Students in this video, let's discuss equilateral triangles in great detail.
So far we have seen that equilateral triangles are triangles where all the sides are equal.
So in this case, let each side be of length A.
All right, now in an equilateral triangle, we have also seen that all the angles are equal in every
angle is equal to 60 degrees, right.
Because we know that the sum of the angles of a triangle are 180 degrees.
All right.
Now, let me draw a line from this vertex to the base such that it is splitting the base into two equal
halves.
Two, I draw the line over here such that BD is equal to D.C. or D, the midpoint of B.C..
All right.
Now you can see that if you analyze triangle abeed, that is this triangle over here and Triangle ADC,
that is the triangle over here.
You can see that these two triangles are congruent Asper as as is criteria.
Right.
Why are they.
So you have this site equal to this side because both is equal to it.
We have seen that this is the midpoint of B.C. So beauty is equal to D.C. and it is a common sight to
all the three sides of these two triangles are equal and hence, as per Essence's criteria, Triangle
Abdeh and Triangle ADC are congruent.
All right.
Now, therefore, we can conclude that Angle and B will be 90 degrees.
Right?
Why?
Because let me right that over here.
Angle A B B is equal to angle E DC because they are corresponding angles of two congruent triangles.
And you can see from the diagram that they form a linear pair.
Right.
And we know that a linear pair adds up to 180 degrees.
So say this is X degrees, this is also X degrees.
That means X plus X or two.
X is equal to 180 degree or X is equal to 90 degrees.
That's why this angle over here will be 90 degrees.
All right.
Now, if side B, C is of length E and we know D, the midpoint of B, C, therefore B is of length
EBITA.
Right now, over here we have a right angle triangle, right triangle beauty.
And the hypotenuse is upside it and another side is of length EBITA.
Therefore we can find that aid, which is the height of this triangle.
It's equal to the square root of a square minus Ebeye to the whole square.
Right.
Why is this.
So this is an upper Pythagoras theorem.
Right.
Because we know is square is equal to it by two the whole square plus it's square and remember it's
over here.
Is it right now from here I find that edge is the square root of square minus two.
The whole square.
All right, let me make some space over here.
All right, now on simplifying this equation, we get that the height of the equilateral triangle can
be described as rude prebuy to a.
So the Heidi's route three by two.
So we have found the height of the equilateral triangle in terms of the side of the equilateral triangle.
All right.
Now let's find the area of this equilateral triangle.
We know that area of a triangle is half base in the height.
Right.
All right.
So let me take B.C. as the base that's of length.
And I know that the height is Route three by two.
So that's half in to a new route three by two.
And that gives me Route three by four square.
And that is the equation which we can use to find the area of an equilateral triangle.
So let's quickly revise.
We have seen that in a little triangle.
All the sides are equal and all the angles are equal and the angles are equal to 60 degrees.
All right.
And we have seen that the line segment, which is dividing the opposite side of Vertex into two equal
halves, is also a perpendicular line segment.
That is we have seen angle.
It is 90 degrees later we will see that this is the median as well as the perpendicular bisecting.
All right.
That will be following up in future video.
Now, let's press on.
We have seen that the height of the equilateral triangle is equal to Route three by two.
So in this way, we have expressed the height of the triangle in terms of the side of the triangle.
And we have seen that the area of the triangle is equal to room three by four, a square.