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QUESTion 30 Ktech olutn tolamm rorIbans Van o tu (Yoatud riot have coud pumn" Uatnd = cwcuatnr 4t ' CMen AlaaS50,Xn 0x0) 50 Nr Fofal 42 046w) Fo( ) OcCit)...

Question

QUESTion 30 Ktech olutn tolamm rorIbans Van o tu (Yoatud riot have coud pumn" Uatnd = cwcuatnr 4t ' CMen AlaaS50,Xn 0x0) 50 Nr Fofal 42 046w) Fo( ) OcCit) SOxq} CaSoyl) Nalnl 02 00 IOskol Oo Wz Nata} Oz1 NOxq}

QUESTion 30 Ktech olutn tolamm rorIbans Van o tu (Yoatud riot have coud pumn" Uatnd = cwcuatnr 4t ' CMen Alaa S50,Xn 0x0) 50 Nr Fofal 42 046w) Fo( ) OcCit) SOxq} CaSoyl) Nalnl 02 00 IOskol Oo Wz Nata} Oz1 NOxq}



Answers

In Exercises $43-48,$ the diagonals of rectangle QRST intersect at $\mathrm{P}$ . Given that $\mathrm{m} \angle \mathrm{PTS}$ $34^{\circ}$ and $\mathrm{QS}$ 10, Find the indicated measure.
$$
\mathrm{RP}
$$

Here, we're told that A is the matrix A B C D B is the matrix P Q R S. And we want to establish the identity that M times I possibly is equal to M A plus. Yeah, B. Now to do this, let's begin here with the left hand side. Mhm mm Times A plus B is equal to m times adding these two major seats together. Just means add the parts that go together. A post P is a post P be prosecutors be prosecuted, then we have seaports are and D plus S. Yeah. Now multiplying a matrix by M. Just means multiplying every part by em as we just distribute the Mn to each component here. Like, so now we can break this apart and so this is the same as M A M B EMC M D plus M P Mq M R m s. And we can pull out an amount of each of these. So this is in times A B c d plus, mm times p q R. Yes, and notice that this is sometimes the original matrix A plus M times the original major to be, and that's what we wanted to show.

Problem. We were given that matrix A. Is the matrix A B C D. And we want to show that M plus in times a as you put it in a possible in a. Now let's begin here with the left hand side. With impulse 10 times a in plus in times a means that we want to multiply every component of A. I am present. And so if you take the top left, eight times in plus in is am plus I am. Yeah, be times in plus and as BM plus being and notice how distributing all these in, distribute the seeing that CM policy in. And then lastly, D. M plus the end. Next I'm going to take and break this matrix apart. And so I can write this as A M BM. Let's see him and GM plus A. M. Being seeing and Dean. Yes, that's first metrics. They all have an M. In them. So I can pull the em out and that's A B C D. The second matrix I can pull an in out Elizabeth. A. B. See. Indeed. Well, A b c. D. Is just your matrix. A. And so this is just the same as M times A Plus 10 times a. And so we showed that impulse, sometimes A is equal to M. A. Wasn't it?

For this problem we're going to be integrating 3 -2 T. Were given the initial condition that when tea was zero that the answer should be negative five. So let's go ahead and integrate integrating a constant. We just add the variable on. So that's gonna be three T. And integrating to T. Where it's going to become T squared. We're gonna divide by two. That's gonna give us minus T squared and then never forget to add the plus C. Now when we plug in zero for T we should get negative five. So that's gonna get us zero minus zero plus C equals negative five or C equals negative five. So then we just put that in, our final answer is going to be three t minus t squared minus five.

So for this particular problem, we're looking at shape J K L M. And each of these letters stands for a different points that we're going to plot on this coordinate right in the middle of our screen. And when we plot these points, we are going to be, uh, connecting the lines to make this shape. And we need to classify it as a rectangle or a rhombus or a square, seeing if it meets the criteria to be called each of these different terms. So let's just go through the different coordinate points and then plot them one by one. So we have J, which is negative four to These are our points that we need to plot. And again the first number is going to be in the X coordinate, referring to the X axis. So it's going to be to the left the negative side four and then up to so it should be just right about here. Beautiful. Now let's do Point K, which is 03 and I'm just staggering these a little bit, so we don't overlap too much on the left side of the screen where I'm running the actual coordinates before I plot them. So on X axis, it's at 0.0 and then it's up. Three. Meaning it should be just about here than we have. Point L, which we're told is going to be one negative one. She can plot over one down, one just fall right there. And then we have finally point M, which is negative. Three negative two. Beautiful. And we can just plot that here. That's over. Three. Down to right about right there. Okay, nice and easy. So now let's try and connect these dots to make them as neat as possible. So we have our that was not a good line. So we have our shape. J k l m. Keep in mind that these are going to be perfectly straight lines. I am subjected to human error as this is a a hand drawn coordinate plane. And then the points are connected by me. So that is our shape. J k l m. So now let's begin to try and classify it as each of these terms. So what is a rectangle mathematically So it has to be a quadrilateral. So four sides quadrilateral, meaning four sides. It needs 4 90 degree angles and keep in mind this isn't 490 degree angles. I'll just do for in parentheses to differentiate. And this would also indicates that there's going to be two sets of parallel sides, although this is kind of implied by the 90 degree angles. It's just another thing that we can keep in mind. So and we don't have We know what the shape has four sides based on the drawing. So beautiful. But what about the last two criterion? Well, let's look at the actual drawing so we can determine if these are going to be 90 degree angles. So let's look at points J and M really quickly. So point Jay is negative. Four and up to point em is negative three and two. So if you look at the difference in the X column, this is to the right one or positive one, and then it is down for okay, down minus four. And if we look at the difference so this again, I'll just do this really quickly in blue. That is this line right here. That is the difference in the points between these two points to make this line. It's over one and then down four. So I'll keep the blue there just for visualization. And now let's look at points K and L. It's okay and l 03 and then one negative one. So if you look at that again, this is actually over plus one, and then it's going to be down for the same thing, right? Because 32 negative one. Subtract 4021 You add one. So the difference between these are the same, which A indicates that they are parallel lines. That's wonderful. And if we're going to connect the other lines by the same metrics, they're going to be parallel as well. So that's a great sign. It means that this indeed is a a rectangle based on our criterion. So let's go to square really quickly. So what's the criterion for? A square has to have four sides again. It needs 4 90 degree angles. But each side is equal length. So equal length sides. That is the difference between, uh, rectangle and a square. Okay, so are these sides equal length? So now we need to look at I'm gonna do this in green here. This line here, which is point Jay. Two point k. So this line here. So it's going to be to the right for and then up one, right? We see it to the right four and then up one. And I'll just do that right for up one. Okay, now let's do the same for points l two m points out to, um All right, let's do em two L Because if you look at the orientation of the plot, this makes more sense because looking from left to right as we go. So this is going to be from here to hear in this direction this line here. Okay, so from here to here again, right for negative 321 is plus four and look at the X axis. So it's to the right. Then it's up. One negative. Two negative one that's up. One plus one is going up in the Y axis, so it's the same change. It's the same change between these two points, meaning that this changes what the actual side length is responsible for. So indeed, this means that it has equal length sides, so this is correct. So as a rectangle, and it is a square, so which totally makes sense. This can be possible, but is it possible for something to be a rhombus when it's at the same time, concurrently being a rectangle and a square, so is cannot be around this. The answer is yes, because a rhombus like rectangle and square. Let's just write our classification here for rhombus like a rectangle and square, we need four sides. Definitely has we need parallel sides, and we have that again, as we've already demonstrated. And we also need, uh, equal lengths, which we just demonstrated. So what is the difference between a rhombus and square? Because so far it seems like it's the exact same thing. So for a rhombus, we do not need for it to have 90 degree angles. This is not a requirement. It can be true, but it does not need to be true. To classify it as because a rhombus. It could be something like this, which you can see. It's not perfectly drawn, but it's a little slanted, so this is not a 90 degree angle here, so that's the difference here. But as we see is indeed is a rectangle around us. The little green check here and a square based on the way we've plotted it and going through the criteria and for it to meet these different terms.


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