Common Core Geometry Unit 1 Lesson 5 - Circles and Arcs

Now in the first exercise, what we're going to be doing is we're going to be trying to get you to really think about what a circle is. Specifically in terms of its center and its radius. Let me read through problem one for you. In exercise number one, we're asked, or we're told, a circle with an unknown radius is shown below. Circles are designated by the point at their center. So below is circle a or that little circular symbol with the letter ray. Now letter a asks us to measure the distance from point a to point B, C and D to round our answers to the nearest 8th of an inch and to reduce if possible. Now, remember these kind of ways that we're laying out these distances, a, B equals AC equals a D equals. So the lengths of the line segments that connect a to B a to C and a to D all right, now we've already reviewed obviously how to use a ruler and you should be pretty good with it. So why don't you go ahead and measure those three distances and then I'll be right back in order to give you the answers. All right, let's take a look. So if we move our ruler up and we put our zero point at point a whoops, let me try to get that a little bit closer. And then rotate our ruler up, what we find is we find that the length of segment AB is one and a half inches. So let's note that. All right, AB is one and a half inches. 

Okay, but what we see is that that's the same length for all three segments. If I simply rotate my ruler around, then I see that the length of AC is also 1.5 inches. Whoops. Of course, that was going to happen. And the length of AD is also one and a half inches. So let's note those. Sometimes I'll probably slip up and even use this symbolism. It's not a slip up. But these double hash mark to show inches. Okay, so we've got a circle whose center is at point a and we've just chosen three random points around it, and we found that the distance from point from the center point a to all those three distances was one and a half inches. This should naturally give rise to letter B of this problem. Let me read it for you. Letter B asks us what would be a good description of all points on this circle. Remember, only the points on the curve. Now this is kind of an important thing to mention right now. When we talk about a circle or a square or a rectangle or a parallelogram, we're really any figure. We're always talking about the points that lie on the boundary of the figure. The space within the figure might be the area of the circle. But the circle itself is simply the points on its outside, okay? So what I'd like you to do now is think about what a good definition of a circle would be and write it down for part B and then I'll be back to give you at least what my definition would be. All right, so what definition did you come up with? 

The plain fact is what we should be noting in terms of the definition of a circle. Is that it's going to be all points. That are a fixed distance. And our circle here that was that fixed distance was one and a half inch, but all points that are a fixed distance. From a center point. Make sure that you understand that definition. If I took a hundred more points that lied on this circle and I measured the distance from the center point a out to any of those points, it would always be one and a half inches. Every point on a circle is equally distant the same distance from the center point. I don't know why I put the same distance in quotes. It's just the definition of equidistant. All right, keep that in mind. It's going to be the key to understanding everything we do with circles and with arcs. So let's keep going on to make sure you understand this idea. All right, it's very, very simple. A circle is the collection or set of all points that are an equidistance from a fixed point. The fixed point is the center and that distance is called the radius of the circle. 

Now we're going to be drawing a lot of circles today and in future lessons. And we're always going to be using compasses. All right? Always be using compasses. Now, the compass that you use in class or at home might be very much like this one. It might be a compass whose radius won't stay in the right place as you draw the circle. It might be one where the pencil leads a little dull or maybe it's a little bit shorter than this one. Using compasses is something that takes a lot of practice and we'll be using them a lot in this course, even on a standardized exam, you'll probably have to use one. Nobody will ever be grading you for exactly how well you use a compass. Just that you understand what you're using it for, right? And keep that in mind if you're constructions or your diagrams, don't turn out to be as good as mine. I'm going to be using this electronic compass up here. I mean, it's going to do a decent job, but even then, I'm not going to be perfect. So don't worry about making a perfect drawing. Make sure you understand the math that you're using. So right now, we're getting rid of this guy. 

What we're going to be doing is using this. And keep in mind, again, the whole idea is if I wanted to have a picture of all the points if I wanted all the points on this circle or sort of all the points that were let's say 5 centimeters away from that point here, let me get a little bit of a better adjustment. Right? I will always have the pointy end of the compass at the center of the circle. And then how wide the compasses is the radius of the circle. That fixed distance. All those points in blue now, all those points in blue are 5 centimeters away from point a, okay? Let's get into the first problem that really involves this. And that's exercise two. Okay? Exercise two is kind of a neat problem that could actually come up if you were doing some mapping work, okay? Like literally on maps kind of work. And let's take a look. Exercise two says locate and mark all points that are both two inches from point a while also being one and a half inches from point B so can we find points that are two inches from point a and one and a half inches from point B well, the way that we're going to do this is we're literally going to use our compass to draw every point that's two inches from point a and then every point that's one and a half inches from point B let me do the one right now to mark all the points two inches from point a I'm going to take my compass, I'm going to bring it over and I'm going to use that ruler that's on your paper. And I'm going to stretch it out to two inches. I'm going to try to get a bit too inches. There we go. 

I'm going to then put the point of the compass right on a because a is the center point of this circle that I'm drawing. And then rotate my compass around and then move it away from point a I had a little bit of a gap there, but that's okay. The plain fact is all those points that you now see in blue are two inches away from point a what I'd like you to do is take a moment, pause the video and draw all the points that are one and a half inches from point B. All right, let's get those points that are one and a half inches away from point B again, we're going to take our compass, bring the pointy part down to the zero mark. Move this back to the one and a half inch Mark. Then locate the center of our circle, draw those points, maybe move my compass a little bit out of the way. And now we have all the points that are one and a half inches from B so what points are we actually asked to find? In the problem we're asked to find all the points that are two inches from a and one and a half inches from B there's two of them. See if you can figure out what points they are. All right, well, they've got to lie on both curves. So that's one of the points. And that's one of the points. Those two points are the only points in this diagram that lie two inches from a and one and a half inches from B we can verify that quickly using a ruler, let's do that really fast. I'm going to bring my ruler up and go to point a, rotate it up, and I find, in fact, two inches from that marked point to point a if I then bring my ruler over, maybe put the zero mark at the point itself, rotate it down. I find one and a half inches to point B all right, I think we'll skip doing the verification on the other point, although I would encourage you to do it yourself. 

All right, let's move on and talk a little bit about more about circles and arcs. Okay. A circle is the collection of all points that are equidistant or a fixed distance away from a center point. An arc is simply part of a circle. The big difference between an arc and a circle is that a circle is all the points where an arc is only some of the points. How many we designate using an angle. So again, let me illustrate this for a moment with the picture that we've got here. I know that this isn't on your worksheet. But here we've got this circle a and I've got a protractor. I'm going to bring the protractor over and center it at point a, or at least I'm going to try to center it pulling a and then I'm going to stretch it out. So if I wanted to now mark an arc on this circle that had the same radius as a, but let's say an angle of 50°, I would just do something like that. And then this arc that's in blue, a little bit hard to see. But that arc that's in blue has exactly the same radius and center. As point a, but it only moves through 50 out of the 360° that it takes to get around a full circle. All right? Arcs and circles, very similar. They both locate points that are an equal distance away from a given point, right? But an arc doesn't have to be the whole circle. Okay, let's keep going. Let's play around with this idea of an arc. In the next problem, you're going to have to be using that straight edge and your compass. 

And your protractor, every single tool that we have, you're going to be using now to construct an arc. Let's take a look at what the problem asks. Exercise number three, construct an arc, BC, with a center at point a, a radius of four centimeters, and a measure of 45°. All right? Well, let's bring out our protractor. You might even want to draw a nice straight line coming out of this. I'm going to make my protractor a little bit bigger. I'll bring my compass over and I'll stretch it out to the four centimeter Mark. Okay? I will bring it up here. And what do I want? I want a radius of four centimeters, an angle of 45°. And that's it. All right. A portion of a circle that is four centimeters from point a, but only takes up 45° of a rotation. Now, the important thing about arcs right now isn't their rotation. We're going to get into those a lot in our unit on circle geometry, but that's way at the end of the course. What is important right now is that you know by looking at this arc that any point on that arc is the same distance away from point a as any other point on that arc. Specifically, four centimeters away. Let's use this idea in the next problem. All right. The final thing that we're going to concentrate on today is how to construct an equilateral triangle. 

Now, as you should recall from a previous course, equilateral triangles are triangles where all three sides are the same length. And we can create them with simply a straight edge and a protractor, something that will draw arcs and circles force. So let me read over this exercise for you. Exercise number four, creating an equilateral triangle. Recall that an equilateral triangle is a three sided figure whose sides are all the same length. The length of segment AB shown below is two inches, and is used as the radius of two arcs. One whose center is located today, and one whose center is located at B these two arcs, DE, and GF intersect at point C draw in segments AC and BC explain why triangle ABC must be equilateral. Well, surely while I'm drawing those two segments in, you can as well. Let me get them drawn in. Then I'm going to pause for a little bit and let you think about why then this triangle must be equilateral. Try to write something down. Some kind of statement about why you think it has to have three sides of equal length. I'm going to draw. I'm going to draw the sides in right now. All right, we're going to get ourselves our line tool. 

There's one segment, another segment. And now take a moment to try to write down why that must be an equilateral triangle. All right. Well, let's go through it. Okay? Simple enough. Let me get a different color. What do we know? Why is this an equilateral triangle? Well, we know all points. All points on the two arcs. Are a length. Of two inches away. From a. Oh, I shouldn't say a and B I should say a or B depending on which arc. All right? All the points on those arcs, because the radius was set to be the length of AB, which is two inches, all points on those two arcs are two inches away from either a if we're on that arc, or B, if we're on that arc. Therefore, I'm going to use three dots often to stand for the word, therefore. Therefore, point C. Is two inches away from both. A and B there, the word and is correct. So AC equals two inches and BC. Equals two inches. And that's it. By setting the radius of our arcs to be equal to this length, we're ensured that that point is the same distance away from the two endpoints of that line segment, and all three sides have the same length as the original line segment. Constructing an equilateral triangle is one of many different shapes we're going to construct using a straight edge and a protractor in this course. So let's make sure that you can do that one with one last exercise. All right. So an exercise 5, specifically you're going to be creating an equilateral triangle. Here we go. Using your compass in a straight edge only, construct an equilateral triangle using segment NN as one side. 

Then verify by measuring each side to the nearest millimeter. All right, so you should have a straight edge right now. You should have your protractor, sorry, your compass, and you should be ready to draw this equilateral triangle. Take a moment and see if you can figure out how to do that from the last exercise, and then I'll come back on and construct it for you. Okay. Let's make an equilateral triangle. So I'm going to take my compass app. I'm going to rotate it just a little bit to make it easier. And I'm going to set its radius to be the length of my line segment. Or I'm going to try to make it the length of my line segment. There we go. All right, that's important. Because that length now will be the radius of the circle or the radius of the arcs that I'm trying to create. I'm not going to draw a full circle. I'm just going to draw an arc. Now my center point shifted a little bit. It might even do that to you. And I'm going to just draw an arc like that. Keep in mind, the arc that I just drew all the points on this arc, all of them are a length of NN away from point M now this is key. Don't change the length of your radius. If you do, you got to go back and reset it to be the length of MN. What I'm going to do is come in and just make it so that oops. So that my point then is over here. I'm going to make the center of my arc now on point N and draw another arc. I now no longer need my compass. 

All right, literally the X marks the spot. On my picture, this point is going to be a length of NN away from N, M sorry, and away from N so all I need to do is connect that to those two points. Let me do that with my line tool really quickly. You'll just be doing this with your ruler, obviously. Here we go. And now I have an equilateral triangle. I'm guaranteed to. I use that compass to locate there's two points. There would be another one down here, but I locate the only point that's on this side of the line segment. That is the same distance away from M and N and specifically a distance of MN. All right? We can verify that it is an equilateral triangle by bringing our ruler up. And we would find that that is 58 millimeters that that is 58 millimeters, and if I bring it over here, I'll find that that is 58 millimeters as well. Now, of course, if you don't have a perfect construction, it won't be a perfect equilateral triangle, and it's hard to do a perfect construction. I was doing my measurements really quickly because quite frankly, I think one of them was 59 instead of 58. That's because things slip a little bit even electronically. But make sure that you know how to do this most basic of all constructions. It's going to be important. We'll get into some trickier ones in a little bit. But for now, make sure that you understand why constructing an equilateral triangle works the way it does. 

All right, let's wrap up the lesson. In today's lesson, what we did is we looked at what the technical definition of a circle was. All points that are located a fixed distance away from a fixed point that fixed distance is the radius of the circle, or how wide you set your compass, right? And that fixed point is the center of the circle, or where you put the point in the end of the compass. Arcs are just portions of circles and oftentimes we'll draw an arc instead of a full circle in order to locate points. We also saw how to use the definition of arcs and circles to construct an equilateral triangle. We'll get into more constructions and further lessons. For now, I'd like to thank you for joining me, Kirk Weiler for another common core geometry lesson by E math instruction. Until next time, keep thinking and keep solving problems.