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Current time:0:00Total duration:9:57

Video transcript

what I want to do in this video is review all of the neat and bizarre things that we've learned about triangles so first we learn so let me just draw a bunch of triangles for ourselves so let's have a triangle right over there the first thing that we talked about is a perpendicular bisectors of the sides of the triangles so if we take so let's take let's bisect at this side right over here and let's draw a perpendicular line to it so this line right over here would be the perpendicular bisector of this side right over here so it's bisecting and it is perpendicular let's draw another perpendicular bisector right over here so we're going that this is the midpoint of that side let's draw perpendicular so this is perpendicular and this length is equal to this length and then let's do one over let's do one over here this looks like the midpoint of that side right over there and then we will draw a perpendicular it's perpendicular and we know that this length is equal to this length right over here and what we learned is the the where all of these perpendicular bisectors intersect and what's neat about this and frankly all the things we're going to talk about in this video is that they do intersect in one unique point that one unique point is equidistant from the vertices of this triangle so this distance is going to be equal to this distance which is going to be equal to that distance and because it's equally equidistant to the vertices you could draw a circle of that radius that goes through the vertices so you could draw a circle of that radius that goes through goes through the vertices and that's why we call this right over here that point that intersection of the perpendicular bisectors so let me write this down just we can keep track of things perpendicular perpendicular bisectors perpendicular bisectors we call this point right over here our circumcenter circumcenter because it is the center of our circumcircle a circle that that it can be circumscribed about this triangle so this is our circumcircle sir Circle and the radius of the circumcircle the distance between the circumcenter and the vertices is the circum or radius so that was a perpendicular bisectors now the next thing we learned and the whole point of this video is just to make sure that we can differentiate between these things and not get too confused so let me draw another arbitrary triangle right over here the next thing we thought about is well what about if we were to bisect the angles so not we're not talking about perpendicular bisector the sides but we're not talking about bisecting the angles themselves so we could bisect to this angle right over here so let me draw my best attempt to draw it and so this angle is going to be equal to that angle we could bisect this angle right over here we could bisect on that's I could do a better version than that so that looks well one more try so I could bisect it like that and then if I'm bisecting it this angle is going to be equal to that angle and then if I bisect this one we know that this angle is going to be equal to that angle over there and once again we have proven to ourselves that they all intersect in a unique point and this point instead of being equidistant from the vertices this point is equidistant from the sides of the triangle so if you're dropping perpendicular to each of the sides so this distance is going to be equal to that distance which is going to be equal to that distance and because of that we can draw a circle that is tangent to the size that has this radius we could draw a circle we could draw a circle that looks like this and we call this circle because it's kind of inside the triangle we call it an in circle in circle and this point we can call which is the intersection of these of these angle bisectors we can call this we can call this the in radius now the other thing we learned about angle bisectors and this was we just had to draw one so let me just draw another triangle right over here and let me draw an angle bisector so I'm gonna bisect this angle so this angle is equal to that angle and let me label some points here so let's say that this is changes of colors let's say that is a this is B this is C and this is D we learned that if AC is really the angle bisector of angle B ad that the ratio between that a b over b c over b c is going to be equal to the ratio of ad to d c ad to d c sometimes this is called the angle bisector theorem so that's that's neat so the next thing we learned is let's draw another triangle here this is just to be an overview of everything we've been covering in the last few videos so let me draw another triangle here so now instead of drawing the perpendicular bisector so let me label everything this was angle bisectors this is angle bisectors and now what I'm going to think about are the medians the medians so the perpendicular bisectors were from the midpoint or Peart lines that bisect the sides and they are perpendicular but don't necessarily go through the vertices when we talk about medians we are talking about we are talking about points that bisect the sides but they go to the vertices and but they're not necessarily perpendicular so let's draw some medians here so let's say this is the midpoint of that side right over there so we could draw a median like that notice it's going through the vertices these did not necessarily go through the vertices vertices this right over here is not necessarily perpendicular but we do know that this length is equal to that length right over there let me draw a couple of more medians right over here so this the midpoint looks like it's right about here the midpoint looks like it's right about there so this length is equal to that length and those that goes through the vertex but it's not necessarily perpendicular and then this one see the midpoint looks like it's right about well right about there and once again all of these are concurrent they all intersect at one point right over here and so this length right over here is equal to this length right over here there was a bunch of neat things about medians when you draw the three medians will write like this that unique point where they intersect we called it the centroid centroid and as I mentioned and you might learn this later on in physics is if this was a uniform triangle if it had a uniform density and if you were to throw it or rotate it in the air it would rotate around its centroid which is essentially it would essentially be its center of mass it would rotate around that as its flying through the air if it had some type of rotational or I guess you could say angular momentum but the neat thing about this is also divides this triangle to six triangles of equal area so this this triangle has the same trying areas that triangle we proved this in several videos ago each of these six triangles all have all have the same area the other thing that we learned about medians is that the where the centroid sits on each of the medians is two-thirds along the median so the ratio of this side of this length to this length is two to one or this is two-thirds along the way of the median this is two-thirds of the median this is one-third of the median so the ratio is two to one another related thing we learned this wasn't really that necessarily about medians but it's a related concept it was the idea of a medial triangle a medial triangle like this where you take the midpoint of each side the midpoint of each side and you draw a triangle that connects the midpoints of each side we call this triangle a medial triangle medial triangle and we proved to ourselves that this when you draw a medial triangle it separates this triangle into four four triangles that not only have equal area but the four triangles here are actually they are actually congruent triangles and not only are they congruent but we've shown that this side is parallel to this side that let me do it use some more colors here this side is parallel actually I shouldn't draw two arrows like that that side is parallel to that side this side is parallel to this side and then you have this side is parallel to this side right here and this length is half of that length this length is half of that length this length is half of that length and it really just comes out of the fact that these are all these are for couldn't grew into triangles and then the last thing that we touched on is drawing altitudes of a triangle so there's medians medial triangles and I'll draw one last triangle over here and here I'm gonna go from each of the vertex and I'm not gonna go to the midpoint of the other side I'm gonna drop up perpendicular to the other side so here I will drop a perpendicular but this isn't necessarily bisecting the other side once again going to drop a perpendicular but not necessarily bisecting the other side and then drop a perpendicular but not necessarily bisecting the other side and we've also proven to ourselves so these are the altitudes of the triangle altitudes of the triangle and these also intersect in a unique is also intersect in a unique point and I want to be clear this unique point does not necessarily have to be inside of the triangle then the same thing was true of the perpendicular bisectors it actually could be outside of the triangle and this unique point we call an ortho orthocenter so I'll leave you there and hopefully this was useful because I know it can get confusing you know wait how is the median different than a circumcenter which is different than an ortho center or an enraged nor any of these type of things so hopefully this clarified things a little bit