Bringing it all together
Review of Triangle Properties Comparing perpendicular bisectors to angle bisectors to medians to altitudes
⇐ Use this menu to view and help create subtitles for this video in many different languages.
You'll probably want to hide YouTube's captions if using these subtitles.
- What I want to do in this video is review all the neat and bizarre things
- that we have learned about triangles
- so first we learned
- 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
- the perpendicular bisectors
- of the sides of the triangles,
- so if we take, so let's take
- let's bisect this side 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's perpendicular, let's draw another
- perpendicular bisector right over here,
- so we're learning that this is the midpoint
- of that side, let's draw a perpendicular bisector
- and this length is equal to this length,
- and then let's do one, let's do one
- over here, this is the midpoint of that side
- right over there, and then we will draw
- a perpendicular - we know that this length
- is equal to this length right over here,
- and what we learned is where all these
- perpendicular bisectors intersect,
- what's neat about this and frankly,
- all the things we're going to talk about
- in this videos is
- 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 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, that goes through
- the vertices, and that's why we call this
- right over here, that point,
- that intersection of the perpendicular bisectors
- let me write this down so we can keep track
- of things, perpendicular bisectors,
- perpendicular bisectors,
- we call this point right over here our
- circumcenter, because it is the center
- of our circumcircle, a circle that can be
- circumscribed about this triangle, so
- this is our circumcircle.
- And the radius of the circumcircle,
- the distance between the circumcenter
- and the vertices is the circumradius,
- so that was the perpendicular bisectors.
- Now the next thing we learn is
- the whole point of this video is to
- make sure we differentiate between these things
- and not get too confused
- 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?
- Now, we're not talking about perpendicular bisecting
- the sides, we're talking about
- bisecting the angles themselves, so
- we can bisect this angle right over here,
- my best attempt to draw it-
- so this angle is going to be equal to that angle,
- we could bisect this angle right over here,
- we could bisect- oh, that's - I could do a better version of 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 drop a perpendicular to
- each of the sides, 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 side
- that has this radius, we can draw a circle,
- we can draw a circle
- that looks like this,
- and we call this circle, because
- it's kind of inside the triangle
- we call it an incircle, incircle,
- and this point, we can call,
- which is the intersection 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 have to draw one-
- so let me just draw another triangle right over here,
- and let me draw an angle bisector,
- so I'm going to 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-
- change the 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 BAD,
- that the ratio between AB over BC,
- is going to be equal to the ratio
- AD to DC, sometimes this is called
- the angle bisector here, so that- that's neat
- so the next thing we learned is - let's draw
- another triangle here, this is to be a full review 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 bisectors
- let me label everything, this is 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 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 they are 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 can draw the median like that,
- notice it's going through the vertices,
- these did not necessarily go through the vertices
- this right over here is not necessarily perpendicular
- but we do know that
- this length is equal to that length right over there
- and we 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 notice it goes
- through the vertex, but it's not necessarily perpendicular
- and then this one,is a 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,
- so this length right over here is equal to
- this length right over here.
- There is a bunch of neat things about medians-
- when you draw the three medians
- like this, that unique point where they intersect
- we call this a centroid, centroid,
- and as I mentioned, and you might learn this
- later on in physics, is
- if this was a uniform triangle, and it had
- a uniform density, and if were to throw it
- or rotate it in the air, it would
- rotate around it's centroid,which is essentially-
- it would essentially be its center of mass,
- it would rotate around that as it's flying
- through the air, if it had some type of
- rotational or I guess you'd say
- angular momentum, but the neat thing about this is
- it also divides this triangle into
- six triangles of equal area, so
- this, this triangle has the same area as
- that triangle, we proved this in several
- videos ago, each of these 6 triangles all
- have all have the same area.
- The other thing that we learned about medians is that
- 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 necessarily about medians,
- but a related concept, was the idea of
- 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,
- a medial triangle, and we've proved to ourselves
- that this- when you draw a medial triangle,
- it separates this triangle into 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's what we do,
- use some more colors here,
- this side is parallel - actually, I should
- 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 that length,
- this length is half of that length, and
- it really just comes out of the fact that
- these are four congruent triangles.
- There's a 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 going to go from each of the vertex,
- and I'm not going to go to the midpoint
- I'm going to drop a perpendicular to the
- other side. So here I will drop a perpendicular
- but this isn't necessarily bisecting the other side,
- once again, we're 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 have also proven to ourselves that
- these are the altitudes of the triangle,
- altitudes of the triangle, and
- these also intersect in a unique- these 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. 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 is
- useful, cause I know it can get confusing,
- you know, how's a median different than a
- circumcenter, which is different than an
- orthocenter, or inrays, or any of these type
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
|
Have something that's not a question about this content? |
This discussion area is not meant for answering homework questions.
Discuss the site
For general discussions about Khan Academy, visit our Reddit discussion page.
Flag inappropriate posts
Here are posts to avoid making. If you do encounter them, flag them for attention from our Guardians.
abuse
- disrespectful or offensive
- an advertisement
not helpful
- low quality
- not about the video topic
- soliciting votes or seeking badges
- a homework question
- a duplicate answer
- repeatedly making the same post
wrong category
- a tip or feedback in Questions
- a question in Tips & Feedback
- an answer that should be its own question
about the site
Share a tip
Suggest a fix
Have something that's not a tip or feedback about this content?
This discussion area is not meant for answering homework questions.