in this video we are going to talk about something called the midpoint theorem and if you can take a piece of pen and paper and follow along with me to see how it actually works so what it says is that if we have a triangle and then what we're going to do so let me actually label this as a b and c what we're going to do is we're going to locate the halfway point of a b so that would be approximately over there then what you do is you locate the halfway point between a and c which will probably be somewhere maybe over there and then you connect a line between those two points now mathematically what that will automatically do and this is what the midpoint theorem says is that these two sides will automatically be parallel so you are allowed to call them parallel and this length so let me call this c and d c d will be exactly half of b c that's how the midpoint theorem works okay so we can try a different triangle and then we can go ahead and locate the centers once again so that will be approximately over there and the center of this side will be approximately over there i'll call that c and d and then we can connect them and once again we can then say well this is what the midpoint theorem says it will tell us that this is parallel to that and the length of cd will be exactly half of bc so remember it's called the midpoint theorem so to use it you need the two midpoints how would we do this mathematically so let's say we had this in a test where we were told that bc is so this length is the same as that length and this length is the same as that length so just remember these lengths here they're not the same necessarily that doesn't have to happen but as long as these are the same and as long as these are the same then it works so mathematically what we would say is that cd because of this cd is parallel to bc and your reason is midpoint theorem so you say midpoint with a little dot so you don't have to go right out the whole midpoint word and then you just say theorem like that furthermore we could then say that cd is going to be a half of bc and that's also because of midpoint theorem now does the reverse work as well let's investigate that now so let's say i have a scenario like this where i have a triangle where i've got the midpoint of the one side which is at c now let me take a line across from the from c to the opposite side if i had to leave this line over here would you say that we have hit the midpoint of ac of course not what about if i drag it to there would that be the midpoint of ac no the midpoint of ac would be and i've just realized that i've got two letters c on my diagram don't worry guys i'll fix that now so the midpoint would be about there and so what that automatically does is it causes these two lines to be parallel but look if if i drag it up there then those two lines are definitely not parallel but as soon as i hit roughly about there where these two are parallel then that automatically means that these two are equal to each other so it means we've hit the midpoint so what i'm trying to say is the following if you are given the following scenario where you've got the midpoint on the left hand side like that or it could be the right hand side and then they've told you that these lines are parallel well then we can say that these two are equal can you see that that's the reverse of what we did in the first one in the first one we started off with two midpoints like that and then from that we said that these are parallel but if they give you the fact that the lines are parallel like that then you are allowed to say that this these two are equal but the reason for that is not called the midpoint theorem so how it would work in a test is the following let's say this is what you were given in a test you could then say let me just call this e you could then say that ae is equal to ec but you're not going to say midpoint theorem because the midpoint theorem you only use that when you have the two midpoints but we don't have the two midpoints now we only have one of them so we can say ae is equal to ec now i've seen some teachers they use the word converse converse means opposite of midpoint theorem other teachers will say ae is equal to ec and the reason is is that you've got a line going through one of the midpoints so a line through midpoint so you've got a line going through the midpoint and that line is parallel to other side like that so just see what your teacher uses but i'm sure you can understand the idea of how this works so let's have a quick go at this one over here so here we've got a triangle where these two lines have been given as parallel and we know that these are equal so we've got them e is the midpoint okay and this length from b to a is 16 and the length of c to b is 20. well what would this mean for us well well done if you realize that it means that cd would be equal to db but now we're not going to call this the midpoint theorem because the midpoint theorem only works if you have two power um two mid points already but what we had was the midpoint and parallel lines so we'll say converse midpoint theorem meaning opposite of midpoint theorem or midpoint theorem in reverse or your teacher might say line through midpoint parallel to the other side so the line that is going through the midpoint is parallel to the other side and so we would be able to say that cd is equal to 10 and db is equal to 10. and then having one last look at this one over here well here we've been given the midpoints of both because they're telling us that this is equal to that and they're telling us that that is equal to that over there and so for example we could say that d e is definitely going to be parallel to bc right the reason for that now because we've been given the two midpoints you can just go straight to midpoint theorem furthermore we could say that de is equal to 6 because it's half of bc and the reason for that is also midpoint theorem