The hidden geometry tricks many don't notice.

Can you find the value of the area of this triangle? That's the only thing that's given that this side is 4. Welcome back to my channel. I forgot to put that this is right angle. Okay, the first thing we need to do is let's name all the points. So we have A, B, C, D. Let's call this side E and this point F. Okay, or since this is a meeting point, let's call it M. All right, this point M. Now, first thing we need to do is this. We need to write the formula for area of a triangle is half base multiplied by the height. Okay, half base multiplied by the height. So if we are looking for area of triangle BEM, we have the base already, which is our 4. And this is the perpendicular height BEM, but we don't know BEM, right? Okay, sorry, we don't know BEM, which is what we actually need here as well. So we are going to focus on finding this length BEM. Now, first of all, let's B to M be equal to X, right? Now, remember that with this, it means that BEM is equal to MT, all right? That means both of them are X, all right? Now, let's call BEY, okay? If this is the first time you have seen us, please click the subscription button, turn on your notification bell so that you don't miss our videos, all right? To upload our videos, always turn on your confidant. There was something for you to watch. Give us a thumbs up if you love what we are doing, all right? Now, back to what I was saying. First, we need to understand something that this is a rectangle, okay? A, B, C, D is a rectangle. Therefore, it simply means that BC is parallel to AD, right? It means that BC is parallel to AD, which implies that BD is a transversal. Therefore, we can say that this angle here, angle B is equal to angle D. Why did I say so? Because if BC is parallel to AD, then we have a Z angle. These are the two parallel lines. This is the transversal, right? So, we know the Z angle. The Z angle is the alternate angle. So, this angle is definitely equal to this angle. They are alternate. So, this angle is equal to this angle. This is our Z, all right? Now, why did I bring that up? It is very important because this is a rectangle. So, here is 90 degrees, right? So, you can see that triangle BEM is similar to triangle ABD, okay? To triangle ABD. Yes. Why did I say so? Since they have two angles in common, then definitely the third angle, which is E and angle ABC, ABD rather, are the same, right? So, it simply means that these two triangles are similar, okay? Now, if they are similar, then the ratio of their corresponding sides are equal as well. Therefore, we can write this as BE is Y. So, I need to clean this because I need that space, okay? Okay. Now, if they are similar, then we can say, let's AD over BD, right? So, if I choose AD over BD for the big triangle, now AD corresponds to BEM, right? We have this and this and we have this and this. So, it actually corresponds to BEM. So, we are going to say is equal to BEM over and BD of the big triangle, BD is actually the hypotenuse. So, this should be BE, right? All right. Now, AD, okay? AD, what is our AD? AD is 12 over BD is X plus X, which is 2X, is equal to BEM is X, right? Over BE is Y. Now, I want to get an expression for our Y first, okay? We need expression for our Y in terms of, okay? Now, let's do that. If you look at this now, you can see that triangle BEM is what? A right angle triangle. It's a right angle triangle. So, we can use Pythagorean theorem. If we use that, using Pythagorean theorem, we can actually write that BEM squared, which is X squared plus ME squared is equal to BE squared, okay? All these are squared. Now, BEM is X. So, we have X squared plus ME squared, which is four squared, is equal to BE squared, which is Y squared. So, we can write that X squared plus 16 is Y squared. Now, if we take square root of both sides, we'll have that Y is equal to the square root of X squared plus 16. So, our Y is square root of X squared plus 16. So, now we have Y in terms of X. Now, let's substitute it into this equation. I need to check something. Okay. I actually checked my camera to be sure that it is working properly. Okay. So, now we have Y in terms of X. So, permit me to rewrite this particular equation as 12 divided by 2X is equal to X divided by square root of X squared plus 16 because that is our Y. Okay? Okay. Now, at this point, I'm going to delete everything on this board because Okay. Now, if we solve this, this will give us 6 over X. So, we have 6 over X is equal to X divided by square root of X squared plus 16. Now, if we cross multiply, we have X times X, is equal to 6 times this which is 6 root X squared plus 16. Alright. Now, we have X squared here. We have X squared here. Let's simplify this by saying let X squared be equal to N. So, if we say let X squared be equal to N, I'm going to write it somewhere because we are going with this letter. X squared is N. So, at this point, we can now write this as N is equal to 6 square root of N plus 16. Okay. N is equal to 6 root N plus 16. Now, because I need to remove this root sign, let me square both sides. Okay. So, if I square both sides, I will have N squared is 6 squared bracket N plus 16. Right? Right. Have you subscribed yet? Because if you are still watching me till now, it means that you love what I'm doing and I wouldn't like to miss you in my next video. So, please subscribe and remember to comment friends. Comment, let me know what you feel about the video. Okay. And what method you would have used in order to get the answer to the question. So, now we have N squared is 6 squared is 36 bracket N plus 16. So, N squared is 36 times N is 36 N plus 36 times 16. We have 36 times 3 is 18 plus 3 is 21. 1 times 6 is 6, 1 times 3 is 3. So, we have 5, 7, 6. So, we have 5, 7, 6 here. So, we can see clearly that we have a quadratic equation here. All right. So, we have N squared is equal to, okay, let's transfer 36 N to this side. So, it becomes minus 36 N. If we transfer plus 5, 7, 6 to this side, it becomes minus 5, 7, 6 is equal to 0. So, we are going to solve this using Poisson law method. I have a video on that in this channel and I'm going to attach it here. So, it gives us the answer to any quadratic equation faster than any other method you can think of. So, looking at this, my N is going to be 18 plus or minus 30. Okay. It's going to be 18 plus or minus 30. Like I said, I will attach the video here so that you can learn how I got this very fast. Okay. So, using Poisson law method. So, my N now will be 18 plus 30 or 18 minus 30. Therefore, my N will be 18 plus 30 is 48 or 18 minus 30 is minus 12. But remember that we have a physical shape here. So, we cannot make use of N. It cannot be what's negative, right? So, we are going with the positive N. But remember what I wrote here. I said, remember we said let X squared be N. So, it simply means X squared is equal to 48, right? Now, X will be square root of 48. That means X is square root of, this is 16 times 3. 48 is 16 times 3. So, X will be 4 root 3. That is square root of 16 is 4 and we have root 3 remaining. So, it means that our X here is 4 root 3. So, we have 4 root 3. And with that, we have found our answer, right? Because our area is half base times height. So, it's going to be half. Our base is 4, right? Our height is 4 root 3. Now, 2 we go into for 2 times and our answer is 2 times 48 root 3 square units. Thank you so much for watching and see you in my next video. Bye.