Geometry: 1-2 Measuring & Constructing Segments



ok today we're going to talk about measuring and constructing segments we talked about the segment yesterday it's pretty much just a line with ends ok the metaphor used was metaphor the correct word to use in that situation okay good deal first time I want to talk about is a coordinate okay yesterday we talked about points a coordinate is just the point on a graph usually your graphs look like this a coordinate is just a point somewhere on there okay coordinates have an X at a while X tells us how far we go over to the right or left why tell us how far we go up or down okay now doesn't always have to be on this you can have a coordinate on a line as well I think that's a and that's baby those are both coordinates just pretty much a point that's later okay begging them drops now distance also heard the word distance before it's just the link from somewhere to somewhere else okay looks like a segment all right but not always just drawing a line fair enough so distance we would find how far it is from A to B let's say a is located at 2 and B is at 8 okay we would pretty much just find how far it is from there to there all right now what we do is go to absolute value that means then all that means is when you get done there's a negative take it off there that's what absolute value means just get rid of the name okay so absolute value of two – hey what would two minus eight be be negative six now those are the lines that means awesome at least two negative signs okay so the answer is six that is our distance you'll never have a negative distance okay and all just this is telling you how far it is from somewhere somewhere else okay good deal yeah next thing we talked about ordered some of y'all might have not a turnover before it's called congruent okay congruent this is the math fancy word for equal okay if two lines are congruent that means they're equal they're the exact same length if two angles are congruent they're equal they're the exact same measure all congruent is the fancy way you're saying goody gumdrops so we say that 2 X is congruent to 8 wanting to find X always to do is put an equal sign in there and then we'd solve it that's it so thanks you say two things are congruent you can just sell them equal to each other all right good next is midpoint all right midpoint is exactly what it sounds like first part sounds like mid which is middle second one is point mid point it's the middle point on a line all right so if I want to find the midpoint of a and B let's say a is that make the 3 B is at 12 okay we're on the line we want to find the midpoint of a and B all I got to do is find the length of a B so we do negative 3 minus 12 that would be negative I'm sorry that would equal negative 15 and absolute value things means that we drop the negative so the answer is 15 now that's the whole link we found the distance okay now we want to find the middle point there we want to find with the exact middle so how do I find the middle of 15 I just cut in half until I divided by two equals seven point five so I know that our point is going to be seven point five whatever's these ours between these two okay so I can just go from this way or this way I could either add seven point five two negative three or subtract from 12 either way I'm going to get the same answer okay it's going to be at four point five okay that's it that would be your answer that's your midpoint we usually handle you for midpoint okay like I said before math people weren't very creative okay look to me out I'm a bit like the destructor point because it sounds cool okie dokie now next word you need to know called boss sex all right if something bisects something that means it cuts in half okay because in the middle if that's the midpoint it bisects lines Amy M bisects that line because it cuts in half then all bisects means is that it cuts in half the end I really don't know what to say about it okay let's do an example of by said I help you pull all your hopes and dreams alright let's say we got a line here let's just make up go crazy you say B and C alright let's say a B is 5 X and B C is 3 X plus 4 so we know the length of this and this let's say we also know that B bisects it what does that make these online a see it makes it the mid one okay so it makes be the midpoint cuz it bisects it so cuts in half now if that's in the exact middle that tells us something about this line and this line if that's the exact middle that means that this line is the exact same as this line which means they are congruent so therefore you can just write out the problem of x equals 3x plus 4 all right that's our problem now we solve it using algorithm all right the first step we got to get rid of the 4 I apologize that's not the first step first step we got to get the X's together so we'd subtract 3x this cancels out 5x minus 3x equals 2x then I bring this down that's 4 and to get rid of that tube we divide by 2 so x equals we found our X now if they were asking for the length of let's say they're asking for the length of AC that's what they wanted to know we're not done yet it okay we gotta plug that in we plug in X their length of a B is now 10 because 5 times 2 is 10 now do the same thing for over here you don't even have to work this side if I trust my work here I know that's the exact same cuz that's the midpoint but just for checking sake we're plugging in 2 times 3 is 6 plus four is ten what's the length of AC altogether now C is 20 AC the whole thing so I put it all together and B 20 a lot of work alright here's a little example problem okay we didn't talk about this but it's also just something you to know okay they gave us the length of km which is the whole thing right and then they gets the link of KL and LM which whenever we put these two together what do we get the whole thing right whenever we put this length with this length we get the whole wing so what does that mean about if we put these two together they are equal to that other one so we'll put that on one side and then if we put these two together are we adding subtracting multiplying your body right we're putting things together we're adding so all we do is we take this all we bring to the table is a plus sign and then 3x minus 2 and now we just solve for X and you're thinking that looks like a long not fun problem but it's not that bad eggs combines with 3x so be for X 0.5 minus we already get rid of that that 0.5 minus 2 is negative 1 point 5 then we'll bring all this down we get the X's over here some subtract 3x from both sides and cancels out and the 1x which you can just drop the 1 and then we add 1.5 on both sides so it is 3 now that might not be what they're asking for all you did was find X if you need to find the length of one of those you got to plug it in

Construct the inscribed circle of a triangle



today we're going to learn how about creating an inscribed circle within a triangle so our first step is that we need to actually have a triangle so I'm going to use my polygon creation tool and I'm gonna create click on three points a B and C and to close the polygon you need to wreak lick on a again and there's my triangle you want to probably make sure that the points are not equilateral or really nice so drag them around a little bit so that looks pretty good to me the next step is to find what's called the in-center of the triangle and the in-center can be found by constructing the three angle bisectors and it turns out that geogebra has a tool for that so if you go under the fourth tool item where it says perpendicular line and you click on the little red drop down it'll give you the angle bisector tool and what the angle bisector tool will do is if you give it an angle it'll give you the line that splits that angle into equal parts so for example if I want an angle bisector at Point C I can click on point a Point C and then point B and that will construct a line that appears to cut the angle formed at vertex C into two spots you can do the same thing at angle for the angle at vertex a by clicking on Point C then a then B and the reason why it's making the line through vertex a is because it was the middle point that we clicked on so if I wanted to make the angle bisector at vertex B I can click on a then B then C or I could click on C then B and then a and now I've got my three angle bisectors and something interesting should be happening though and that is these three angle bisectors seem to be intersecting at a common point in this common point we're gonna give a name called the in-center and it's something called a point of concurrency the point where all three lines intersect at the same spot so we'd like to find where that point is and if you go to the point tool and click on the drop-down there's something called the intersection of two objects tool and would allow you to do is click on two things and find where they intersect now it's a little finicky you can't select all three it gets confused so make sure you only have two of the lines selected and then click on it and it shouldn't make a point now what's supposed to be special about this point is its distance from the sides of the triangle should be the same in any direction you go and if you just kind of eyeball it it doesn't look like that because this distance doesn't appear to be the same as that distance and the issue is that the distance we're talking about should be the perpendicular distance and right now this is not a perpendicular line to the side so we actually don't want to look at the angle bisectors anymore so I'm gonna hide them and the way you do that is by getting the arrow selection tool and by right clicking on the object and you can turn off the show object option so I'm gonna hide my angle bisectors so they are now gone what I want instead now is to create the perpendicular distance from this point to the size of the triangle and I can do that by going back to the construction tool this one right here where we got the angle bisector and do the drop down you can see that there's a perpendicular line tool and I'm gonna go ahead and create a perpendicular line between this point and that side you can see that does appear to form a right angle and so what I'm going to do now is create the point of intersection between the two sides and now what I should be able to do is using the circle creation tool is make a circle with D as my center and E as a point on the circle and it might be a little hard to tell this but the distance from D to E is the same as from D to this side of the triangle and from D to this side of the triangle we know that because each of these links is a radius so it does appear to be a circle that is inscribed within triangle with D being or in center to test it out I'm going to go ahead and get my arrow tool and I'm going to drag around a vertex and just see how the circle moves on the inside you'll notice that no matter how or where I move the vertex C and I can move any vertex the circle stays within the triangle and the circle stays tangent to each of the sides of the triangle so it is indeed this is indeed the inscribed circle in D is the in center

Construct Congruent Triangle SSS.m4v



all right so we're recording so the first thing you want to do is be able to construct a triangle congruent to triangle ABC using the side side side congruence postulate so the very first thing we're going to do is copy one of the sides so I'm just going to copy slide AC first and I'm basically going to just draw a ray that would extend past the length of segment AC let's make that point X right there now what we want to do is copy AC and so we open our compass way up hopefully your compass is big enough it opens fr okay we just make a little mark there and then we copy the segment and the call-out point Z so so far we have one side cup now since we're going to be using side side side we need to copy all three sides we're going to shrink this down a little bit we're going to copy segment BC now so we open up our compass length of BC and then from point Z which is the corresponding point or to make an arc make it a little bit bigger here because we don't know exactly where that that corresponding point for point B is going to be it's somewhere however on that arc then we also copy segment a-b here okay so now we have the length of a B we go to the corresponding point X and we draw an arc and we know that the congruent segments going to be somewhere on this arc and in fact it's going to be right at the intersection we make that point why there and now just connect the endpoints that's how to copy a triangle using side-side-side

Geometry: 1-3 Measuring & Constructing Angles



geometry and constructing angles all right it sticks in one batch three men constructing angles all right the world ruler in this section is pretty much just about angles okay then one of the basic things in geometry example exhibit this right here that's an angle okay made up by two rays from meet at a youth vertex that point right there it's called a vertex all right now in an angle some of the basic things you need to know inside that angle call the interior outside everything outside of it all called the there's stuff thanks to you exterior is very good all right like I said before the point where they meet is called the town virgin all right good deal next thing you know about angles it's called measure mostly I've heard the word measure before it's talking about how big or small some is right here how far it's open it's called the in a world thanks for being in small measure comes into play all right professor all right now we measure vegetables in degrees okay a degree most time you think about what temperature oil guess what it's by this like it was 45 degrees that's the degree symbol okay that's how open or why these things are now all the way around would be how many degrees my circle is under 360 360 is all the way around so pretty much every degree is one out of that 360 okay well we have an angle and went all the way around be 360 there were times I equals there's a cute which means that it is less than 90 all right it's less than 90 degrees they begin in there listen cute is less tremendously right right angles you'll see these a lot and do a lot of stuff with them right angle means it is exactly 90 degrees in cynically 90 degrees all right now we usually mark it by putting that little box there because guess what I'm a corner piece of paper that's 90 degree angle boxes perfect squares are oh nine degree angles so we put a little square there for 90 degrees all right and then obtuse is take a log s greater stop it please greater than 90 degrees all right so you got a cute write that up too lovely now a straight angle is the last one you won't see it a lot called straight usually just called the line but that's what it is exactly all the way across so therefore it's 180 degrees okay so you got a cute which is less than I mean the way love you'll remember that is it's a cute blue angle okay you think of a cute little puppy alright Sam not at all but that's the way some people memory okay smaller right we just crack away it's perfect okay it's 90 degrees I'm too sis bigger and if it's a straight line it's always 180 degrees that all these angles one solution do the work good deal yeah last thing we talked about the words congruent and bisect yesterday congruent means it's the exact same which means easily so two angles are congruent you can just set them equal to each other all right so current means equal bisector means it cuts it in half so if you have an angle please stop all right if you have an angle and let's say the angle is 60 degrees all right you have an angle bisector if it box of X and it cuts it in the middle okay therefore 60 will become half so it be learning and urgent very good all right heart regime problems they never succeed in this math game find out on the next episode of Dragon Ball Z new label angels that is how you do it all right say we want to talk about this thing right here ABC okay we just right angle a b c the important thing when you label angles you put the little angle front of it put all three points that are involved the important thing is the middle point it has to be the vertex it has to be where you know it changes directions see we're changing directions at the B that means it has to be in the middle okay that pretty much labels how bangle is examples how do you do them all right pretty much if they just give it to you in words you got to draw it out they said L is in the interior of angle of jkm all right so first control jkm okay he has to be the vertex because it's in the middle and we know L is an interior that means it's somewhere inside this okay so we'll just draw help all right now they wants to find JK m 2j k m so we're trying to find the whole thing it's a measure of JK l is 42 degrees please don't say that every time you say next one is l km is 28 degrees now obviously if we're trying to find the whole thing and they gave us some number one part and another part you can put it together and find the whole thing's all you do is you had it 42 plus 28 now equal 70 degrees you know the whole thing is 70 degrees and you do the same thing say they gave us the whole thing and apart and we want to find the other part we were just subtracted like we do the whole thing with 70 and we knew that part was 42 we wanted to find this right here we would just do 72-42 and we would get 20 degrees good job