Geometry - Pearson 7.3 & 7.4 Review

Hi guys, if you missed the zoom today, this video will give you a nice little review of 7 three and 7 four. So diving right in. This one says that my triangle ABC has measures whose measures are 58 and 58. So I know that my triangle right here that I'm talking about. Both of those measures. So I'm just going to pick random ones right there. I'm saying that a and C are both 58°. So given that, I know that I can figure out it's missing angle, since I know things about a triangle. Given that my triangle, all of the degrees add up to a 180. I know I can take the two angles that I have. Subtract them from that total, and that gives me that be missing angle there is 64°. Which is going to be helpful in what we are trying to prove, because now my next triangle, it's saying that, okay, I have this other triangle that measure is 64 and 58. So again, I'm just going to say, okay, this one's going to be bear with me learning a new program. So this angle here can be 58. And just because I want to, I'll make this angle here 64. So again, knowing that the sum of all the angles is a 180, I can take a 180 subtract 64, subtract 58, and that actually gives me the missing angle is 58. Degrees. So looking at these two triangles, I had to prove if triangle ABC was a similar to DEF. Remember that this symbol here is my similarity symbol. So by looking at them and everything that I found out, since the angle, the triangles have two pairs of congruent angles, I know that I can prove this by my angle angle similarity. Triangle, ABC, and triangle, DEF, are similar bye bye bye angle angle. Similarity. Okay. Onto my next problem here. A little bit more information, just like how we've been always talking in class once we have information in our problem. Just start marking that. So it's telling me that in triangle ABC, my a is three, so that means a side here is three. My B is four. And my seat is 6. In my other triangle here. It's telling me that my D is 9. IE is 12. And that might F is 18. So looking at this triangle, again, I'm trying to prove that they are similar. I don't have any angle measures. That can help me with this. I can't find out any of my angle measures since they're not marked, and we're not going to sit here and use protractor and try to figure that out. So I have to be able to prove that this is similar by using my size. So by looking at my signs, I need to create proportions and see if they remain true for the whole triangle. So if I'm looking at this, I'm saying, okay, I have my side AB here. I also have the side that corresponds with it is D so I know that if I create that portion of AB, which is 6 over DE, which is 18, I take that fraction, I simplify it as much as possible. 6 over 18 simplifies to one third. Okay, so that was my one side that I got my ratio. Now I don't know my other sides. So my BC here is three. My EF here is 9. So I have three over 9, that's simplifies to one third. You do have to check all sides. And just be careful that you're keeping the same format. So like I'm using my triangle on the left first and then my triangle on the right second. And then last but not least, I have my AC, which is four. And then I have my DF, which is 12. And that simplifies to one third. So by looking now at everything we have here, I know that all my sides are proportional. So I know that triangle, ABC, and triangle. DEF are similar. And they are similar because of my side side side similarity. So we can prove by angle angle. We can prove it by side side side. And now we're going to talk about the other way that we can prove it. So looking at these two triangles, again, we're trying to prove that they're similar. And if they are similar write a similarity statement. So looking at this, I know that angle C right here is 61°. And angle F is also 61°. So I know that angle C and angle FR agreement. I also had to compare the sides though. So going back to comparing my sides, I need to make sure that the ratio between the sides holds true. So looking at this, if I have BC and EF, so if I have 6 over ten. That would simplify to three fifths. And then looking at my other side, I have my AC, which is three. And my DF, which is 5. So it is already three fifths. So I know that they have two sides that have proportional sides, and they also have a similar angle. So given that, I can prove that triangle, ABC, and triangle. DEF. Or similar by my side angle side, similarity. So those are my three ways that we can prove triangle similar. That was everything you kind of did in a 7.3. So moving on to our next question here, this is going to start talking about all the stuff new stuff for 7.4. So if I'm given my triangle of PQR, so that is my PQR. That's the bigger triangle there. And I want to prove that it's similar to QSR. That's my smaller triangle. I'm trying to figure out what this side, my QSS. So first thing I'm going to do is I just like to kind of look at these triangles separately. I think it's a little bit more helpful to visualize the proportions. So like my QSR, I know that it's hypotenuse is 15. I know that this leg is 9, that's my S that's my R that's my Q. My bigger triangle here if I made it kind of like visually looking the same as QSR. I had that. Well, here's my 90° angle. Across that 90 grain, I don't know what that measure is. But I do know that the shorter leg is 15. And my longer leg is 20. And that would be P that would be R that would be Q making sure everything checks out. Good to go. So from here, if I want to figure out the missing side, so I want to figure out the whole question is asking what QS is. I need to, again, set up a proportion. So if I want to figure out what QS, I'm just going to say that's X right now. Aside that will correspond with that is 20. And then it's going to equal or I was dealing with this triangle first. So I'm going to have 15. I'm not going to have 15. The 15 second help me actually because I don't know this measure here. So I'm going to use my 9. And my 15. So once I set up my proportion, I'm then just cross multiplying. So I'm looking at my 15 times my X switch to which gives me 15 X and then I have 20. Times 9. With me. And that would give me a 180. So from there, again, I am just solving for X at this point. So I would divide by 15 on both sides, a 180 divided by 15 gives me 12. So to make sure I'm answering the question that's given, what is QS? QS is 12 units. Okay. The other next example, gotta stay there. It tells me that PQR is similar to those exactly the same probably interested. Yes, it was. Okay. So if I have triangle a, C, B, and I want to figure out what this length is. We need to use one of the theorems that we were talking about in this section of 7.4. So the theorem said that we can notice that CD based off the triangle there is the longer leg of CDB. So like this is longer than that side. So this leg is longer than that side there. But this leg is also shorter than that side there. Since we see this, it's called theorem 7.4. It says that CD divides the right triangle ACB into similar triangles. So we know that those two triangles are similar. So from there, we can actually use that knowledge and create a proportion. So I know that my AD, I'm just going to call this one X for right now, because that's what we're trying to find. So if I have AD, that's 6.4. That's going to be over X. And then I would have my X messy D over my BB, which is 3.6. And now again, I'm just going to cross multiply. So X times X gives me X squared. 6.4 times 3.6 gives me 23.4 .04. In order to get XL by itself, we need to take the square root of both sides. So that is how we get X is equal to 4.8. So I know the feed length of the altitude CD is 4.8. Sure. So looking at this next example, it says that AVS 64, so I'm just going to mark it up as it gives us information in the directions. And that AC is 80. We want to find what a B is. So this whole side here. So again, I'm just going to use my knowledge of the geometric mean. So I know that AC this side here is my geometric mean of both triangles because it's my longer leg on the one, and then my shorter leg on the other. So I can set up my proportion as AB over a C and that's going to be equal to AC over 8 D so AV is what we are looking for. So again, I'm just going to use X there. My AC that told us was 80. Again, my AC was 80. And my AD is 64. So from there, cross multiplying 6 four times X gives me a 64 X 80 times 80 gives me 6400. I then want to solve for XO 60 6400 divided by 64 gives me X is equal to a hundred. So making sure we answer that question. AV is 100. Okay. So that was just like a nice little review of the two sections that we've been working on. I just pulled random examples from various things that I found to kind of go over. So if you want, you can kind of test yourself right now. You can pause it before I go over the answer, or you can just use it as extra kind of like video lesson. So if I want to prove if these are similar or not, and then if they are similar, how, and again, if they are somewhere writing a similar statement. So looking at this again, I'm seeing all I have all sides. I have no information on angles. I need to prove somehow that these are similar by using my side. So I know that I have to kind of figure out my ratios to see if they work that way. So the one thing that I'm going to do for this number 7 is I'm just going to take this lovely triangle here. And just kind of make it look like that one. So I know my shorter leg is on the ground here, like this side, and that's I would match up. And then my hypotenuses are like that. So I'm taking this triangle. That would be my leg. That would be my hypotenuse and my other leg. So my hypotenuse here is 30, a shorter leg is 15. And this would be 25. I'd just like to do that just so I can kind of see better which one's going to be matching up with what, so if we're rotating it or something, that's just I think personally a nicer way of looking at it. So now I'm going to set up some proportions. So I'm seeing that my 80 and my third, my 84 over 30. And then if I simplify that, 84 over 30 gives me 2.8. And I'm also going to look at my other side, so I have 70 and 25. So 7 8 over 25. That also gives me 2.8. And then last but not least, my 42 over 15. 15. And that also gives me 2.8. So based off that information there, I know that this can be proved similar. Or my side side side. I'll stick with what they're doing for T, UV is similar to triangle. So this is why I think rotating. It's super helpful. If that was 15 and that was my hypotenuse, so this should be S that would be R and that would be Q so now I can match up and say, okay, if I have T I need to put my Q first and then my R and then I guess so triangle U TUV is similar to triangle QRS. By side side side, similarity. Okay. Looking at my next problem here. A nice line down my screen so I don't mess up the work. I can use the line tool this time. There we go. Okay. So now looking at this number 8, again, same thing. I want to prove that they're similar and if they are right a similarity statement. So one thing we're looking at is I know I'm giving a couple angles here. If there's anything else that I can kind of go from there and learn. So I know that this angle here that this angle here, they would be congruent since they are vertical angles. But that is the only thing that I'm able to prove. I can't prove anything else like they use the one tick mark here to here in three here. But I know nothing about this one, so I have not enough information. I can not prove it by any means, so these two triangles are not similar. Let's go stay there. And then number 9 here. So looking at this one, it gives us that this angle measure 61. I also see that angle measure is 61 over there, which is awesome. Looking good so far. And then looking at my figure, anything else that I can point out, well, I know that this angle measure here is congruent to that angle measure there because they are vertical angles. So I can actually prove that triangle HDF. It's similar to triangle. So if I have H G F, the convert to triangle H. TS all right angle angle similarity. Okay. Making my line here. And looking at number ten. So again, if you want to like test yourself, pause the video now. Otherwise I'm going through it. So we see that. This angle and that angle are congruent. I don't have any information besides those angles so I can't use angle angle. I don't have all three sides. So I'm going to have to try to prove this by side angle side similarity. So looking at this here, if this 42. It's my longer leg that would be matching up with my longer leg of UV. So if I do 42. 42 over 18. And I simplify that for two or 18, about 2.3 repeating two. Repeating. And then my other side there, I have, I want to make sure I did the 42 first, so I have the 21 first. I'm dividing that by 9. 9, that looks like a force, so that's going to drive me crazy. There we go. And when I set up that proportion, it also gives me 2.3. So now I can say, okay, I can prove these triangles similar. So number ten, I have triangle. I'll start with the smaller one. W, UV is similar to triangle. I'm going to match it up. So I did W so it was the 9. Okay, so I'm going to do triangle H FG. So triangle WV is similar to triangle. H F G by side angle side, similarity. Okay. So now just four problems looking at these. So if I'm given this figure here and it doesn't know what this I'm trying to find this question mark. So I know that they can be similar. By side angle side. So I know that my angles are there. That means that I also know that these sides should hold a proportion. So I'm going to set up if I have 28, that's going to be over 42. That's the proportion of my hypotenuses. And then I'm going to set up X over. 33. So they're just cross multiplying. Let's switch to my text box. So 42 times X just gives me 42 X 28 times 33 gives me 924. From there, I would divide by 42 on both sides, and that's how I get that X is 22 units. I don't know if the next one here. So this is a prime example of when I'm going to break up my problem. So what I want to find out is this little sliver here. I know that this whole side is 12. So I know I'm just going to change colors, so not confuse it with the other problem. I have this big triangle. That is not a triangle. So this is T this is S and this is U we know that this side is 12. And I know on the bottom there, I have 18 plus 6, that's going to give me the length of 24 there. So that represents your bigger triangle. And then looking at the smaller one here, so this is U this is one thing that I'm just going to do for this is I'm going to call this side Y right now because I'm actually not trying to figure out what this measure is. I want to figure out what this little sliver here is. So the first thing that we do have to do is set up our proportions here. So I have my 24 that's going to be over 6. And then I have like 12 over my Y. So again, just cross multiplying that would give me 24 Y is equal to 6 times 12 gives me 72. But then would divide by 24 on both sides. So Y is equal to three. So now, given the information that we have, I know that this right here is three. And I want to figure out what X is so I can take my 12. I can subtract the three from it. And that's how I get that X is equal to 9 on this problem. Okay. So I want to find the missing length indicated. Leave your answer in the simplest radical form if needed. So looking at our problem here, again, we're using RN geometric mean to kind of help us out there. So given this smaller triangle, I know that my 12 here is my altitude. And then give them a bigger triangle, this one here, I know I can set up the proportion of this side X over 12. And that's going to be proportional to my 12 over my 16. So from there, cross multiplying 16 X is equal to a 144. I've been want to divide by 16. So that's how I get X is equal to 9. And that was all possible. I was able to set up that proportion. Because we realized that this 12 here, like the side length of 12, is the geometric mean. That allows us to set this up. Okay, looking at so I know that I'm trying to find again the psilocybin here. I know that my altitude like geometric mean here is 48 and the other side is 64. So I can set up that 64 over 48. Is going to be equal to my 48 over X and then from there, just cross multiplying so 64 X is equal to 48 times 48 gives me 2000, 304. I would then divide by 64. And that's how I get X is equal to 36. Okay, so that was my little review of our two sections. I hope you guys enjoyed. If you have any more questions, you just let us know. Thanks.