looking to construct a perpendicular line. This one is amazingly important for one that we're going to be doing in the next lesson that's quite challenging. 

Before we get into these, though, let's review a few things that are important in order to understand these constructions. All right. The key, the key to all perpendicular line constructions is the following. A point lies on the perpendicular bisector. The perpendicular bisector of a line segment, if it is equidistant from the endpoints of the segment. All right, now we've proven this actually a couple different times. We proved it back when we were looking at rigid body motions. We also proved it earlier in triangle congruence, but this is an amazingly important thing. So important that I want to illustrate it a little bit using a geographer widget. All right? So let's take a look at this. In this picture, what I've got is I've got basically an isosceles triangle. All right? So sides AC and BC are equal length, which means that point C here point C is equidistant from point a and point B and if I asked geo gebra to create the perpendicular bisector of line segment AB, notice that it goes directly through point C now, I've constructed this thing so that no matter how I move these points around, apparently I can't move these points around. But no matter how I move them ah, I see why? Apparently I'm not the best with Joe gibber right now. 

If I move these points around, notice that no matter how the triangle is oriented, as long as C stays equidistant from points a and B, it always lies on that perpendicular bisector, always. And that's going to be the key. Because if we can construct the perpendicular bisector, right? Then we've got a perpendicular line to another one. So keep in mind, though, equidistant from the endpoints of a line segment, and that point will lie on the perpendicular bisector of that line segment. So now, let's review a construction that we've seen before. So in exercise one, we're going to take a look at one that we've done before. Way back when this was actually back in, I think, unit two that we saw this construction. But it's been a little while. So let's take a look. Exercise number one, in this exercise, we review how to construct the perpendicular bisector of a segment. So just what we were talking about. Given, given AB shown below, do the following. All right, so let's just kind of follow along. We're going to draw an arc centered at a above AB that is more than half the length of AB. 

I should really say that whose radius is more than half the length of AB. And draw the same arc also centered at B so I'm going to take my compass, put the sharp end at a, and I'm going to extend it so that it's more than half of the length of AB. All right? You could actually make it the entire length of AB. You could even make it longer than AB. I tend to not quite go that long, but it's kind of up to you. And now what I'm going to do is I'm going to draw an arc kind of up here, there's my arc. And then what I'm going to do is I'm going to draw exactly well, I'm not exactly the same arc, but an arc with exactly the same radius who's centered at B if I can just get my point to actually lie at B there we go. All right? There it is. Okay. So I've drawn these two arcs that have the same radius, but have centers at a and at B all right, says mark their intersection point C let's do that. Okay. Now, letter B says do the same except with a different radius, although it could be the same below AB. Labeled this intersection D now again, I want to emphasize something. The key here is not making the radius different down here. It could be exactly the same. And in fact, when we do the constructions later, I'll likely leave it the same.

 But I want to make a point for why it doesn't have to be the same in a moment. What does have to be the same is that these two arcs have to have the same radius and the ones down here have to have the same radius. But let's get those other two arcs drawn. All right, so what I'm going to do is I'm going to swing this thing all the way around so that it's upside down, I'll put it at a, maybe I'll even stretch it out so that it's all the way or more or less all the way to B yeah, and now I'm going to draw an arc somewhere down here, okay? Flip my compass so that I have the point at B again, let's get that arc drawn. Okay, and get rid of my compass. Says mark that point D, which I will. Label this intersection point D now, we've got a couple questions to answer. Get rid of my ruler. Letter C says, why must point C and D both lie on the perpendicular bisector of AB? Why is that? I claim that C, which I can't see now. It's off the diagram. And D, both lie on the perpendicular bisector of segment AB. Why is that? Pause the video now and write something down. All right, well, hopefully. You said both C and D lie equidistant. From a and B and by that, what I mean is that C lies the same distance from a as it is from B and that's because we kept those two radii the same. And D lies equidistant from a and B because we kept the radii of those two arcs the same. All right? That's key, right? Again, these radii could be the same as these. That's fine as well.

Now it says draw in CD and verify that it is both perpendicular and equidistant and equidistant. Or perpendicular and a bisector, sorry. All right, let's draw that in. Look at that. Not bad. Let me really quickly do the piece to show that it bisects. Right here, I can see that that is 26 millimeters. 26 or 25 millimeters, I'll go 26. And that is not quite 26. It's more close to 25, but pretty close. And what certainly seems to be the case is that it's perpendicular, but I can verify that really quickly. With my protractor and I'm getting a nice 90° angle there. All right. So the key. To doing any real perpendicular line work. Identify one point that's equidistant from the endpoints. Identify another point that's equidistant from the endpoints, draw a line segment through them, and you are guaranteed that that line segment not only bisects this one, but it is also perpendicular to it. Let's see how much you process that. Let's take a look at the next problem. All right? Exercise number two. Given line segment EF below, construct its midpoint and label it M all right, pause the video now, construct its midpoint, label it M, leave all your construction marks. Take a few minutes. All right. Let's go through it. So to find the midpoint, what we have to do is bisect the segment. We have to create the perpendicular bisector. And then where it intersects this line segment, will be the midpoint. So let's do it. 

All right, and I'm going to do it the fast way, and I'll show you what I mean by that. All right, I'm going to put my compass sharpened at E I'm going to stretch that radius out so that it's more than half the length of EF. All right, I'm going to bring it up. I'm going to draw an arc up here. And just so that the picture doesn't get too bad, I'm going to also draw an arc down here, but I won't draw an arc all the way through. You absolutely could. All right? Now, key without changing the radius here. And that's probably even harder for you than it is for me. I'm now going to put the sharp end at F again, bring it up, draw that arc, try to draw that arc. Somehow draw that arc there it is. You think you have problems with Dole pencils. There it is. Move it away. I don't even need to label the intersection points of the arcs. Okay, what I know is that the point up here is equidistant from E and F, the point down here is equidistant from E and F, therefore, both that point, and that point line the perpendicular bisector. 

So if I bring my ruler up. Rotate it around, maybe do a little bit of this. And that and what that means is because that is now the perpendicular bisector of that line segment. This has to be the midpoint. All right? You could easily be asked to construct the midpoint on a standardized exam. And the way that you do it, strangely enough, is almost kind of like overkill, right? We're creating the perpendicular bisector of this segment, which is very, very easy to do, right? Equidistant, equidistant, draw the line, boom, we have it. But because it bisects EF, we know that that must be the midpoint. Okay, now let's get into the real constructions for the day. Perpendicular line construction number one. Okay, here we go. I've got a line. I've got a point on the line and I want to draw or construct a line that is perpendicular to the original that passes through that point. Let's see how to do that. Now, in exercise three, we're going to go through that construction. We want to be able to construct a perpendicular line through a point on a line. Below, we have a line with M, M, with point a, not at its midpoint. A is not at its midpoint. We can create a line perpendicular to M through point a we can create a line perpendicular to M through point a, right? So that's the idea. I want a perpendicular line here, goes through point a let me pause for just a minute and tell you what you'll never get any credit for on something like this. I bought it. 

So if you come over here, even with a straight edge and you go, well, let's see, there it is. There's nothing. Right. Because that's not the point. The point is to demonstrate that you know how to use the tools of constructions to create this perpendicular line. So don't go there. There we go. All right, anyway. So let's walk through this construction. It's kind of cool. Letter a draw a circle around point a so that it intersects the line segment below twice. Mark these intersections points B and C all right, so I'm going to put my compass at point a, all I have to do is draw a circle. And you don't even have to draw a circle. You could just draw an arc and an arc, right? So I could just do this, right? And this, you can draw the whole circle. All right? But if you're worried about the picture getting a little bit too busy, then don't worry about it. All right, we're going to mark these points B and C all right. Now, before we finish the construction, we want to try to understand why it works as we're doing it. Letter B says explain why a must be the midpoint of segment BC. Take a moment, pause the video and try to write something down. Why should a point a be the midpoint of this segment of segment BC? Take a moment. 

All right. Well, here we go. A is the center of a circle. With diameter. BC. I didn't draw the whole circle in, right? But if I did, then BC is a diameter of that circle, a is the center of it, and the center, there's that weird red, will be. The midpoint, and then abbreviation there. Of any diameter. And that should make all the sense in the world to you. Okay. So what we've got is we've got a circle here. And interestingly enough, right, we know exactly where it's center is. By construction. All right, let's keep going. We'll go back to the picture, but construct the perpendicular bisector of BC. All right, so in other words now, I've got this segment. I've kind of created segment BC. And I want to now create the perpendicular bisector of that segment. That's the construction we just did two times in a row. So take a minute and do the perpendicular bisector of BC. All right? Let me go through it really quick. Remember how we create the perpendicular bisector of a segment? Okay, I'm going to take this. I'm going to bring it over to point C, stretch it out so that it's more than half of the length of BC. I'll bring it up here. I'll draw an arc, try to draw an arc still trying to draw an arc. There we go. Maybe rotate it down below. 

Oh, it's going to get right into my wording. Draw an arc there. Bring it back up. Don't let that radius change. Bring it over here to point B all right, come up here. Make sure that we get the arc, there it is. And down here, one of these days, it's going to work on my first time. And move this away. All right, let's now take our straight edge. Bring it over. And. Draw that. Okay. Lots of mess here. All right, since a is the midpoint of BC, we now have a perpendicular line, passing through a, wait a second. That was exactly what I was just trying to do, right? The whole purpose of this construction was to create a perpendicular line that passes through a point, right? And it's perpendicular to the line containing that point. Now how did I do that, right? Let's go back to this for a minute. You draw back for a second, right? I know how, when I have a segment, right? I know how to construct the perpendicular bisector of that segment, right? The problem is that perpendicular bisector of that segment will always go through the midpoint of this segment. So the key is, with my first step in this problem, IE drawing an arc here and drawing an arc here, I make a segment for which a is the midpoint. And therefore, when I then perpendicularly bisect segment BC, that perpendicular bisector must go through point a now you might say, well, Kirk, you missed. And I did miss a little bit, right? It didn't quite go through a it should have gone through a if I'd used computer software and things like that. 

Well, I guess I did use computer software. But if I use a program like Joe Chopra, it would have completely gone through point a I still would have gotten full credit for this quite frankly because I've shown everything I need to and what you absolutely have to show in this case is that arc that arc, these two, these two, and then obviously the line segment itself. But really, the kind of ingenious piece of this is we've now created a segment that makes the point we care about its midpoint. And we do that by drawing a circle with that as its center. That's kind of cool. All right? So let's keep going. The next perpendicular line construction is one where we've got a line and we've got a point that's not on the line and what we're looking to do is create the line that's perpendicular to this one that passes through that point. Okay? And what we're going to see is a trick that's fairly similar to the last one. All right? So let's see how this works. Here we go. Exercise number four, okay? So given line N, shown below and point a, right? We want to construct a line that passes through a and is perpendicular to N simple enough, again, don't eyeball it, no credit for just kind of going like that, nothing. It's great if you know what perpendicularity means and what it means to pass through point a, but still no credit. So let's start. Let's step by step. Let array draw an arc centered at a that intersects N twice. And then we're going to label those intersections B and C now this is trickier than before. 

For me, especially. Because what I've got to do is I've got to put the tip of my compass all right, at point a, and this is actually this much trickier for you as well. And then what has to happen is now when I swing my arc through, it's got to intersect line N twice, okay? It's got to intersect line N twice. And it's a good thing if those intersections are relatively far away from each other. But really quick. Let me call that B and let me call that C okay? Now, letter B says explain why a. Must lie on the perpendicular bisector of segment BC, right now again, keep in mind. When I say segment BC, I'm not talking about this entire segment. I'm talking about only the portion that starts at B and ends at C why must a, lie on the perpendicular bisector of that. It's pretty easy, right? Because a is equidistant. From B and C. Or almost better yet, right? B and C are equidistant from a and their equidistant from a because they lie on a circle, they lie in a circle that has a is its center, and any points that lie in a circle are exactly the same distance as any other points that lie on the same circle away from the center. But that's kind of cool because that means now all we have to do is find one more point that lies equidistant from the endpoints of B and C and then connect it to a watch. Letter C construct the perpendicular bisector of BC. 

This will now pass through a and B perpendicular to N now, you know, you could just start and be like, all right, I'm going to construct the perpendicular bisector of BC and I'm going to bring this over and stretch it out so that it's make an arc up here and another arc down here and another couple over here. And that would be fine. But you don't need to locate a point up here that's equidistant from B and C because you already have one. Point a so now all I really have to do is locate a point down here. So I'll bring my compass down. And I will draw that arcade. It worked at that time. Bring it up, switch it, continue to have exactly the same radius, bring it down, draw the arc, attempt to draw the arc, bring my compass back up, throw it out of here, really. Get another saucy there. All right, bring my ruler down. And oops, bring my ruler down. And over. And hopefully. Draw that line. Look at that. We probably don't even need a protractor to really know that that particular line is perpendicular to that one. Isn't that cool? And again, think of how similar it was to the last construction, right? What we're doing is we're taking this line that we care about. And we're taking a segment of it. We're creating a segment. 

Last time we were creating a segment specifically with this as its midpoint. This time we were creating a segment that we knew its endpoints lied equidistant from the point that we cared about that was off of the line. Once we did that, all we had to do was find one other point that was equidistant from the endpoints of B and C, connect that to a and now we're guaranteed that because both a and mystery point down here are equidistant from B and C, that that must be the perpendicular bisector of BC. Again, the fact that it bisects BC is not nearly as important as the fact that it's perpendicular. Because that's what we were looking to do in the first place. All right? Let's wrap up this lesson. Okay. Two of the most classic constructions in all of geometry were the ones that we looked at today. IE taking a line with a point on it and constructing another line perpendicular to it that passes through that point. As well, we looked at how we could take a line with a point not on the line, draw a line perpendicular to the original one that passes through that point. Both of them relied heavily, though, on the original construction, which was just how do we construct the perpendicular bisector of a segment. Make sure to go back and watch those two last constructions as many times as you need to to really get them down and practice them on the homework. There is almost no better example of where practice makes perfect than on constructions. So don't skimp on those really work hard on them. For now though, I'd like to thank you for joining me for another common core geometry lesson by E math instruction. My name is Kirk Weiler, and until next time, keep thinking and keep solving problems.