I students in this video, let's discuss a very important and useful property in right angle triangles
called the Pythagoras's theorem or the Pythagoras's property.
All right, so let's take an example of a right angle triangle.
This angle over here is a 90 degree angle.
And we call the side opposite to the angle that makes a 90 degree as the hypotenuse to this side over
here is opposite to the 90 degree angle.
And we call that the hypotenuse.
All right.
Now, the other two sides are legs of the right angle triangle.
Now let's move on.
We have defined these terms.
Remember what the hypotenuse is?
It's the side opposite the 90 degree angle.
All right.
Now let's take a right angle triangle.
We have a triangle with sides A, B and C, and this is a right angle triangle with a 90 degree angle
over here.
All right.
Let's make three more copies of this right angle triangle.
All right.
And let's change the arrangement of these four triangles in this manner.
Now, let's make a copy of this over here.
OK, now you will see that this area over here is equal to a into a that's a square, right?
Because over here you get a square.
Let me name this figure a, b, c, d, e, f, g, and it's now B, B if it is A square.
And each of its sites is of lengthy.
We know that the area of a square is a into a you can check more about it under quadrilaterals.
Now, the area of a square is site square lower here.
Every site is equal to eight.
Therefore, the area of this square is equal to a square.
All right.
Now the site, the angles of a square are 90 degrees.
Now, this angle will be equal to 90 degrees.
Why would that be?
Let me quickly show you that let's construct a line that's parallel to AC.
And over here now, we can see that this angle will be equal to this angle.
Let me call this angle one.
So this is also angle one.
And you will see that this angle over here will be equal to this angle.
Let me call this angle to angle to why is that so?
Because these are alternate interior angles, right?
This line over here, which we constructed, let me call it, as Lynelle is parallel to G and it's parallel
to AC.
Therefore, this angle and this angle are equal.
Similarly, this angle and this angle are equal.
Right.
And we started with the right angle triangle.
Right.
This angle will be angled to and this is 90 degrees.
So we know that as our angles and property angle one plus angle two plus this angle over here, which
is 90 degrees, is equal to 180 degrees.
Therefore angle one plus angle two is equal to 180 minus 90, which is equal to 90.
Right.
That's why this angle is 90 degrees and each side is of length.
That's why BATF is it square.
And we know that the area of a square is equal to a new A or side in the side.
And in this case, the area is equal to a square.
All right.
Now let's move on.
We have made a copy of this shape over here and we have a square as the uncovered area in this case.
Right.
All right, now let's make some arrangement shifting in the case of these triangles and see what happens,
I shift this triangle in this direction.
And it goes over here.
All right, now the boundary is the same, you cannot all right, now this triangle is shifted in this
direction.
And this triangle over here, I shifted down.
All right, now we have maintained the boundary as the same right there for the area of uncovered,
which is uncovered over here as to be equal to the uncovered area over here.
What do I mean with uncovered area?
It's the area that is not occupied by a triangle.
Over here.
You can see this is occupied by a triangle.
This is occupied by triangle.
The remaining area is what we call the uncovered area over here.
We have shifted the triangle such that this is the area covered by a triangle.
This is the uncovered area.
All right.
Now, what is this area?
This is equal to being to be which is equal to be square.
Right.
Because you can see that this site is B and this side is B, so we are getting a square of side B over
here.
Similarly, you can see that over here, this site is of lengthy and this site is of Benghazi and we
are getting a square of sight to see.
So the area over here is equal to C Square.
Now, because the uncovered areas have to be equal and the uncovered area over here is equal to a square
and the uncovered area over here is equal to be squared plus C squared, therefore is square, is equal
to B squared plus C square.
And remember, what is a BNC?
These are sides of the triangle with which we started, which is a right angle.
Triangle is the hypotenuse, which is the side opposite the 90 degree angle.
Therefore this is what we call the Pythagoras theorem that is in a right angle triangle.
The square of the hypotenuse is equal to the sum of the squares of the other two sides.
That is a square is equal to B squared plus C square.
And remember, the Pythagoras property is valid only for the right angle triangles and this is the right
angle triangle.
That is why this is true.
So let's quickly summarize.
You take any right angle triangle.
Let's take an example or let it be five.
Let this be three and let this before this and ninety degree angle, you will see that five square is
equal to three square plus or square.
What is five square.
That's twenty five.
What is three square.
That's nine.
What is four square.
That's sixteen and nine plus sixteen you will see is equal to twenty five.
So in a right angle triangle the square of the hypotenuse is equal to the sum of the squares of the
other two sides and that is what we call the Pythagoras property.