# Geo Ch5 L4 - Medians and Altitudes

## Math

Thanks for watching. Got another lesson here. This one is medians and altitudes for chapter 5 lesson four. The objective is to use the properties of medians and altitudes of a triangle. Nice. So when you build the medians, you're actually creating the centroid of the circle, because they meet here at the centroid. I am a centroid, Sent here to destroy you. I'm just kidding. Medians are, we miss the definition of it. But medians are where you take you connect the vertex to the midpoint of the opposite side. So it's a vertex to midpoint vertex the midpoint. So it is cutting the sides in half so you go from vertex to the midpoint, which is obviously cutting in half. And it shows that by giving you a little dash marks here like that that teased you or split into two equal pieces. There's special properties, the special properties is that there's always a long piece that cuts it up into three pieces. These little ones are always one piece. There's like one of them in there. And that is equal to this one piece here is equal to two of this one. There's two of those pieces in here. So this one, this little tiny one has one, the other two have two pieces. Inside of it. What we can do is do what we can dive into it, you know. The other one is the concurrency of altitudes. Altitudes is, I guess I should have backed up and told you, hey, how do you find the altitude of this mountain? Well, you wouldn't say, oh, altitude is like how high it is, right? If you talk about altitudes and mountains, you're talking about how high it is. You wouldn't measure how high that mountain by looking at this line. That makes some sense. How tall is a mountain? You'd be like, well, it's measured by like putting the ruler on here this way. That makes no sense. You can do that. You'll be making perpendicular line. So you have to measure the altitude by putting a straight line in there like that. And it's got to be perpendicular to the ground to measure how tall it is. So that's the most important part about. Altitudes is you're making a perpendicular line. So notice that the 9 lines that contain the altitudes of their concurrence and the concurrent that's called the orthocenter. So in a right triangle, if you connect, if you make altitudes, notice the altitudes you're always the corner. That's key. It's going to come up later. This is always the orthocenter. Right there. Obtuse ones and the cute ones are a little bit more confusing, but they won't really come up as much. Let's jump into it. Oh, here's all the summary of all four. Perpendicular bisectors made at the circumstance angle bisectors make the incenter beatings make the centroid altitudes make the orthocenter and I also have notes on the right, like I showed in the last video. If you like those notes, you can download them. They come in handy. I'm just going to show them. This is nice because it goes through here's the enchanter. How is it made? And go buy sectors, same for the circumstance, you know? It shows all that, but what's nice about this one is it gives you key properties like the location, right? Location forum, the special properties down here. So this is important. It comes in handy. If you like it, do go ahead and download it. Nice. All right, let's jump into questions. So number one, and this is the kind of the quick check that you would do in the book. So right away, the first four problems in the last inter hallway is the best because they really hits home and whether or not you know what you're doing. So is AP a median or an altitude, let's take a look. AP to P now if it's an altitude, you'd have to be making a perpendicular to a certain side. If you look, you might get distracted that you think that. Right angle has something to do with the AP, but no. If you look for a right angle here or here or you don't see it, notice that you're going from the vertex to the midpoint here because you're splitting this in half and that was actually called the median. Because you split these two in half. Since it's median, we were dealing with medians or we made a centroid is what we made. In the words of centroid, with centroid is here where the medians intersect. So we have a centroid point right there. That comes in handy. So it can help us solve number two. If AP's 18 in order to KP, so AP, this whole entire thing is 18. So the idea is that there's three pieces in there. This is one piece. And then there's two pieces in here. There's one piece two piece and three pieces in there. There's three pieces all together. And they're all equal. There's one, two, and three pieces. They're all equal. So the way usually the best strategy is to take the 18, the whole, take the whole thing, and chop it up into three pieces. The 18 and cut up in three, and you get 6, so what would KP be? Well, KP is just one of the pieces. KP just has one piece. If you go from K to P, that just has one piece. So that would already just be 6. Done. Now I'm going to delete some stuff because I got like callers on here. So deleting the Android still K though. So centroid is still here. Nice. If BK is equal to 15, this BK is equal to 15. Then how long is K Q? Well, the idea is that if this BK is 15, it's got two pieces there. It's got one piece and two pieces. So that means that you'd want to take that and cut up into two. There's two pieces in there. So take the 15 and cut that up into two pieces. And then what do you get? Well, you get 7.5. So each one of these is 7.5. So you can put that down that each one of these is 7.5. But then I also means then that that piece is 7.5 here. This is 7.5, and then that one also here is 7.5 too as well. So the answer would be 7.5. Strategy is just like if they give you the whole thing three pieces, it cut up in the three. They give you the one piece up here. You cut it up into two pieces because that one has two. Number four, which two segments are altitudes while altitudes are the ones with a 90° angle. So if you see here, really, if you see it, it's this one because it's attached to the 90° angle. So AC is one of them. And another line that's attached to 90° angle is AB. Since those are actually measuring the height of the triangle. So both AB, and aC. Because it's attached to 90° angle. Let's do it. In the triangle DEF, the midpoint of the side opposite the vertex D is M and the centroid is C, if DM is 21, what our DC and CM? So we got, you can set it up really quick. Got D here. And then it's going from as the vertex D is going to and which makes sense midpoint, right? And we have a centroid C usually centroid is somewhere closer to the midpoint over here. So C is in centroid. So how long is this piece? So I didn't really cut up. I didn't really draw perfect, but remember this one has two pieces in each one of these is like equal so you can say like X and this one also has to be an X because it's a three pieces together. And if it tells you the DM is 21, rather than you could say 21 is equal to, well, we have three pieces. They're all equal. One, and two, and three, would that be equal to three X and you solve and divide by three, divide by three, and you get X equals 7. So what are D.C. and CM? What DC, DC? D.C. is two of them. Two 7s. You have 7 and another 7. Well, D.C. then would be equal to 14. And then CM is just one of them, so that's just one X, which would be 7. Bam, all right. Find the order goes center of ABC. We have to pop the points. Supply A, B, and C. A is three 7. That'll be here go over three up 7 one two 7 right there. That's a now for B it's 9,7. 9,7 severe right here. And then for C be 7, 3. In here. Then we combine these together? Then combine these together to see what we got here. Nice. And so what we want to do is find the orthocenter, so ortho centers, making perpendicular lines. From the vertex, and to the opposite, you start at each vertex and you make a line perpendicular to it. So this one's nice because you just go straight vertical. And then you would have a perpendicular line here. Like that. These ones are a little bit trickier because you have to kind of guess a little bit. So you might have to guess, all right, looks like it's perpendicular here. And this one is perpendicular. Here, so we're going to meet. Um. It's a tough one, isn't it? Well, it's a tough one. You can see that they all meet here. It's tough to be able to particularly see it. So this is where Gerggia comes in handy, okay? Let's use Gerggia . So what you want to do is plot the points with three first point with three 7 enter. Next one was 9,7, oops. Get the column there. Ding. 9 karma 7 and your next last one is three comma. 7. Oops. 7 comma three backwards. An inch here. And then we got the points over here. So you can create, you can connect these, connect them. And then so what you do now to be able to find you can make perpendicular lines. That's what you do. You make perpendicular lines. You have to kind of go backwards though a little bit. You can actually have to click on the side first to click on this side and then I connect it to this vertex. I click on this side now and click on that vertex. And click on this side and connect it to this vertex. And then indeed, we kind of got close when I did it myself, is that the point where they all intersect is right there. Right there. And that point is 7 5. 7 5. Bam. Now let's just go back, let's see if my picture was pretty accurate. Let's see here. It's a close, it's a promise. It's a little close to the law. It's not easy to make these. But it is that this is the 7 mark here, and then this is right here would be the 5 mark. So yeah, it is close to 7 5. The little off though. That's the problem with these. They're kind of tough, right? So Georgia brought helps out. Let's move on, last question here. The figure at the right, C is the centroid. If HD is equal to 30 X plus 12 Y, what expression represents CD? Well, in the theorem, if you looked at the theorem, this CD is CD is two thirds. CD is equal to two thirds HD. The whole thing. So this one's kind of complicated, but what you had to do is substitute. Take two thirds of that expression of HD, which is the 30 X plus 12 Y. And now when it comes to fractions, which usually do is this top number means the multiply. And this dot bottom number means divide. So maybe what I want to do is just divide by three for each of these first. 5 by three on each of these first. If you do that, you would still be left with multiplying by the M you still need to multiply by you still need to multiply by the top to two. So I'm just going to keep that down here, but I'm going to divide each of these by three. So I get ten X plus four Y but then still add the multiply by two. And then that would make so two times 20 is excuse me. Two times ten, it makes 20 X and then two times the four makes 8 Y. So fractions are that hard. So if you see a fraction, just think the toggle number just means multiply the bottom number, which means the