A tricky geometry math puzzle/ find the area of the unknown portion.

It is said that only three percent were able to solve this particular question. Welcome back to my channel. Note that this is a freehand sketch and so it's not drawn to scale as well. In this particular question, we are told to find the area of this portion of the rectangle. The area of this triangle is four and we are asked to find the area of this portion of the rectangle. So first things first, we need to find out something. Let's name this A, B, C, D. So if we should draw out these two triangles, let's draw it out. I wish one of my students will see this particular question because I just taught her similar triangles so that she can see how these questions relate from simple to complex questions. So these two triangles are similar. Now I'm going to prove that to you. First of all, you will see that opposite angles are equal. Vertically opposite angles are equal. So it means that this angle here is equal to this angle here. Also because these two sides are two sides of a rectangle, it means that these two sides are parallel to each other. And if they are parallel to each other, remember that your alternate angles are equal. So when you have two parallel lines and a transversal, Z angle is formed. I call it Z angle because it's a Z shape. So we have this, we have that, we have that. So you are going to notice that this angle here will be equal to this angle here, Z angle. Similarly, this angle here will be equal to this angle here, Z angle. Alright, so they are called the alternate angles and they are equal. If this is the first time you have seen us, please hit the subscription button and turn on your notification bell so that you don't miss our videos. Tune in at your convenience to check for new uploads. Always tell us how you feel about our video in the comments. Thank you very much. So going back to this, so you are going to see that if we curve our Z this way, then definitely this angle is equal to this angle and this angle is equal to this angle. So that proves that these two angles are similar. Now, why are we bothered whether they are similar or not? Notice that the areas are given. So we have 1 and 4. Now for similar triangles, the ratio of their sides is equal to the square root of their areas. The ratio of their sides equals the square root of their areas. So if the square root of this is 1 and the square root of this is 4, then their sides should be 1 is to 2. So if we have 1 is to 4 as area, then the sides should be the square root of the areas. So the square root of 1 is 1 and the square root of 4 is 2. So it means that the sides are in the ratio 1 is to 2 to each other. The corresponding sides are in the ratio 1 is to 2. So if that should be the case, I'm sorry if I'm rushing my words, if that is the case, then if we should call this side x, then definitely this is what 2x, right? Yes. And if we call this the height, if we call this height h, then the height of the bigger triangle is what? 2h. Now does it really matter? Yes, big time it matters. Now I want us to note that if this is x and this is our h, then this is our 2x and this is 2h. Now if I call this, this is ABCDEFG. Now for triangle EFG, let's find the area of triangle EFG. Now the area of triangle EFG, the area, remember the area of a triangle is half base times height. Half base times height. So it means this will be half multiplied by, now the base, the base is what? x. For this, the base is x and the height is h. Now the area is already given as 1. So we can say 1 is equal to half xh. So we can say that if we cross multiply, we will have 2 times 1, 2 is equal to xh times 1 is xh. So we are going to keep this aside and call it equation 1. So I'm going to write it somewhere that xh is equal to 2. Alright, we are close to our answer. Yes, we are close to our answer. Alright, now moving forward, I want to be sure this mic is still on. Okay, it's on. Alright, then moving forward, you need to note that this line here, okay, is equal to AB. So this implies that AB is equal to h plus 2h. So we can say that AB is equal to 3h, right? So permit me to write 3h here. Now, this is, we're almost at the end of this video. Yes, because remember the area of a rectangle, okay, is equal to length times width, right? So for this rectangle ABCD, for rectangle ABCD, the area of ABCD will be equal to the length, which is 3h, times the width, which is 2x, right? So the area will be equal to 3 times 2 is 6, 6hx, alright, 6hx. But hx is equal to 2 from here. You can see our xh is equal to 2. So you can say that the area of ABCD is equal to 6 multiplied by 2. And this will give us 12 units squared, right? So the area of ABCD is equal to 12 units squared. So we are going to leave that and then try to find the missing area. Now, the missing area can be found with what we have now. The missing area, we already told that the whole of this place is 1. The area, the whole of this place is 4, right? So that means that 1, let's call the missing area x. No, let's not call it x so that we don't confuse ourselves. So if we call the missing area, let's call it anything. So let's call it x. Note that 1 plus 3 plus 4 plus x, all of them are supposed to equal what? 12. Yes, it should be equal to 12. So if we add, we have 1 plus 3 is 4, 4 plus 4 is 8. So we have 8 plus 6 is equal to 12, okay? So we take away 8, we have x is equal to 4 units squared. So that is the area of the unknown portion of the rectangle. Thank you so much for watching and see you in my next video.