00:01
So as per the required question, we have to write three most favorite theorems of high school mathematics.
00:10
So my first theorem is this, that between two parallel sides, okay? that between two parallel sides.
00:21
The area of triangles.
00:27
Okay, so first line is this and the second line is.
00:33
This for example and this is our triangles okay so i wanted to create a triangle with the same base okay so triangles with the same base okay so these two we have to add this two okay so triangles with the same base for example here the triangle a bc so the triangle a bc and triangle a bc and triangle bdc.
01:13
Okay, so these two triangles have same area.
01:17
What is the theorem? the theorem states that the area of triangles made between two parallel sides with this same base have equal area.
01:29
So now what we have to prove? so we start here as the triangle.
01:36
So we have to draw a perpendicular from the point a.
01:39
So this is perpendicular and a perpendicular from the point.
01:42
D okay so for example let us suppose this is m and this is n okay so now first we know that the area of the triangle a b c okay so what is the area area of this triangle okay so area of this triangle is nothing but half base what is the base length here base length is bc into height height is nothing but am okay so base into height so half base into height so this is the area of the triangle a bc now area of the triangle b dc so what is the area of the triangle bdc okay so area of this triangle is nothing but half of base base length is nothing but bc into height height is nothing but d n okay so this is the area of this triangle now as we see here that these two sides were which two sides the side am is equal to a n sorry dn okay these two sides are same hence this quantity here and this quantity is same so this whole quantity must be equal to this whole quantity okay so we say that so area of triangle a bc area of the triangle a bc is equal to area of the triangle b dc okay so this is our proof this is is our theorem first.
03:20
Now the second theorem.
03:22
The second theorem is nothing but the midpoint theorem which states that inner triangle, okay? in a triangle, okay? in a triangle, if inner triangle, so this is our triangle, okay? in a triangle a, b, c.
03:57
For example, let us suppose this side.
04:01
So this side is de.
04:05
And the important point here is this.
04:07
This is this.
04:07
Is midpoint okay so d is midpoint of a b and e is midpoint of ac so we have to prove that de is parallel to bc and de is equal to nothing but half of bc okay so we start with what so we start with the proof so we know that by the converse of thales serum what by the converse converse of thales theorem okay thales theorem so by the converse of thales serum what we know that we know that in a triangle if as you know that a d upon d here it is one ratio 1 is equal to ae upon ec okay so these two are one ratio one so if these ratios are same, then what we know here? we know that de must be parallel to bc.
05:25
This is the converse of thalesram by the converse of thalesram.
05:29
This is a theorem which is known as converse of thales theorem.
05:33
Now as d is parallel to bc, so this line is parallel to d .c.
05:38
So this angle here is equal to this angle by the corresponding angle theorem and this angle here is equal to this angle by the corresponding angle theorem.
05:46
So the triangle a -d -e is now similar to triangle a -b -c.
05:53
Okay.
05:54
So now what we have to say? we have to say that that ad upon a -b is equal to a -e upon a -c is equal to d -e upon d -e upon d -e upon d -e upon bc.
06:22
Okay.
06:23
So what we know here that ad upon ab.
06:26
So what is the length of ad? it is nothing but half of ab.
06:31
Okay...