A Nice geometry math question| find the area of the triangle

We are asked to find the area of this triangle. Welcome back to my channel. So what we are going to do is simple, alright? Let's consider triangle A, B, D Okay, let's call this length This is A and we call this length B. Alright, or let's use X and Y. Yeah, let's use X and Y This is X and this length is Y Now from here to here E and G. Let's call the length P That is from here to here. Let's call it P Alright. Now area of a triangle is half of the base multiplied by the height Half of the base multiplied by the height. So if we should consider triangle A, B, D Alright So in triangle A B, D The area Which is 6, right? Should be equal to half multiplied by the base is Y multiplied by The height is X So it means that 6 is equal to XY divided by 2 Okay Then the next thing will be To actually cross-multiply So cross-multiplying 6 times 2 is 12. So we have 12 is equal to XY All right. Now we call this equation 1 then similarly similarly in triangle A, D, E in triangle A, D, E Let us extrapolate this length. Okay, so let's extrapolate it Okay. Now having done that remember that B, D will be equal to A. Let's call this F Let's call this place F Okay, so A, F will also be equal to Y. So this place is also Y All right, and that is actually the height of triangle A, D, E. All right So in triangle A, D, E now the area which is 2 will be equal to half multiplied by Now the base is here, D, E, which is P, right? P multiplied by The height which is A, F, which is also Y Okay So cross-multiplying cross-multiplying 2 times 2 is 4. We have 4 is equal to P, Y All right. Now since we have Y here and we have Y here Let's make Y the subject of the formula from equation 1 So from 1 We have that Y will be equal to Y will be equal to 12 over X. 12 divided by X Okay, all right, so let's substitute that into equation 2. This is our equation 2 All right, let me clean this so Let's substitute that into equation 2. I also have something important to me So our equation 2 is what? 4 is equal to P, Y. So this means that 4 is equal to P Multiplied by our Y is 12 over X. 12 over X All right So if we cross-multiply, we have 4X is equal to 12P Okay. Now, let's divide both sides I'm trying to form an equation. So let's divide both sides by 4. Dividing both sides We are going to have 4X divided by 4 is equal to 12P Divided by 4. So we have X is equal to 3P. Now, let's get a ratio of X over P All right We will have X over P Is equal to 3P over P So this implies that X over P is Equal to 3. So we're going to keep this aside. This is very important All right. Now when you look at these two triangles, remember that this is 90 and this is 90 Okay, if you look at triangle A, B, D and triangle E, D, C Okay, so I'm going to write this here that X over P is equal to 3 So we need to clean everything so that we can have enough space to work with Hope you have subscribed to this wonderful channel If not, please do that. Okay. All right Now, this is what we have. I want to come to this side so that you understand the comparison. Now, I'm comparing Triangle. Notice that triangle A, angle A, B, D So angle A, B, D is equal to angle E, D, C Angle E, D, C. Both are equal to 90 degrees Okay Similarly, you can see that both of them share this angle. They have the same angle in common. So angle A, C, B is also equal to angle E, C, D. Both are what? Unknown. We don't know But they are equal because they are common. This angle is common to both angles Now, I want us to remember that the ratio of the corresponding sides of two similar triangles is equal to the square root of their corresponding areas Okay is equal to the square root of their areas. Now the ratio of their corresponding sides If I take A, B, it's going to be A, B over E, D, right? So it simply means that A, B, D Triangle A, B, D is similar to triangle E, D, C. That is what we concluded. A, B, D is similar to triangle E, D, C So from here we can say that the ratio of their corresponding sides, which is A, B divided by E, D is equal to the area, square of the area, the square root of their area. Now area of triangle A, B, C is 6 plus 2 plus 8 which is what? 8 plus 8 divided by the area of the smaller triangle which is A. Now A, B is X E, D is P is equal to square root of 8 plus A all over A Okay, now let's solve Now X over P is 3 is equal to the square root of 8 plus A all over A Now let's square both sides So if we square both sides, we are going to have 3 squared will be equal to the square root of 8 plus A all over A all squared Okay All right. Now 3 squared is 9 is equal to, this we remove this we have 8 plus A over A Now cross multiplying cross multiplying We have 9 times A is 9A is equal to 8 plus 8 All right So if we bring this to this side, we have 9A minus A is equal to 8 So 8A is equal to 8 divided by 8 divided by 8 And we have that A is equal to 1 1 Square unit. So it means that here is 1 Therefore the area of triangle A being C Will be equal to 6 plus 2 plus 1 which is 9 units squared. Thank you for watching And see you in my next video