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Let $O$ be the origin and let $A(2,0), B(0,2)$ be two points. If $P(x, y)$ is a point such that $x y>0$ and $x+y<$ 2 , then(A) $P$ lies either inside the tria...

Question

Let $O$ be the origin and let $A(2,0), B(0,2)$ be two points. If $P(x, y)$ is a point such that $x y>0$ and $x+y<$ 2 , then(A) $P$ lies either inside the triangle $O A B$ or in the third quadrant(B) $P$ cannot be inside the triangle $O A B$(C) $P$ lies inside the triangle $O A B$(D) none of these

Let $O$ be the origin and let $A(2,0), B(0,2)$ be two points. If $P(x, y)$ is a point such that $x y>0$ and $x+y<$ 2 , then (A) $P$ lies either inside the triangle $O A B$ or in the third quadrant (B) $P$ cannot be inside the triangle $O A B$ (C) $P$ lies inside the triangle $O A B$ (D) none of these



Answers

$A, B, C,$ and $D$ are noncoplanar. $\triangle A B C, \triangle A B C, \triangle A C D,$ and $\triangle A B D$ are equilateral. $X$ and $Y$ are midpoints of $\overline{A C}$ and $\frac{\text { and }}{A D} . \quad Z$ is a point on $\overline{A B} .$ What kind of triangle is $\triangle X Y Z ?$ Explain.

So this is the pre image I have on DIT says the triangle is rotated 90 degrees counterculture clockwise about the origin. So let's start with the easy points and that's A and B. They're exactly on this X axis here on. If they were to go 90 degrees counterclockwise, they would just go on the y axis here they're just rotate like that. So point a would be over here and point B would be over here. Ah, to be specific, Uh, it's called this a prime and be prime. A prime is gonna be a 01 and be prime is gonna be at zero Fi. Okay, Now for point C see is ah two comma three. So this means that from the origin it goes to to the left and three up. Now, an easy way to figure out how this location would affect these points is just rotate these arrows 90 degrees counterclockwise. So after the rotation, this point would go to up and three to the left. I've just rotated the arrows 90 degrees counterclockwise. Now, translating this back, I would have see prime, uh, negative three and two, right, because three to the left is negative. Three on the X axis and two up is positive. Two on the Y axis. So marking those points I have seam prime over here. Can I just joined the points. So that's the image of this triangle.

Okay, Since when? We have unequal triangle on a coordinate plain hurr a co ordinates, uh, excite is to a long and it tells us where two of the points. So I plotted out on the information that we know, and we want to know what are the coordinates? Appoint a up here. Okay, not too bad. So we need to find the x and Y corn and appoint a but some of the easier one r X coordinate. If we you go straight down here, call this perpendicular. Okay, this has to be the midpoint between B and C because it's equilateral. We know this is a perpendicular by sector so that these two have to be the same. So really, the x coordinate is whatever the midpoint of zero and two a is to be and those together so zero plus to a divided by two. And that's gonna work out to simply just hey, so our X coordinate. If we put a point here when, uh, this would be at a zero, and more importantly, this is a comma, something we're not quite sure. The why is just yet. We're gonna get there him. And here we go. Let's find this wife. So there's nothing really about the height, just what we want it. We want to know what is this number right here. But we do have is we know each side. Each side length is to a long so this side over here is to a This is the by sector, which means this just gets to be one A or a. And now what technical we have is we have two sides of a right triangle so we could use path A grant. Their or you could use the distance formula. Chef works better for you. I like staggering. They're better to be fair, which says, and since for some for a short sign, let's say a square equals C square minus V square. That's not confused. A with the the A in the formula where a is just a short sight society filling. So you want to know what's X square? If C squared is to a square minus E square in this case is just a I know that actually comes out is a little bit confusing because we have a in the form of than A is one of our very that's gonna be our. So if we simplify this out to a square, it's going to be or a to the second power minus a second power. And then, for a it's like a part minus eight is in par is three a seven hour and then I just want to take the square of all of that. It's a square of export is just X. The square root of a square is a the square of three can song. It's any rational numbers. We just leave it as squared three. So we're left with this height here as being a square heard of three and that becomes the wife Cornet are missing them. Says is a square over three that seems a little bit confusing, but believe it or not later on in the book, you realize how special that triangle this is and you're gonna do these and you're gonna be like, Oh, this is so easy now. But that's foreshadowing something to look forward. All right,

In this question, we are given ordinance of the Vergis ease of a right triangle in the coordinate plain and asked what other coordinates might be used to make a coordinate prove easier. So this is kind of an open ended question there, certainly multiple possible answers. Let's just take a look at two possibilities, so we'll actually will begin by plotting be even points or the given coordinates. So we have one vertex at the origin 00 a second Vertex zero comma two s. So notice we're on the y axis here because the exporting to zero and that would give side lengthier of two s units and then the third given point ordinance are negative. Two s positive to us. So that's gonna be over in quarter to and again. The side length here. The length of the other leg would also be to us from zero to negative. To us, that's a distance of two s. So what we see right here is basically a right triangle with congressional legs or another term for that. Sometimes we hear 45 45 90 triangle. So let's consider some other possibilities where assigning the coordinates to and I saw Seles, Right triangle. Ah, 45 45 90 Triangle. Well, most often when we place a right triangle in the coordinate plain, we placed the right angle at the origin because then we can put one like along the X axis and one leg along the y axis. So if we were to do that, we would still have a Vertex. At 00 we could keep another Vertex along the Y axis at the 0.0. I'm a to us. So that's still two s units on one leg. But now, if we place the other vortex along the X axis two s units away from the origin our second set of coordinates Sure, I third set accordance would be to s comma zero. Okay, so we've kept the same expressions for the length of the legs to s. But notice now, our coordinates are simpler. We don't have any negative values. And we have another coordinate the equals zero. So that's one possibility you might wonder. Well, why would you want to use to s with length of aside, Why not just a plain old expression like and And we could certainly do that as Well, sometimes we use expressions like to us or a side link if we know we're gonna have to cut the value in half. But if we wanted to create this, I saw sleaze, right triangle with the vertex of the order at the origin. But again, just use a simple expression like end for the length of each side. We could also do that. In that case, our second vertex would have coordinates and comma zero, and then our third vertex up along the Y axis would have coordinates zero comma n. So here's two possibilities or assigning coordinates to are a sauces right triangle that would be a bit simpler than the original ones given.

Right, let's say on the X Y axis I have a point A at the origin and that I have a point. Be over here. That is length A that has points a B. So therefore, if I were to connect these two points, the horizontal distance would be a and the vertical length would be be okay. And let's say I wanted to create a right triangle. Let's say I wanted to create a right triangle. Ed B. How would I get a right triangle here? Well, my slope here rise, Overrun. How much of going up and over is going to be Be over a positive. Be over a So, if I were going to do that to see what would the slope have to be for this line to be perpendicular? Will it have to be the opposite? Reciprocal. So negative a over B. So therefore, from here it have to be going from sea. I'd have to be going down a and then overwrite be so this point here I'm going down so I don't need a negative. See would be a plus B zero to make these coordinates a right triangle


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