Intro to medians of a triangle

if I take the random triangle like this and if I ask what would the midpoint of this side BC be just for fun and there are many ways to find that maybe I can take a paper and fold it do something or the other to find the midpoint of this side and let me try and guess the midpoint for our purpose here so maybe it's over here now you do that and you name this side this this point rather D so D is the midpoint of the side BC now if you take this point D and go to the point opposite to this side which is a and if you connect the two then you get a line segment here now this line segment is what we call a median that are at a median now that's a name given as you can see to the line segment that connects a point to the midpoint of the opposite side and you might guess that from the name a median is usually used for something in the middle so D is right in between or it right in the middle of B and C now you can think of many questions one of the questions that comes to my mind is can this median ever lie outside the triangle can it ever go outside this triangle any median that you draw now take a minute to think about that question for a line segment to ever go outside this triangle it may be maybe something like this or something like this for it to go outside ever from starting from a the other point that you draw that you connect it to must be outside this triangle because one point is on the triangle unless the other point is outside you will never be able to draw the line outside but the midpoint of the opposite side is always gonna lie within this triangle the midpoint of BC can never be outside here or over here it has to be right in between so the median of a triangle can never go outside of that triangle and now that you know that the medium will always lie within the triangle how many medians can you draw for a given triangle now one median will start from a point and join it to the midpoint of the other side and there are three such sides with three such points so you've drawn one I want you to pause the video and maybe take a piece of paper and draw the other two medians and see if you notice something viewed beginning to happen if you want to take quite a few triangles and try it let me draw the second median the one that connects B to this side and let's let's guess what the midpoint of this one would be maybe it's somewhere over here that looks like close enough so now let me connect this point to this point over here one side to the midpoint of the other side and now maybe I can call this point E is the midpoint of AC now there's only one more median I need to draw or I can draw in fact and that's the one that connects C to the midpoint of a B and there it is and now let me name this point F and we know that this side EF is going to be equal to F V now if you notice what's going on over here something should pop out to you I drew my median ad and then I drew my median B both will be within the triangle and they'll meet at some point that's not surprising they have to meet at some point but the third median that I drew seems to be meeting at the same point the third median is not going somewhere there or somewhere like this even though it looks a little not exactly like the meeting over a point of it that's because I did not find the midpoint exactly but if I had and you can verify this they would meet exactly at the same point over here so where ad and B E meet is exactly where C F will also meet these two lines now if you think about it that's mind-blowing ad and B are drawn the third line knows that it should pass through this point how does it know that and you might say very correctly that maybe we got lucky here maybe this is a coincidence this might be an accident when for this particular triangle that will happen but it won't happen for any triangle and I and I challenge you to draw as many triangles as you want they could be a cute they could be a use you know whatever you draw the three lines always know that they must meet at one point right in between and whenever something like this happens we all get really excited and we give it some name the point at which all the three medians meet is called the centroid of a triangle the centroid of a triangle it's the point that you get by drawing any two medians it's enough because we know the third one is also going to pass through that as you get to know triangles better and become their friends you start noticing these properties even more and just to give you a taste of it one of the interesting properties of a median is that if you draw this median do you notice that you've divided this triangle into two pieces I'm drawing I'm shading one of these pieces over here that's one piece and the other one is over here now one of the interesting properties of a median is that it these two pieces will have the same area so a median divides a triangle into two pieces of equal area now think about why that must be true