Geo 9.1 Area of Parallelograms and Triangles

All right here we go let's see what we can just have with your 9 chapter 9 segment great one. It is area when I think area I think of the classic video game Area 51 such a great game. I love it. So let's take a look at area. Whole chapter is about finding area of two dimensional shapes. So let's make sure we're cool on that. Excuse me, let's start off with parallelograms here. And let's just draw one in here. So parallelogram, I think the easiest one to see area would be something like this, maybe go one, two, three, four, and maybe I'll go one, two, three, four, 5. Let's go 6. So if you check this out, so I'm just drawing a rectangle in here. And the reason I'm doing it is because what is this? This is four units this way. And this. This is a four by 6 rectangle. And let's pretend that these were centimeters. Each one of these boxes were centimeters. So this would be 6 centimeters wide. This would be four centimeters wide. So what can you do? I want to know the area. I want to know what is the area of this. Well, what does that mean? You have to count up all these boxes. Well, how many do you got? You've got 6 rows of four. So if you count them all up, what do you get? You got 24. So why do I have 24? Well, I mean, I have 24, and these are centimeter. What? We'll look at this. What is this? This is now a squared centimeter. So this is centimeter square because it's one centimeter by one centimeter. So the area of this is one centimeter squared. So I actually have 24 centimeters squared. So that's what areas, that's why it's squared. We're talking about the square little blocks inside of there. So what's the formula for a parallelogram? A lot of times you may think, yeah, it's length times width is the classic rectangle. We're going to refer to it from here on out is base times height. So what was the base of this, the base was 6. The height was four. It was 6 times four. So it's great for rectangle parallelogram. That is a parallelogram. Opposite sides are parallel. But what about when I slant it up like this in a parallelogram? Is it still work? Yes, it still works. Just be careful. Where's your height? Your height is you can think of a couple of different ways. You can think of this as your height out here. And then this is your base over here. Or a lot of times we like to draw the height. I actually on the inside. So there's the height and the base. So this works for parallelograms. In a rectangle is a special case of a parallelogram. Excellent. So that's half of the section right there. Let's try some mixture we're good with this. So I am doing base times height. And this is a rectangle. So this is going to be three times 6. Just make sure you label everything. So this is what this is 18 meters squared. So we got to put that label in there, especially in metrics, and whatnot, or else we're not going to count it. Make sure you have it in there. Does this freak you out? So now I have really got a parallelogram in here. So I can tell the base is just happens to be on the side, the height is this dotted line through the middle of the right angle on it. So not a problem, this is going to be what. It's going to be bases 11. The height is 7.9. Bust out your calculator here real quick. Let's clear that out. We've got 11 times 7.9. So we're going to have some nice decimals in this one. We got 86.9. And again, what was our units we're talking about yards squared, fantastic. So we're just cruising. Let's do one more of these. Make sure we're good to go. Does it freak you out when the heights on the outside out here? This is the height of the parallelogram. Oh, and this is a little bit different. I don't know the base. So now if I have a equals base times height, the area is actually 12, I don't know the base. I'm going to call it X in the height is two. So can you solve this? And maybe you'd like to look at it like this. X times two is the same thing as two X and can I solve this? Sure, just divide both sides by two. What am I looking at here? 6. And what is this? This is actually 6 meters. So it's not square because I'm talking about this distance here, 6 meters. So maybe I'll give you the area. You've got to find a missing side, or maybe I'll ask you to find the area. Fantastic moving on. What about triangles then? So triangles are really the formula comes straight from a rectangle. You know, remember, this was base times height, is the area of this. We'll check this out. If I draw this across here, what do I have? I have two triangles, don't I? Here is a triangle here. So what is the area of this triangle? Well, the area is base times height, but what happens? I only want half of it. There are two triangles here, one, and two, I only want half of it. So really, you've probably seen this before. It is base times height. So we have something called an altitude, the altitude is just the height. It's when I draw in that right angle. The key is that must be a right angle right there. So what in the world does that look like? For most triangles. Most triangles, it's going to be, we'll draw it right in the middle. We'll say, here's the base. Draw your altitude straight down from up here and put the right angle in. And there is our height. So our base in our height. But with obtuse triangles, it can look a little crazy. And it's a lot like the parallelogram. If you are okay with the parallelogram, you can draw it any way you want. But if I want to draw from here, it's straight down. And I kind of have to continue this to show that right angle. So here's the base that goes with that height. And you can change it. I can draw that same obtuse, and I can draw another one. I could have actually drawn the height in this way if I wanted to. So you draw in any altitude you want. That would have been a height, but now this is my basis long side over here. So you may see it any one of these ways, but we're just doing one half base times height. Awesome. So I know if I see this formula, let's just make sure we're cool with it. In this case, which one's the base? Well, if that's my height, it's got the right angle. What's got to be the base this has to be the base down here. So we're looking at one half base is 6 height is 4.7. It's just a plug and chug. I'm going to go straight to calculator. I'm going to call one half just .5. So I'm going to make it a decimal right off the bat, and I'm going to times it by 6. Times it by 4.7 boom there it is. So that gives it to me 14.1. And I should label this 14.1 14.1 yards squared. Fantastic. Moving on, just make sure we got it. Can we see which one is the base and height? Well, I know there's my height, who's the base, this is the base then. So no problem, this is just going to be one half base is 7. Height is 5.4, and I know I'm going fast, feel free to pause and slow me down. And let's make sure we have this. Again, I like to type in one half as .5, so I'm going to go one half times 7 times 5.4. And you get 18.9. So we get 18.9 meters squared. Boom. I love it. Love it. One more here. What do we got back here? Hiding back here. Excuse me. So now, oh, it's one of these where I don't know the what part do I know. I don't know the height of this triangle. So they give me the area 35. It's still going to be one half base, the base in this case is 11.3. But I don't know the height. So can I solve it? Sure, let's simplify this a little bit. 35, what is half of 11.3? So you can do 11.3 divided by two if you want. Or if you just want to stick with it is always decimal. You can say .5 times 11.3. So either way you like to think about it, I'm going to stick with .5. Keep it consistent. I've got 5.67. So this is really 5.65. Is the height. So what do I do to solve both sides? I want to get H by itself. So I'm going to divide both sides by 5.65. And that'll give me an H of what is that. Let's close this out here. 35 divided by 5.65 boom. 6.19. And I'm going to round a two decimal places here. So 6.19 looks great and we're in meters here. Now meters squared is a distance 6.19 meters. Fantastic. So there are triangles, so this should be a pretty nice section. So those were kind of all the normal ones. Then we're going to mix in some of the stuff from chapter 7. Make sure to keep that fresh for us. So what if I don't give you all the information? I could see, okay, here's the base. Here's the height, but I don't have any of the info. Well, this is a special right triangle. If you remember back to 30, 60 90, we can go back to a similar triangle where the small side is one, the big sites too, and the medium side is radical three. So if I want to find the height, what do I do? I say ten is to two, ten is the two as H is to one. NH over one is just one. So this is really 5, so I can say, okay, no problem, H is 5. So if I want to do that because H is 5, how about the other one? For part B, I can do the same thing. I'm going to put up here. I can say ten is the two. The long side matches alongside as B matches radical three. And we can do our little cross multiply, we can say two B equals ten radical three. And what do we do by both sides? Got to divide by two. So really I can find B let me get rid of this. B is going to be what ten radical three divided by two is 5 radical three. So a little bit of work there to fill that in. But now I can do my formula. We could say area is one half base times height. The base happens to be 5 radical three. The height happens to be 5. Can I multiply this all out? Sure. One half times 5 radical three times 5. So the whole number is I can multiply so I can actually do 5 times 5 is 25, and I'm going to times it by half is 25s halves. So this would be the exact possible answer. That's a lot. If you want, that's great. If you want to leave it in the simplest radical form, that is a fantastic answer. If you want to make this all decimals, you're more than welcome just to type it all in and say .5 times 5 radical three. Where's my radical? There it is. Radical three. And we're going to times that by 5. And if you want to put this answer, that's great. 21.65. So you can approximate it as 21.65. It's a pretty good approximation. Just make sure you label it. We're talking meter squared. So either one of these is cool. And we're good to go. Fantastic, very nice. What else do we got? So we had special right triangles. What about Pythagorean theorem? So back to the parallelogram. So I've got the base. Can you tell the base the base is all of this? This is the whole entire base. So the base is 20. I don't know the height. So I need to find the height, but if you look at just the top here, there is a little triangle here. It's this, this and this. So I've got Pythagorean theorem where I'm going to say H squared plus 5 squared is 13 squared. And maybe you recognize this one. This is one of the special Pythagorean triples. If not, you can just solve it though. We've got H squared plus 25 is one 69. And subtract 25 from both sides, you get one 44. And then we're going to take the square root of both sides. So H is really 12. So if H is 12, now we've got the height. Now we can fill this in and say, parallel gram is what area is base times height. So it's going to be 20 times 12. Fantastic. And what's 20 times 12? That's going to be 240 meters squared. So that's number two. So maybe you have to use a little Pythagorean theorem in there. I love it. And the last thing that can happen is maybe we'll do a little trig function action. So it's not a special one like 30 60 90 or 45, 45, 90. It's just a trig. So what do I need? I need the base here. What trig function is going to relate these? Well, remember, this is what? This is opposite. This is adjacent. So I'm going to say opposite over adjacent. That's got to be tangent. The tangent of 61 equals what? Opposite over adjacent. And now let's solve this so you're going to multiply both sides by B do not like being in the bottom there. So we're looking at B tan 61, and 61 is just some decimal. So don't freak out about that. These cancel, and I want to get B by itself, so I'm going to divide by this decimal, which is ten 61, whatever you do to one side you got to do to the other. So those cancel. So really B equals, now I got to go calculator. Double check your mode. Oh, I was in the wrong mode. Got to be in degree mode for this to work. So we were looking at what was that? 19 divided by tangent of 61. It enter and I get 10.53. So again, two decibels, please 10.53 and what is this? This is, oh, there's no units here. If that happens, you can put units squared. So I don't know what they are. I'm going to put units squared in there. I'm going to pretend like I meant for that to happen to show that example, and there it is right there. If you want to use trig for this special right triangle, you could have used trig right here. You'll still get this decimal. You could have done some kind of sine and cosine to find these sides here, and it would have worked out great. So if you do not like those, feel free to go straight to trig. Awesome, we've got one more. All right, let's bring the pain here. Wrap this bad boy up. Love it. All right, so a lot going on this problem right here. It looks like just our normal area equals base times height problem. So the area is given to us in this case is 16. The base is this whole thing up here, X plus two. So make sure you put in parentheses as X plus two, the height of this thing again has to be in parentheses is X minus four. So why we got a little bit of work cut out for us to solve it, but I love it. This is great. So I'm going to distribute this, double distribute something called foil, but we like to double distribute, so I'm going to go X times X is X squared. X times four is minus F four X, then distribute the two. Two times X is two X two times four. So this is why we do in that algebra review. Make sure this stays fresh so you got it. So this is no problem for you. Now what could I do? I can combine like terms here. So add your X's together. I've got minus four X plus two X is minus two X and now I'm ready to solve this. Well, how do you solve this? Well, it's a quadratic. I get this X squared here, so I want to set it equal to zero. So I'm going to subtract 16 from both sides. And now it says zero equals X squared minus two X, negative 8 -16 more is -24. Fantastic. Now we are ready to factor it. So we're going to factor this bad boy on bottom, negative 24. What multiplies the negative 24 adds or subtracts to negative two. I think we're looking at four and 6, and who's got to be the negative. It's got to be the 6 because four -6 is negative two. Four times negative 6 is 24. Fantastic. So when I factor this, I've got X plus four. X -6 in this equals zero. So we factored it. So what are the actual solutions? What makes this zero? Well, if X is negative four, and if X is 6, now when I go back to this problem, is negative four really going to make sense. Can I really put negative four in for X? No, because negative four plus two is negative two. Can this distance be negative two? No, that's impossible. So negative four is actually impossible. Just to make sure you may want to make a note of that. That doesn't work. But this 6 definitely does work when you put it in 6 minus four is two, 6 plus two is 8. Multiplies 16 in your good to go fantastic. We start with a little alienated farm. We're going to wrap it up with alien ant farm because we got the whole alien theme rock and good luck on the mastery check. Peace out.