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More coordinate proofs with quadrilaterals

In geometry, a quadrilateral (from the Latin quadri, "four", and latus, "side") is a polygon with four edges and four vertices or corners. Sometimes, the term quadrangle is used in mathematics for a quadrilateral with no "special" angles. A quadrilateral with four equal length sides is a square, and a quadrilateral with four right angles is a rectangle. A parallelogram is a type of quadrilateral.


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all right. We're here to do some more examples of coordinate proof specifically with Quadra laterals and recall that we have. I keep saying this, but we have three major tools. The two of the three major tools would be using distance in slope distance is gonna prove things congruent. Slope is gonna prove things in the parallel or perpendicular. And then if we need to talk about by section, then we'll use the midpoint midpoint formula. Now we're gonna prove that we have a given or a certain specific quadrilateral and recall all of the different tests for a quadrilateral. So, for example, if we have a parallelogram and we want to prove prove that we have a parallelogram, we could do one of many things. One is we can prove both pairs of opposite sides of parallel. We can prove both sides of opposites. Both parents up, the sides are congruent. We could prove that one pair of opposite sides is both parallel and congruent. We can also prove that the diagnosed by seeing each other, so that would just be for different ways to prove that you have a parallelogram. Nothing but what you need for a rectangle or for a rhombus or for a square. It's some combination of showing sides are equal opposite sides, recall or all sides are equal consecutive sides. Or that you have right angles. All of those fall into the category of using distance, slope and mid point. So we're gonna combine the tests for Quadra laterals and the tools that we know in coordinate geometry. We're gonna bring them together to prove or disprove. So let's go ahead and tackle some examples like this one we have given the figure below proved that it's specifically a rectangle and not a square. What we need to show. Okay, well, we could do this a lot of ways. We're gonna kind of we're gonna kind of, like, lead through different. We're gonna do different steps to show that we have a rectangle and not a square. So what? Some things we could do is show that yeah, all the angles air, right. But not all the sides are equal. We could do that. Okay, We could show that it's a parallelogram with one right angle, which is going to force the thing to have four right angles, by the way, because you're gonna have equal opposite and supplementary consecutive. Um, we could show that the diagnose are congruent, which is what a which is true to a rectangle and that the advice it each other but that you have two sides that aren't equal because well, for a square you have to have all four sides equal. Or you can show that it has all four angles. But the diagnose aren't perpendicular, which would make it a rhombus. And if you have florid angles and perpendicular diagnosed, well, then you have the square because a rhombus and erecting a big square. So let's dive into these things. Lots of ways. We can approach this problem so we have this quadrilateral A B C D. Let's go ahead, and it's probably very important to label all points here. This looks like it's at seven. Negative four Looks like. See, as at 80 looks like B is a negative 43 It looks like a is at negative five negative one. And let's not forget the but looks to be with the diagonals intersect each other, and that looks like Point E, which maybe it's 15 common Negative. 15 I'm gonna put a question mark there for now. Okay, so let's do this first bolt show that we have a parallelogram with one right angle if I have a parallelogram with one right angle. Okay. Remember, parallel programs have equal opposite sides, and they also have consecutive supplementary angles. So if you have a quadrilateral that you know is a parallelogram and you prove one of the English to be right, well, it's going to force the opposite to be right, but also the consecutive angles to be right as well, because we have consecutive supplementary angles. So if we can show that to be true, but show that we have two sides that are not congruent well, then we don't have a square, because obviously, all sides of a square have to be equal. Okay, so let's get to get and prove we have a parallelogram. So what I'm gonna do is I'm gonna use slopes. I'm gonna show that the opposite sides or parallel. So I'm gonna find the slope of a B and hopefully show that the slope of D. C is equal. Okay, well, a B here we go. We're gonna have three minus negative one and negative for minus negative. Five. It gets us a four over a one. Or in other words, before you can probably see that in the picture. And if you go up for over one that we land at point B for D. C, we're gonna have zero minus negative. Four all over. Eight, minus seven. Well, that's 4/1 again, which is four. So sure enough, we've proven that Sides A, B and D C or parallel. Okay, then let's go ahead and look at the slope of BC and the slope of a D. Well, that would be three minus zero and negative. Four minus eight. That's gonna get us three over. Negative 12. Or, in other words, negative 1/4. Pardon? All right, for a D, we're gonna have a negative four minus negative. One over seven, minus quote unquote negative five. Well, it's going to get us a negative three over a 12, which, sure enough is negative. 1/4. So we have the other pair parallel. So we've now checked off. Is it a parallelogram? Yes, it iss Okay, well, we've also proven that we have right angles everywhere, because if you look at consecutive sides their slopes or opposite reciprocal. Okay, that's gonna prove that they're perpendicular to each other. So indirectly, we've went ahead and found that every one of these angles is 90 degrees. Okay? And all you need to do is show one of them because a parallelogram with one right angle is automatically a rectangle, so we know it's a rectangle. So now we need to do is show that two sides are not congruent. And that would have to be two consecutive sites because we already know it's parallel grams. We know the opposite sides are growing so off to the side here, I'm gonna try to show two consecutive sides are not equal. And sometimes to prove something true, you wanna prove something isn't equal. So I'm gonna look at side the distance of side BC, and I'm also gonna look at the distance of let's see here of D. C. Okay, so B C, which is right here. That's the square root of negative four minus eight quantity squared plus three minus zero quantity squared. Negative for minus negative. Minus eight is negative. 12, which is 1. 44. This is gonna get you nine. So you're gonna get the square root of 1 53. All we have to show is now D C. This side over here isn't equal to the square of 1 53. And so that equals negative for minus zero. That quantity squared plus eight minus seven. Quantity squared. Hope that you can already see it. We get 16, you get the square to 17. Those aren't equal. Okay, BC doesn't equal d. C. And so we've just taken care of showing that this is indeed a parallelogram with one right angle, aka It's a rectangle, but rectangles could be squares of all sides are equal, but the way show that two consecutive sides were not okay. We do not have a rectangle. Sorry, we don't have a square. We have a rectangle. Okay, let's try another one. Show the diagnosed or congruent advice, eat each other, which would make it a rectangle. But then again, the two sides are equal. So we've already shown that the two sides are not equal. Okay, This part has already been taken care of in the yellow, as you see up here. Okay? So I'm not going to reinvent the wheel. So what? I'm gonna do is show that we have congruent and bisecting diagnosed. So we need to do is we need to show that the distance because we're gonna show congruence e the distance of B de is equal to the distance of a C and they bisect that's gonna require midpoint. So BD that would be the square root of the square root of negative for minus seven squared plus three minus negative. Four quantity squared. Let me go ahead and get a red. Get rid of this. You come back to hear squared. Okay, so that's gonna get us negative for minus seven NATO. It's 1 21 and then you have a seven squared. So you have 1 21 plus 49. You end up with the square root of 1 70. So let's go ahead and show that the distance of a C is the same. That's gonna be the square root of negative five minus eight quantity squared plus negative one minus zero quantity squared. Um, well, that ends up being negative. 13, but squared is 1 69. This is one. So you end up with 1 69 plus one, which is well radical 1 70. So what we've done is we've showed the diagnose are congruent. Okay, Awesome. Now that's so that they bisect each other. That, combined with what we showed in yellow and green above that the consecutive sides are not equal is going to give us a rectangle. So let's go ahead and find our mid points. So we have to basically show the midpoint of B d and A C are equal, and that will tell us that we have bisecting, uh, diagnosed. So we'll do this. And let's say green. So the midpoint of B d okay is gonna be negative. Four plus 7/2 comma, three plus negative four over to that gets us three halves. Cama Looks like negative one half. And if you looked at the picture, that is very believable that were three halves over or, in other words, 1.5 and then a half down. That is very, very blue. Okay, so let's find the midpoint now of a C. And that would equal negative five plus eight over to come a negative one plus 0/2. I mean, can you see it? You get three halves and you get negative. One half for the white coordinate. Yes, they have the same midpoint. So therefore they bisect each other and we have already shown that we do not have congrats on consecutive sides. So we therefore, we've proven once again that we have a rectangle Okay for this third bolt showing that we have four right angles were to do that The first bullet we've shown that we have consecutive sides with opposite reciprocal slopes, so we already have that part. So now we want to show that the diagnose aren't perpendicular, so we had to find this is the last part. We gotta show that the slope of be de doesn't equal the slope of a C. And if we show that we're gonna prove that the diagnosed aren't perpendicular therefore it's not a rhombus and a combination and directing on a rhombus would make a square. So if we can show that we don't have the perpendicular diagnosed, then we're good. So the slope of B D. This is going to be negative. Four minus 3/7 minus negative four, which is negative. Seven over 11. The slope of a C is negative one minus zero over negative. Five minus eight. That's negative. One over. Negative 13. Or, in other words, 1/13. That doesn't equal these air. Not equal. Okay, so we've basically shown three different ways how this is a rectangle and not a square. All of which started off approving It was a parallelogram. Um, So here's how you can use your three major tools with corded geometry and all the things you know about quadrilateral. And, of course, this is not all of them. We could we could prove things around us is We could prove things are sausages, Trappers awaits. There's tons of rules that would indicate what quadrilateral dealing with all of them are approached the same way. Use your three fundamental tools Slope formula, distance formula, midpoint formula. Okay, I'll see you later. Thanks.