Unit 2 Lesson 2 Midpoint of a Line Segment MPM 2 D

next we're going to take look at the midpoint of line segment first of all line segment is part of line between two specific points including the points themselves so you can see two line segments in example one and for example one we want to determine the midpoint of each of the following line segments so starting with the first line if it starts at 0 and goes to 15 we see that the midpoint occurs here at 7.5 and to get that answer we just take 15 and we can divide that by 2 which we know equals 7.5 for we're not starting at 0 so what is the midpoint between 7 and 22 the best way to find the midpoint between two numbers is to take what's called the average the average and the midpoint are equivalent to each other when we're dealing with two points so if we want to find the average we would take 7 plus 22 and we would divide that by 2. well we know 7 plus 22 is 29 and then we want to do 29 divided by 2 so that's 14.5 so you may have done different way to get this midpoint but this would be the most efficient way take the average of the two numbers 7 and 22. so now let's get into some geometry what if want the midpoint between and which are now two points so let's plot these two points so point is at one and eight and point is at five and two connect these points with line we now have line segment which is just part of line if want to find the midpoint using the diagram could just count squares and can see if go two over and three down hit this point and if go two over and three down get to that end point so do have midpoint right here and the midpoint if count over is three units to the right and five units up but let's say we don't want to use graph every time what we can do is take the average of each of the components so let's start with the component if look closely see that have two values one and five let's take the average of those two values if do one plus five and divide it by two see that get an value that's six over two which equals three which matches my graph similarly if take the average of the values so now take eight and two do eight plus two and divide it by two that's ten over two which equals 5 and that matches my graph again so we see we have midpoint that's 3 comma 5. and that is how you find the midpoint of line segment on an plane given any line segment with endpoint x1 y1 and x2 y2 the midpoint is the point with the coordinates and we can see here the average of the x's one plus two divided by two and the average of the y's one plus two divided by two so again if we want the midpoint we just take the average of the x's and the average of the values let's go to an example let's determine the midpoint of the line segment with endpoints five and seven so there's five and seven and four negative five and here's four negative five want to find the midpoint of this line segment so let's do it without the graph this time let's start with the x's if want the average of the x's i'm going to take five and four add them up and divide by two so get five plus four over two that's nine over two which is four point five or nine over two let's take the average of the y's so the y's we see are seven and negative five so get seven plus negative five over two that's two over two which equals one so this tells us that we have midpoint that is nine over two or four point five comma one let's go see when we plot this if go over to 4 4.5 and count up 1 hit that point right there and that is indeed the midpoint of and so again we're just taking the average of the components and the average of the components and that'll help us find our midpoint let's change the question around now let's say we want to find an end point knowing the midpoint and one of the other end points so first let's plot this we know that one endpoint is at negative two and five we'll call that and we know that the midpoint is at one and three we'll call that we need to find the other endpoint so so far we know half of the line so we need to find the other half quick way if you have graph is to count squares so go 2 down and 3 to the right can continue that pattern 2 down and 3 to the right that'll be our other end point say so we can do that with graph but how do we do it algebraically so let's start with using the average concept again there is the for the endpoint there is the for the midpoint remember the is equal to x1 plus x2 divided by 2. remember the x1 and the x2 are the endpoints and we only have one endpoint so this here we'll call x1 the without the number on it is the midpoint so we're going to call that as well we know that is going to equal 1 plus 2 divided by 2 and we have these two values the first one being an endpoint so that's y1 and the second one being the midpoint so that is the so all we're going to do is sub these values into the corresponding equation so for the we take the from the midpoint that's one we know one of the endpoints is negative two and then we don't know the other end point so that's two and we divide it by two get rid of the 2 from the bottom so we're going to multiply both sides by 2 and we end up with 2 equals these two cancel because 2 divided by 2 is 1 so we get negative 2 plus sub 2. if we add 2 to both sides we now have the endpoint value so let's see what we get we get 4 equals x2 now let's go to our graph notice our graph here that value for we count squares is 4 1. so we got the value right we've got the four let's try it with the component let's sub in our values for so we see from the point that one or sorry we see from the point that is three 1 is 5 because that is from the end point the other end point we don't know so that's 2 divided by 2 following the same steps i'm going to multiply both sides by two to get rid of the fraction here because this two and this two two divided by two is one but two times three is six and on the right side we get five plus two if we minus 5 from both sides we end up with our answer 1 equals y2 so putting the and together we have that point is four and one which is what we see from our graph as well all right last thing want to talk about is applications using the midpoint the first application we want to look at is what's called median line and median line joins vertex of triangle to the midpoint of the opposite side so vertex is like the corner of the triangle so here's vertex and here's the midpoint of the opposite side so that dotted line there is what we call the median line from vertex and we could also create median line from vertex it would look something like this and we could do meeting line from vertex it would look something like this and in fact all three would join at some point but we're not going to talk about that today so let's look at how to find the equation of median line there's three steps to do this first step we need to find the midpoint and you have to find the midpoint of whatever side you need to use so that's step one once you know the midpoint you can find the slope of the line so you can find the slope of this line here and once we know the slope we can get the y-intercept so with the midpoint with the slope and the y-intercept you'll end up with some equation equals mx plus which will be the equation of that median line so let's look at an example and let's try to find the equation of median line so example three we're going to determine the equation of the median line from vertex in triangle abc if the coordinates of the vertices are negative 3 negative 1. so there's is 3 and 5 and is 7 and negative 3. let's connect those lines together so there's to to and to which forms our triangle we want the median line from vertex so we want to start from vertex and get to the midpoint of the opposite side so as we said in step one to find the equation of median line we need midpoint and for this example we're going to need the midpoint of bc ybc because we're starting from vertex and we're trying to get to the midpoint of the opposite side so let's find that midpoint so using the points and we're going to find the average of the x's and the average of the y's so let's start with the x's the x's are 3 plus 7 and then we're going to divide them by 2 and 3 plus 7 is 10 divided by 2 gives us 5. we take the average of the values so 5 plus negative 3 and again we're going to divide by 2. so 5 plus negative 3 is 2 2 divided by 2 is 1. so we find the midpoint of bc is going to be 5 comma 1. so let's go check if go over to 5 and fits nicely on this triangle there is the median line from but we want the equation of the median line so we've drawn it now let's come up with an equation so to find the equation of line first we need the slope and we need the slope of the line and the line is so from point to point remember that is negative 3 negative 1 and we just found out that is 5 and 1. so the first thing is we need the slope the slope of am remember is delta over delta taking the two y's so i'm going to start from the so 1 minus negative 1 and then subtracting the same way 5 minus negative three if we do the subtraction get two over eight which is equal to one quarter if look at my graph could check that if go up one and over four that looks correct the green line has positive slope and we saw that our slope is one quarter so now we have part of our equation we know that is equal to one quarter plus now we need to find the y-intercept and in order to find the y-intercept we need point on that line lucky for us we have two points we have negative three negative one and five one so we're going to use one of these points i'm going to use 5 and 1. so put 1 in for equals 1 4 5 in for plus remember that 5 is 5 over 1. so now have 1 equals 5 quarters 5 times 1 is 5 4 times 1 is 4 plus i'm going to move the five quarters to the left side or subtract five quarters from both sides so get one minus five quarters equals remember that one is one over one the equivalent fraction would be four over four minus five over four and therefore would equal negative one quarter so now we know that the equation of this line is equals one quarter minus one quarter so let's go up to our picture and check we should have intercept that's roughly negative quarter and we should have slope that's one quarter so if you look at this that's not bad there's the y-intercept and if count 1 and go over 4 that's not bad as well so it looks like our answer is correct and that the equation of the median line is equals one quarter minus quarter try these questions and see how they go