In this video, let's discuss something interesting about equilateral triangles, we have learned what
we mean with median angle bisect perpendicular by sector and altitude, and we have also seen their
respective meeting points in a triangle.
All right.
Now let's see those in the case of an equilateral triangle.
Over here, we have an equilateral triangle led the side of this equilateral triangle.
We have linked it now in an equilateral triangle.
You will see that the median angle by sector, all the different perpendicular bassnectar are all the
same line.
Let's try to understand why.
Or here I have a triangle right now.
Imagine let this be the median.
Let's just take this as the median.
So if you observe this triangle over here and this triangle over here, you can see that they are congruent.
Why would that be?
This side is equal to decide model of length.
If we have constructed this side over here as the median right there for this.
And this will be equal because the median joins the vertex with the midpoint of the opposite side.
But these two sides are also equal and this side is a common sight.
Therefore, you will see that this triangle over here and this triangle over here are congruent right
now because these triangles are congruent.
What are the things that we can derive out of it?
We can see that this angle.
And this angle has to be equal, right, because there corresponding angles of two congruent triangles
and you will see that these two angles form a linear bent.
If I say that this is X and this is, again, X degree, we have seen they are equal now because they
form a linear pair, X plus X is equal to 180 degrees.
Therefore X is equal to 90 degrees.
Right.
We have seen that X will be equal to 90 degrees.
And because X is equal to 90 degrees.
Right.
You can see that this line over here is the perpendicular by sector as well, because it's bisecting
this side and it is perpendicular.
So we have seen that the median and the perpendicular by sector are the same now because this triangle
and this triangle is congruent.
Therefore, this angle over here and this angle over here will be equal.
Right, because again, there corresponding angles of two congruent triangles.
Therefore, this line over here is also the angle by sector.
Right.
And because this angle over here is 90 degrees and it is passing through this vertex, it's also an
altitude because we have learned that an altitude joins the vertex and it's a perpendicular drop to
the opposite side through a vertex.
So this and this side over here is also an altitude.
Therefore, in an equilateral triangle, the median angle by sector altitude and perpendicular by sector
are all the same line.
All right.
Now, having established this, let's move on to we have drawn one median over here.
Similarly, let's draw the other two also and we will see that they intersect at this point.
Let me name these points so it'll be easy for us.
Let this point be point G now, because this line this line in this line is a median angled by altitude
and perpendicular by sector.
Therefore, point the G is the center, sock'em center, centroid and auto center.
So in an equilateral triangle, all these points going side.
So it's the same point.
The center is also the subcommander, it's also the centroid and it's also the auto center.
Now, let me mark these points, SPQR Now we know that in the case of a centroid, right, it divides
the median in the ratio, two is to one.
So this is very useful, especially in the case of an equilateral triangle in a lot of problems.
Right.
Because we have seen that in an equal triangle.
All these four are the same line.
Right.
Therefore, I can see that over here.
Let's take, for example, this line a bit so I can see it divided by g.p is equal to two by one, because
this is also the centroid.
And we know that the centroid divides of media in the ratio two is to one, therefore aided by GPS equals
two by one.
Now let this be the height of the triangle.
We know that this angle is 90 degrees, right?
Therefore, AP is also an altitude.
So therefore AP is the height from point A to B, B.C., let's call it as it's now.
We know that in an equal triangle, we have seen in a previous video that it is equal to Route three
by two it.
Where is the side of the equilateral triangle?
All right.
Now we've seen that divides up in the ratio two by one.
Right?
So I can say that a G is equal to two by three times its right.
This is two weeks and this is X-Rite, because everybody GPS to buy one so I can see it is two weeks
and GB's X now to explore X is three x three X is equal to X, which is the altitude.
Right.
Therefore I can see that two X that is agea is equal to each divided by three in the two, that's two
by three into X.
And remember AGEA is the sock'em radius.
Right, because there's also the Sarkhan Center.
And remember, if you joined the circle center to a vertex and that distance is called the Sarcone radius.
All right.
Let me make some space over here and we proceed.
So we have seen that the Serkan radius in an equilateral triangle will be equal to two by three X,
and we already know it is Route three by two.
So let's not combine these two.
If it's required in a question, you can combine it like you can write it as two by three in the root,
three by two.
All right.
Now let's proceed.
We can see that DP's X and that would be one by three.
It's right the same manner.
Three X is equal to each.
So X is equal to X by three.
So X by three is the in radius.
Remember, if you take the in center and you take this length over here, because when you draw an inner
circle, all the sides are tandem's.
Right.
Therefore this distance over here will be the in radius and the radius in an equal triangle is equal
to one by three X.
So we have learned a lot of interesting properties about the equilateral triangle.
Let's quickly revise the first properties that the median angled Bicego.
All the different perpendicular by sector are all the same line.
Secondly, we have seen that the center satcom center centroid, an auto center is the same point and
we have seen that the circle radius of an equal triangle is equal to two by three each.
And we have seen that the radius of an equal triangle is equal to one by three.
And you know the logic behind this, right?
So it's very easy for you to remember.
And we collect and apply in questions because you're not just building it, but you know why the circumference
is equal to by three X and you know why the radius is equal to one by three.
It's in an equilateral triangle.
And remember, in an equilateral triangle or these four points coincide.