Common Core Geometry Unit 6 Lesson 7 Squares

Now right away, I want to kind of push you a little bit to then think about what other types of quadrilaterals squares are. All right? So let's take a look. Exercise number one. Based on the definition above, a square is which of the following special quadrilaterals. Give a reason for why it is or isn't below each. All right, so what I'd like you to do is think about which of these four things a square also is. So is it also a rectangle? Is it also a rhombus? Is it also a parallelogram? Is it also a trapezoid? All right? And if it isn't, fine. But if it is, give a reason why. Okay? Pause the video now and think about this a bit. All right, let's go through it. Is a square a rectangle. Yes. Yes. Why? Because it has four right angles. That's what makes a rectangle, a rectangle, is that it has four right angles, since the square has four right angles. It is a rectangle. All right. Is it a rhombus? Yes. Because it has four congruent sides. Oops. I don't know what happened there. It has four congruent sides, so a rhombus is any quadrilateral. 

Let me get rid of random red dot. Rhombus is any quadrilateral that has four sides of equal length since the square has four sides of equal length. It is a rhombus. Is it a parallelogram? Yes. Yes, and there's many explanations you can give for why it's a parallelogram. But maybe one of the easiest ones is all let's go with the rhombuses. All rhombi. Are parallelograms. You could also say yes, because all rectangles are parallelograms. And finally, is it a trapezoid, and the answer is yes. Why? Well, that's because all parallelograms. All parallel grams are trapezoids. By the way, as some of your teachers have probably pointed out to you, the idea that all trap that all parallelograms are also trapezoids is a new concept, all right? There are probably even some of you who are watching this video. Who live in states where perhaps they haven't swung over to that definition. 

It's what's called the inclusive definition of a trapezoid. The inclusive definition of a trapezoid says that a trapezoid has to have at least one pair of parallel sides, meaning that it could have one pair or two pairs as a parallelogram does. Other states go with the debt what's called the exclusive definition of a trapezoid stating that a trapezoid has exactly one pair of parallel sides and therefore no parallelograms or trapezoids. In the common core generally speaking, we go with the inclusive definition, which says that all parallelograms are also trapezoids. But again, you should talk to your teacher and find out which way your particular state looks at it and also how they test it because that's pretty important too. All right, let's keep going. Oh, actually. Let's go back to one last thing on the screen, right? Because this is what it's all about, right? A rhombus, plus a rectangle, gets you a square. Now, the reason that that's important is that rhombuses, rhombuses had a lot of properties to them. And we saw those in the last lesson, properties like the diagonals, bisecting the angles, the diagonals being perpendicular to each other. 

The square gets those properties as well, because it's a rhombus. Remember rectangles, they have a nice property, and that's the fact that the diagonals are the same length, right? That's the big one with rectangles, is that the diagonals are the same length. And again, squares get that as well, because all squares are rectangles. So squares get all of this stuff that both rhombi and rectangles have. And we're going to use those in problems today. All right, so let's jump right into a design problem using squares. Squares come up a ton in the real world. And so we have to be prepared to use their properties to be able to answer various questions. This one's a little bit wordy. Let's go through it together. Exercise two. Chemi is designing a Square Garden whose diagonal is exactly 16 feet long. She would like to place fencing around the perimeter of the garden so that it is completely surrounded. If the fence in she likes comes in 12 foot long rolls, how many rolls will she need and how much fencing will be left over when she's done? Show all calculations that lead to your final answers. All right, so this is kind of cool, right? And obviously, it's helpful if we draw a good picture. Okay? What I'd like to do though, you know all the math that you need to do this problem. 

All right, it's kind of lengthy. But I'd like you to do is pause the video now and take a shot at it and then we'll work through it. All right, let's do it. Good picture. Let's start with that. Now notice it says that it's diagonal is 16 feet. It doesn't have to say it's shorter diagonal. It's longer diagonal because the plane fact is, due to the fact that it's a square and squares are rectangles, the two diagonals are the same length. So what do I know about this square? All I know is that this is 16 feet long. And yet I want to put fencing around the square. So I need to know what its perimeter is. Okay? Now what that means is I need to know what a side length is. Maybe I'll call it X but of course, because a square is also a rhombus, all side lengths are the same, and because the square is a rectangle, all of these angles are 90°. So what I'm now dealing with is I'm now dealing with an isosceles right triangle. That kind of looks like this. And because it's a right triangle, I can now use the Pythagorean theorem to figure out how long the side length is. 

Now generally speaking, in the Pythagorean theorem, you can only use that if you know two of the side lengths and you're missing the third one, whichever one it is, the a, the B or the C in this case, because the two sides that are missing are the same length, we can use the Pythagorean theorem. Now obviously, the Pythagorean theorem is a squared plus B squared equals C squared, but in this case, I'm going to have X squared plus X squared is equal to 16 squared. That's going to be X squared two X squared. Be careful there. And of course, 16 squared, we can do on our calculator if we don't have it memorized 256. Now be careful, of course, you know, this isn't a course in algebra, but you need to know things that like X squared plus X squared is two X squared. It's not X to the fourth or anything like that, right? And now we need to solve for X so we're going to divide both sides by two. All right, and that's going to give us 128 X squared is one 28. And now we need to solve for X, so we're going to take the square root of both sides. 

Now, a 129.8 is not a perfect square. It's kind of an ugly number. So we're going to have to go over to our calculator right now. That's no big deal. I just got to get out of this mode and go over to here, our nice TI graphing calculator. What was I doing? Well, I had had my two 56 and I had divided it by two, right? To get my one 28. And now I have to take the square root of that. I could have just done square root of answer or whatever. So the actual side length of my square is a little bit messy. It's 11.313, et cetera feet. Right? Remember, that's the side length of a square. I'll go with 11.31 for right now. Let me just pop back into pop back into here really quick. And I'll just put 11.31. Now, I need to have the entire perimeter. I need to go back to my calculator for that really quick. But one of the beautiful things in all I have to do is take this and multiply it by four, right? Answer times four, and now this is my perimeter. My parameter is 45.25, 45, and a quarter feet. All right, so the parameter, we'll just use a big P for that four X is 45 .25 feet. 

Now remember, the question was or the question asked us, how many rolls of fencing would we need if the rolls came and 12 foot sections? Now for many of you, the numbers are small enough here that you could actually kind of just work this out for yourself. But of course, if I needed to, I could just take that answer and divide by 12. I need to know how many 12 foot rolls I need to buy. All things perfect, I would need to buy 3.77 et cetera, rolls of this fencing. I can't do that. Stores don't allow me to do that generally speaking. So I'm going to have to buy four rolls, right? I'm going to have to kind of round up no matter what. I'm going to have to buy four rolls of fencing. And of course, four rolls of fencing at 12 feet per roll is 48 feet of fencing, okay? So let's go back over here. I'm going to go into view, full screen. So I'm going to need four rolls of fencing. That's part of my answer. All right, that's how many I'm going to have to buy. It magic red dot. Four rolls of fencing. It also asks me how much fencing will be left over when she's done. 

Well, the four rolls of fencing give me 48 feet. But I only needed 45.25 feet, so that's actually when I subtract is going to leave me with 2.75 feet. Left over. So almost three feet of fencing left over, not too bad. It would have been a shame had we needed, let's say, 37 feet of fencing, right? And we could have gotten 36 feet with three rolls, but we would have had to buy four rolls. And then we would have had a lot of waste here. We're just wasting 2.75 feet of fencing. We need the four rolls to get us above that 45.25. That gives us the 48 feet of fencing, and then we have 2.75 left over. Let me step out of the way so you can take a look at this. Remember a lot of this we did on the TI calculator. All right? Okay. Let's move on. Let's keep going. Let's see what else we have. All right, exercise number three. Okay, now we're going to get back a little bit into reasoning and proof land, although I think I'm going to stick with kind of a paragraph proof on this. Let's take a look at a kind of a neat problem involving the reasoning with squares. XR says number three, and the diagram to the right square ABCD is drawn. And the midpoints of its sides have been marked and connected with segments to form quadrilateral. Given explanation for Ye FGH must also be a square. 

Now again, it probably seems rather obvious that if I take this larger square, ABCD, and I mark off its midpoints, E, F, G, and H, and then I connect them. Oh yeah, of course, that's a square. It's obvious it's a square. All right? But why? This is actually rather complex, given how much it takes to be a square. So what I'd like you to do is pause for a moment. And think about how you would go justifying that this inner figure, EF GH is also a square. And it's a little bit of a challenge, okay? And we'll go through it. Yeah? Let's go through it. Again, that little pause I gave, that's not the amount of time it should have taken you to think about it. That's the amount of time it should have taken you to pause. And then unpause the video. You may have taken quite a bit to think about this. So let's talk about what we know. All right? What I know, definitely. As I know all of these angles are right angles. Now, I also know that all four sides of the larger square, the larger quadrilateral, are congruent. And then, because of those midpoints, I also know that this is congruent to this. But of course, that means this must be congruent to this. And this must be congruent to this, and this must be your own to this. Et cetera, all right? I didn't get my right angle here. 

Let's get that one in. And so what can I do with all of that? Well, I can say that these four right triangles, right? BFE, et cetera, right? I can say triangle a, E, H, is congruent to triangle BFE, which is congruent to triangle C, G, F, which is congruent to triangle, D, G, H, I, side angle side. So those four right triangles are identical to each other. By side angle side. Now, what does that get me though? What that gets me then is the fact that EH EF FG and GH, those four sides, HE must be congruent to EF must be congruent to FG must be congruent to GH. And that's by CPC TC. If you will, they are the hypotenuses of those four congruent right triangles, and since they're all hypotenuses, they are corresponding parts of congruent triangles, so they must be congruent themselves. And of course, this is very important. Because there's two things that make a square a square. Number one, all four sides are the same length. And we just established that. That is the same as that is the same as that is the same as that. But the other piece is somehow, somehow, we need to justify, I'm going to do it in red. We need to justify that those four angles are right angles. 

Now, we haven't done that yet. And that's a little bit trickier. That's a little bit trickier. Okay? But let's reexamine maybe one of these little right triangles. So if we're thinking about triangle E, AEH, go back to blue. If we're looking at triangle AEH, then this thing is an isosceles right triangle. An isosceles right triangle, which means these two angles have to be the same measure. The base angles of an isosceles triangle are congruent. And what are they both have to be? Think about that for a minute. Well, these two angles, right? Along with that 90° angle, must add up to a 180. And because of that, these two angles must add up to 90. And of course, 90 divided by two gives me each one of those angles being 45. So that's a 45° angle. That's a 45° angle. And then I can go on and say all of those must be 45° angles. Because these are, after all, congruent right triangles. 45. 45. 45. And 45, right? All those have to be 45° angles. But the final little piece, one more time, let's go in maybe green this time. These three angles must sum to be a 180°. As well as these three angles and these three angles, et cetera. Well, I've got a 45, and I've got a 45. If I add those together, I get 90, since all three of them must add up to be a 180, right? I now can do 180 -90. And that, of course, is another 90° angle. And that is what gives me the four 90s, right? So I can establish, this isn't exactly a proof. But I can establish using facts that I know about isosceles triangles. Isosceles right triangles and the rest of it, I can now use to establish not only that that inner figure, that inner quadrilateral has four congruent sides, that's by CPC TC, but also that the four angles are 90°. 

All right? Let's keep going. Exercise number four, okay? Back to the coordinate grid. Quadrilateral ABCD has coordinates of negative four two four 8 ten zero two negative 6. Using coordinate geometry, prove ABCD is a square by showing that it has four sides of equal length and four pairs of perpendicular sides. All right, let's do it. Now there's many, many different ways to prove something as a square in the coordinate grid, and that's because there's many ways to prove that something is a parallelogram, to prove something as a rectangle. A rhombus is pretty straightforward, but there are different ways to do that. When I'm trying to prove, though, that a figure is a square. In the coordinate plane, I pretty much go brute force, which means I want to show that all four sides have the same length. And that I've got four pairs of perpendicular sides, in other words, four right angles. So, so let's first talk about how we determine that all four sides have the same length. Which tool would we use from coordinate geometry to do that? We're going to use the distance formula. So let's take a look. All right? All right, let's go through not AD. Let's go through a B okay? Let's find the length of side AB using the distance formula. And remember, the distance formula says, we're going to take X two minus X one, four minus negative four. Square that. 

The Pythagorean theorem, right? Then we're going to do 8 minus two squared. Four minus negative four is 8. Squared. And then I've got 6 squared, add those two together. I get the square root of a hundred. And that's ten. All right? Now what I'd like you to do is use the distance formula to find the lengths of sides BC, CD, and DA, or AD, however you want to look at it. And hopefully, hopefully each one of them will also turn out to be ten units. But why don't you pause the video now and take a little bit of time to do that. All right, let's go through them. And of course, they do all end up being that ten units, but we should verify it. Let's go through BC. The distance formula there. We're going to have ten minus four squared plus zero -8 squared that's going to give me the square root of 6. Squared plus negative 8. Squared again, the square root of a hundred. And that's ten. Let's do CD now. The distance formula there. All right, we've got two minus ten. Squared negative 6 minus zero. Squared, lots of squaring here. Negative 8. Squared plus negative 6. Squared again, giving me the square root of a hundred. Which is ten. And finally, DA. And what would we have there? We'd have two minus negative four squared plus negative 6 minus two squared, that's going to be 6 squared, although that doesn't really look like a 6, so let's see if we can make it look a little bit better. 

Yeah, that looks great. 6 squared negative 8 squared. We've got all these 6s and 8s coming out in this problem. All right. Now, what we've really done, of course, in this part of the proof is we've shown that ABCD is a rhombus. All right, that's what we've done. We've shown it's a rhombus. Now what I need to do is also show that it's a rectangle. Now again, there's different ways to do that. One way to do it is to actually just calculate the lengths of the two diagonals. And if they're the same length, you're actually done. It's then a rectangle. And a rhombus, which means it's a square. What I'm going to do is I'm just going to establish the fact that there are four pairs of perpendicular sides. So let's go through that. And we do that, remember, by using slope. Okay? So I'm going to find the slope of all four sides, or maybe I'll find the slope of side a, B first. Then have you work on the others? All right, so for AB. Its slope, remember, will be Y two minus Y one. Divided by X two minus X one. All right, that's going to be 6. Divided by 8. And that's a slope. Of three fourths. All right, so there's my slope for AB. Why don't you go ahead and calculate the slopes of BC, CD, and DA. And then trying to make some statements about perpendicularity based on those slopes. Take a minute. All right, let's go through it. All right, so we have AB. It's got that slope of three force. Maybe I'll put my BC in here. 

Let's see if I can squeeze that in. BC has a slope of zero -8 divided by ten minus four, be careful with your signs. That's going to be negative 8 6. And that will reduce to negative four thirds. Now, before we even move on, though, notice, I got a slope of three fourths. I have a slope of negative four thirds. Those are called negative reciprocals. And because they have negative reciprocal slopes, that means that a, B and B C must be perpendicular to each other. What that immediately does is gives us that right angle. All right? Let's keep going. We have two more slopes to calculate. Let's calculate the slope now of CD. All right. The slope of CD is going to be negative 6. Whoops, wow, that was one very red 6. Negative 6 minus zero. Divided by two minus ten, be careful here. We're going to get negative 6 in the numerator and negative 8 in the denominator and a negative divided by a negative is a positive. Specifically here a positive three fourths. Now the fact that these two are both three fourths means that those two sides are parallel to each other, not a surprise given that a square must be a parallelogram, but we're not really too interested in that. Let's do one last one. 

Let's find the slope of DA. All right. Specifically, let's see the slope of DA will be negative 6 minus two in the numerator. And two minus negative four in the denominator. Again, be careful with your signed numbers here. That's going to be a negative 8 in the numerator, a positive 6 in the denominator, which is going to reduce to negative four thirds. Again, not surprising that we've got a negative four thirds and another negative four thirds, given the fact that it's a parallelogram. But, but again, not really the point. What I want to now say based on those slopes and let me kind of move this out of here is that AB is perpendicular to BC. BC is perpendicular to CD. CD is perpendicular to DA. And DA is perpendicular to AB. All right, notice how I have to do that, because each one of these statements is establishing the fact that I have a right angle. And you want to give a little reason here, where you can say, because slopes. Are negative, I'll just abbreviate this negative reciprocals. So let's wrap it now. Let's make a final statement. A, B, C, D, is a square. Because it has four congruent sides. And four pairs. Of. Perpendicular sides. And you could also say that it's got four right angles. Four pairs of perpendicular sides almost looks like the word perpendicular. 

Now, again, there are definitely other ways of proving something as a square in the coordinate grid. But to me, this is the most straightforward. We established that all four sides are the same length using the distance formula. Then using the slope formula, we establish the fact that we've got four pairs of perpendicular sides or four right angles, however you want to think about it, and therefore we have a square. All right. Let's go on and do one final exercise. See, because what the last one gets at is that there are definitely many, many different ways to prove something is a square. What I want to know is whether or not particular scenarios give us enough information to conclude that a quadrilateral is a square. So let's take a look at that in the last exercise we're going to do. Exercise number 5. For each of the following sets of conditions, carefully explain why they are enough to show that a quadrilateral is a square. Why they are enough to show a quadrilateral is a square. Think about each piece of each condition and what it tells you. Okay, so in each one of these situations, we're given enough information to conclude it's a square. But why? Okay? So let's do the first one together and then have you think about the other three on your own. Let array, a quadrilateral with four equal sides and one right angle. Right? Why is that? Why would that be enough to know that it's a square? 

All right, well let's talk about it. It's enough to know it's a square because four whoops equal sides makes it a rhombus. And. Thus a parallelogram. Okay, so the fact that it's got four equal sides means it's a rhombus and the fact that it's a rhombus means it's a parallelogram. But any parallelogram. With one right angle. Must be a rectangle. All right? And any time we have a quadrilateral that is both a rhombus and a rectangle, it is a square. All right? So that's it. You take a look at the other three scenarios and see if you can explain why it gives us enough information to know it's a square. All right, let's go through them. Letter B, a parallelogram, so we're starting off by knowing it's a parallelogram. With equal diagonals and two adjacent sides of equal length. All right. So this is enough. If I tell you you have a parallel gram that has equal diagonals and two adjacent sides adjacent means touching sides of equal length, then I know I have a square. So why is that? Well, apparently I'm going to have to write a lot smaller here. But a parallelogram, maybe abbreviated parallelogram with equal. With equal diagonals. Is a rectangle. 

All right. And we proved that a while ago. Anytime you have a parallelogram, it's got to be a parallelogram. Anytime you have a parallelogram whose diagonals are the same length, then it is a rectangle. Okay? Now can we get the rhombus part of it? So any parallelogram. That has two adjacent. Sides equal. Must be a rhombus. All right? That may actually some additional explaining, but if you think about it right, it apparel gram, opposite sides are always the same length. So therefore, if we had a parallelogram that had two sides that were touching in the same length, well then all four sides would also have to be the same length, and therefore it would have to be a rhombus. Let's take a look at letter C a parallelogram with perpendicular equal diagonals. Awesome, a parallelogram with perpendicular equal diagonals. Well, this one's really great. If a. Parallelogram. Has perpendicular diagonals. It is a rhombus. A rhombus. If a parallelogram. Has equal diagonals. It is a rectangle. Okay. And again, that's the key notice in each one of these. We come back to the fact that it's a rectangle for this reason. It's a rhombus for that reason, put a rectangle in a rhombus together, and you have a square. 

All right, last one, letter D a quadrilateral, whose diagonals are perpendicular bisectors of one another, quadrilateral, whose diagonals are perpendicular bisectors of one another, and one right angle. Okay? And this one requires a little bit more work, all right? Because we're not starting with a parallelogram. We're starting just with a generic quadrilateral. So it kind of goes like this. If the diagonals. Of a quadrilateral bisect each other. It is a parallelogram. All right, so the first thing that we know are the first thing we can get is the fact that the diagonals are perpendicular bisectors, not so much the perpendicular, but the bisector, if they're perpendicular bisectors, just bisectors, then it's got to be a parallelogram. Since they are also perpendicular. It is a rhombus. All right. But. Since it is a, give me a self a little bit more room. Since it is a. Parallelogram. With a right angle. It is also a rectangle.