N Gen Math Geometry Unit 5 Lesson 8 The Midpoint Formula

foreign hello and welcome to another geometry lesson by emath instruction my name is Kirk Weiler and today we'll be doing unit 5 lesson 8 on the midpoint formula so there are many things in coordinate geometry that are important so far we've seen slope formula right that's that can be used to tell us things about whether two lines are parallel or perpendicular we've seen the distance formula which allows us to tell us how far two points are away from each other in the coordinate plane and also allows us to figure out whether or not two line segments have the same length today we're going to figure out formula that will give us the coordinates of the midpoint of line segment if we know the coordinates of its endpoints okay so let's get into that right now now midpoints we begin with midpoints on simple number lines the midpoint of line segment right is the unique Point meaning that there's only one for every line segment on any segment that divides it or partitions it in half or into two congruent segments it is simple feature to find in the coordinate plane very very easy but first we'll start with finding the midpoint of one-dimensional number line or on one-dimensional number line so exercise number one segment is shown below with end points at 3 and 13. letter what is the length of segment how can you find it using calculation rather than Counting all right we'll pause the video now and see if you can answer this question well the length of is 10 units long how can you do that without using it with using calculation rather than counting just by subtracting right so we've talked about this quite few times now subtraction is pretty much the key to most distances so can say that is 13 minus three which is 10 units long now of course that is exactly what you'd get if you counted but we don't want to have to count let's take look at letter where's the midpoint of the segment plot it as Point how do you know it's the midpoint all right well pause the video now and go ahead and figure out where the midpoint is I'm sure that you won't be that won't be much of problem for you plot it and then how do you know it's the midpoint all right well remember the midpoint is the point that will divide the segment into halves right half of 10 units is five units so if go five units above or five units below I'm going to be at up place of eight right and how do know it's the midpoint well can know it's the midpoint because AC is five and is five right and these are equal all right now let's take look at letter find the average or the mean of the positions of points and show your calculation below right so is at 3 is at 13 just want you to find the average of the numbers 3 and 13. pause the video now and go ahead and do that well obviously the average of two numbers the mean of two numbers is adding the two numbers together and dividing by two so if simply do three plus thirteen and divided by two I'll get 16 divided by two and I'll get eight so the mean of the two numbers is eight now letter what do you notice about the average of the two endpoints well hopefully what you notice about the average of the two endpoints is that it lies at the midpoint or it gives you the midpoint of the line segment right whoops here it is right what'd we find we found that we were at midpoint at eight so it is it is the midpoint location right and that should make lot of sense right in fact right what we should know is that the average or the mean of two numbers will always fall halfway between the two numbers always since the average always balances the number of units above it and the number of units below it and that's true whether or not you're averaging two numbers three numbers four numbers or whatever but in our case we'll only ever be averaging two numbers and so the amount of distance above the mean has to be equal to the amount of distance below the mean that means the mean must be at the midpoint of the two numbers cool thing is that this not only works on simple one-dimensional number line but it also works with segments in the coordinate plane so let's take look at that in exercise number two on the grid below segment has been plotted and has endpoints at two comma 3 and 14 comma 11 letter find the average of the and coordinates of the endpoints show your calculations below all right well let's do the average and the first in fact maybe I'll put with little AVG for average down there right and what would that be that would be 2 plus 14 divided by 2 which would be 16 divided by 2 and there's 8 again complete coincidence 8 won't always be the answer why don't you find the average of the two coordinates all right that should be simple enough so average that's going to be 3 plus 11 divided by 2. it's 14 divided by 2 and that's 7. all right let's take look at letter plot Point using the average and coordinates from in other words want to plot point of eight comma 7. let's do that right we come over here eight up seven and we're at Point all right now it definitely looks like Point is the midpoint of right but let's take look at letter give an argument for must be the midpoint of well keep in mind right that something is going to be midpoint if it produces two segments that are the same length so really the most straightforward way of showing that is in fact the midpoint of is to find the length of segment AC find the length of segment BC and show that they're the same length how do we find those lengths with the distance formula so let's do it let's find the length of AC together using the distance formula and then maybe you'll find the length of BC so right what I'm now doing is I'm now finding the length between and right and so the distance formula will tell me that I'm going to do 8 minus 2 squared plus seven minus 3 squared all right so taking our time we'll get 6 squared plus 4 squared that's going to be 36 plus 16. and that's going to be the square root of 52. now don't need to simplify square root of 52 or anything like that just want to verify that BC has the same length same length now just to be clear about that maybe I'll put right here that's 14 comma 11 maybe should have put right here just want to be able to see the points all right so now what I'd like you to do is calculate the distance between point and point go ahead and do that right down here if is in fact the midpoint we should also get the square root of 52 but take little bit of time to do that all right well you're going to see right away that it is but let's go ahead and go through it right for BC using the distance formula we're going to have 14 minus 8. quantity squared there's an squared plus 11 minus 7 quantity squared so BC is going to be the square root of 6 squared plus 4 squared that's the square root of 36 plus 16 again which is the square root of 52. and there it is right and that's really all we need to know that is the midpoint of in fact it's the very definition of midpoint right it is point that divides or partitions segment into two segments that have the same length in this case both of those segments have length of square root of 52. all right so let's formally introduce the midpoint formula now the midpoint formula if X1 comma y1 and X2 comma Y2 are the two endpoints of line segment then the midpoint lies at the average coordinates of the two endpoints in other words it lies at X1 plus X2 divided by 2 comma y1 plus Y2 divided by two all right and just kind of illustrating it with this you know picture it's very very simple right I've got X1 comma y1 X2 comma Y2 if want the midpoint take my two coordinates average them take my two coordinates and average them the only thing that have ever found that students find tricky about the midpoint formula is that sometimes they want to subtract the two coordinates and subtract the two coordinates before dividing by two now why do students sometimes want to do that well notice the other two important formulas that we've seen in this unit so far have been the slope formula and the distance formula and in both the slope formula and the distance formula you end up subtracting coordinates from one another and subtracting coordinates from one another but you're doing that in order to find the distance between the x's and the distance between the Y's in this case we're adding the x's and dividing by 2 adding the Y's and dividing by two because we're trying to find the average coordinate and the average coordinate which happened to fall halfway between our first you know between our two sets of coordinates anyway it's pretty darn easy formula to use let's use it in exercise three and find bunch of midpoints find the coordinates of the midpoint of line segment whose endpoints have the coordinates shown all right got four exercises here but they're all done in the same way so let's take look in exercise three letter right want to find the midpoint of this so I'm going to take the two coordinates add them together and divide by two take the two coordinates add them together and divide by two again just like all my other formulas want to be careful here 9 plus 13 is 22 divided by 2 3 and 7 is 10 divided by two like it when they end up being even 22 divided by 2 is 11 10 divided by 2 is 5. and our midpoint is at 11 comma 5. you just can't really get much easier than that right because all I'm doing is finding the average of the x's and the average of the Y's that becomes new coordinate point that is halfway in between now of course where things can become little bit dicey is when you have negatives involved let's do one like that and then I'll have you do and on your own I'm sorry and on your own so let's do together again to find the midpoint I'm going to take my two x's add them together divide by two take my two y's add them together divide by two take my time negative four plus two is negative two divided by two eight plus twelve is twenty divided by two negative two divided by 2 is negative one twenty divided by two is ten easy peasy lemon squeezy there we go negative one comma ten all right what I'd like you to do now is pause the video and go ahead and figure out the midpoint of this segment and the midpoint of that segment all right let's do it here we go negative seven plus one divided by two negative two plus fourteen divided by two negative seven plus one is negative six negative two plus fourteen is positive twelve negative six divided by two is negative three and twelve divided by two is six all right for letter this is probably the hardest one here because it's going to have decimal involved eight plus three divided by two negative 4 plus negative 10 divided by two eight plus three is eleven no and odd divided by two and negative 14 divided by two that one's not so bad 11 divided by two is five and half or five point five and negative 14 divided by two is negative seven that's it right there's not much more to it than that that is the midpoint formula average of the X's average of the Y's and that puts you right in the middle of the two and you have your midpoint now we can end up using the midpoint formula quite bit let's take look at exercise number four does the line whose equation is 5 equals 5x minus 32 bisect segment EF whose endpoints lie at the points 4 comma 3 and 8 comma negative seven justify your yes no response all right well the first thing that's really important is to understand what it means for line to bisect line segment so just for minute right like if had line segment right and let's say that had line that was bisecting it and I'm not talking about being perpendicular bisector right even tried to draw that so that it wasn't perpendicular to the line segment but what does it really mean right if line bisects segment what that means is that line goes through the midpoint right of that line segment goes through the midpoint of that line segment well that was interesting had little bit of black show at this time instead of red cool okay so pause the video now and see if you can figure out whether or not line that has equals 5x minus 32 will bisect segment whose endpoints are given pause the video now see if you can play around with this well in order for this line to bisect that segment it's got to go through the midpoint of that segment so the first thing I'm going to do is figure out the midpoint of segment EF all right so let me just lay that out the midpoint of EF let's just figure out where that is all right well can do that by just doing what we just did add the two x's and divide by two add the two y's and divide by two all right that's going to be 12 divided by 2 and negative 4 divided by 2 and that's going to give me midpoint at 6 comma negative 2. all right so know know that the midpoint of segment EF is 6 comma negative 2. the question is does this line pass through this point well think about all the times in eighth grade math Algebra 1 and even little bit in this course that we've asked question does this point lie on this line right well can just check that want to see if this point lies on this line and to do that I'm going to put negative 2 in for I'm going to put 6 in for I'll get negative 2 is equal to 30 minus negative 2 equals negative 2 that's true and so the answer is yes the line contains ef's midpoint so bisects BF right that's the whole point for line to bisect and segment means for it to contain the segment's midpoint or to pass through the segment's midpoint and we can tell whether particular point lies on particular equation or particular line in fact maybe I'll put that equation right here by simply substituting the point into the equation of the line and seeing if it makes the equation true all right real cool we don't need any coordinate grid or anything like that on here figure out what the midpoint is see if it lies on the equation of the line if so the line bisects the segment all right let's take look at one more exercise number five on the grid below segment PQ is plotted with endpoints at negative three negative four and five comma eight letter find the coordinates of the midpoint of PQ plot it as Point awesome no problem there why don't you pause the video now and go ahead and figure out the midpoint of segment PQ all right let me do it here we go so I've got negative three plus five divided by two comma negative four plus eight divided by two that's going to be 2 divided by 2 and 4 divided by 2 and that will be 1 comma two that's nice little midpoint there so 1 comma two let me put it in right as Point for midpoint awesome all right let's take look at letter find the slope of PQ in simplest form awesome so this has absolutely nothing to do with the midpoint right now just want you to find the slope of PQ in simplest form you could probably do it graphically but you could also do it by using the slope formula either way go ahead and find the slope of PQ all right I'm gonna do it with the slope formula it never hurts to see the slope formula again right slope is going to be Y2 minus y1 just have to be little bit careful there divided by X2 minus X1 8 minus negative 4 is positive 12 5 minus negative 3 is positive 8. can divide both of those by 4 and reduce my slope down to three halves awesome all right let's take look at letter draw the perpendicular bisector of PQ State an equation for it below in point-slope form all right well let's think about this just for moment the perpendicular bisector now in the last problem we worked with bisector all right but perpendicular bisector is little more special right not only does it go through the midpoint but it goes to the midpoint and is perpendicular to the line segment in question now know that the line segment in question has slope of three halves so part of this is going to be to think about what the slope of the perpendicular bisector is pause the video now and by thinking about that slope right try to draw the perpendicular bisector and then we'll come back and we'll talk about the point-slope form of its equation all right well the slope of the perpendicular bisector now right I'll put it down as slope perpendicular must be negative two-thirds right so we're going to want change in of negative two for change in of positive 3. that looks little bit better that'll allow me now to go three units to the right and two units down three units to the right and two units down and then can go backwards and really should have my ruler here but don't so I'm just going to do that that's not too bad all right there's my perpendicular bisector now I'd like to write its equation in point-slope form right this was something that was brand new to us just few lessons ago remember point-slope form basically says look if have the slope and any point that lies on line can write the equation of it as minus y1 equals times minus X1 right where is the slope of the line which is sitting right here and X1 y1 is any point that lies on the line now granted just by plotting this line saw lots of different points that lie on it but one point that might be very convenient to use would be just the the midpoint and again you don't have to use that point any one of the points just plotted you could use but think I'm going to use that one and therefore will have minus 2 equals negative two-thirds times minus one again any point could be used though that the perpendicular bisector goes through all right let's take look at one last little piece of this problem and of this lesson letter pick point on the line you drew and see that is not Point label it as point use the distance using the distance formula find the distance from to both the end point to both endpoints of PQ what do You observe about the distances now what love doing with this problem when I'm teaching it in front of class of many students is literally say pick any point along the perpendicular bisector that's the one in red now right any point at all you know make it not one of the nicer points right just not the midpoint and then using that point find the distance from that point to or sorry that distance from that Point to and the distance from that point to or if the point's down here right and then kind of compare them all right here we probably just want to use one particular point so I'm gonna think I'm gonna go here I'm going to use this particular point that is the coordinates negative 2 comma 4. Now again you could use that point you could use the point negative 5 comma six four comma zero would work great let's see that would be seven comma negative two ten comma negative four any of those points any of those points were great the only one would want you to stay away from is the actual midpoint itself which was at one comma two all right so let's just make sure we've got the the two the two coordinates down is at negative three negative 4 and is at 5 comma eight all right what I'd like you to do now is I'd like you to find the distance from to and also find the distance from to pause the video now and go ahead and do that all right time to do our distance formula here we go from to okay and I'll call this PR right so it's the length of that segment from to we can do the subtraction in either order I'm going to do negative 3 minus negative 2 squared plus 4 minus negative 4. squared lots of negatives there maybe should have picked different one all right negative 3 minus negative 2 is negative 1 just being careful 4 minus negative 4 is 8. squared all right 1 squared negative 1 squared is one 8 squared is 64. so get the square root of 65. nothing really interesting about that I'm not going to simplify it I'm not going to put in terms of decimal whatever so the length of the segment that would connect to is the square root of 65. now we want to find the length of the segment that would connect to I'll call that QR because it is at the right little bit smaller here but think can get it in there to I'm going to have 5 minus negative 2. squared plus 8 minus four squared 5 minus negative 2 is 7. squared eight minus 4 is 4 squared all right 7 squared is 49. 4 squared is 16 49 plus 16 is 65. and QR is the square root of 65. what observation can we make right point is equidistant from point and and we knew that already or we should have in other words many times in this course maybe even too many to count although for mathematician there's almost nothing that's too many to count right but many times in this course we've talked about the fact that any point that lies along the perpendicular bisector of segment will be equidistant from the endpoints of that segment now literally what we just found is we found this length and this length and we found that those two lengths were equal but if we had found let's say this length and this length those would have been equal or this length and this length they would have been equal or this length and this length etc etc all right and really what you're getting is you're getting all of these little isosceles triangles granted they're kind of like turned little bit to the side and they give us cool looking picture here but one thing just wanted to reiterate with this is that result that we saw earlier not in the coordinate plane right in the euclidean plane we now see again in the coordinate plane and we can really verify it with things like the distance formula this problem had almost nothing to do with the midpoint formula except for the first problem but it was pretty cool all right let's wrap this up so today we saw the last of our major coordinate geometry formulas and there's three of them there's the slope formula the change in over the change in there's the distance formula which is essentially just the Pythagorean theorem in the coordinate plane applied to the distance between two points and finally the midpoint formula the one that we concentrated on today which said if we want the coordinates of the midpoint of segment we simply average the coordinates of the two endpoints average the coordinates of this to the the two endpoints and those are the coordinates of our midpoint and hope that that may that means from what you already knew about the average of two numbers which is whenever you have two numbers and you find their average the number you find is at the midpoint between the two numbers that you averaged all right so not much more to say about the midpoint formula other than the fact that we'll use it quite bit as we move forward so get some practice on it even though it's pretty easy for now just want to thank you for joining me for another geometry lesson by emath instruction my name is Kirk Weiler and until see you again keep thinking and keep solving problems