A Nice but tricky geometry math question| can you find the area of the shaded region?

Welcome back to our channel. Happy New Year. We have an amazing math question to solve. So in this geometry question, we are asked to find the area of a shaded portion. First of all, we are going to work with what we have. This is a right angle triangle. Area of a triangle is half base multiplied by the height. In triangle ABC, the area should be equal to half multiplied by the base. The base is 9x plus 6 multiplied by the height. The height is 6 plus 2x. That should be our area. Moving forward, we'll have the area of triangle ABC is equal to half. Now let's open these brackets. 9x multiplied by 6 should give us 54, right? So multiplied by 54x, 9x multiplied by 2x will give us 18x squared. So it should be multiplied by this plus 18x squared plus 6 plus 6. 6 times 6 is 36. 36 plus 6 times 2x is plus 12x. Moving forward, I just hope I did not make a mistake. Let me cross check first. 9x times 6 will give us 54x. 9x times 2x will give us 18x squared. 6 times 6 is 36 and 6 times 2x is 12x. So we have the area of triangle ABC to be equal to half bracket. 54x plus 12x will give us 6x. So we have 66x plus 18x squared plus 36. Therefore, area of triangle ABC will be equal to... When 2 divides 6, we have 33x. When 2 divides 18, we have 9x squared. When 2 divides 36, we have... That should be 18. So plus 18. Now this is the value of the area of triangle ABC. We are trying to get the area of the shaded portion, so just stay with me. So let me write this somewhere. The area of triangle ABC is equal to 33x plus 9x squared plus 18. All right. Now moving forward, have you subscribed to our channel yet? If not, please do that. Hit the subscription button and turn on the notification bell. Now let's find the area of this unshaded portion. So area of triangle BDE. Half multiplied by the base, and the base is 9x, multiplied by the height. The height is 2x. So we have the area of BDE to be equal to half multiplied by 9x squared. So we have 9x squared as the area of the unshaded portion of the diagram. Now let's write it somewhere. So area of BDE is equal to 9x squared. Please... We are almost at the end of this video. Yes. But we need to get the value of x to get our value for the area of the shaded portion. Now area of the shaded portion, which is AS, should be equal to area of this triangle, which is triangle ABC, area of triangle ABC minus area of triangle BDE. All right. So area of the shaded portion will be, we already have area of ABC as this. So let's copy it out. 33x plus 9x squared plus 18 minus area of BDE, which is 9x, right? Minus 9x squared. So we have plus 9x squared minus 9x squared. Both of them will cancel out. So area of the shaded portion is 33x plus 18. But we can't leave the answer this way. We need to get the value of x. And that is where there is a twist in this question. How do we get the value of x so that we can get the actual value of the area of the shaded portion? So to get x, we need to use the Pythagorean theorem. So using the Pythagorean theorem in triangle ABC. Using Pythagorean theorem in triangle ABC. In triangle ABC, the Pythagorean theorem will be AB squared plus BC squared is equal to AC squared. Okay. Now AB is 6 plus 2x. BC squared is 9x plus 6. And AC is 13x squared. All squared. Okay. All right. Now, moving forward, moving forward, we can see that, recall that A plus B squared is equal to A squared plus 2AB plus B squared. Okay. A squared plus 2AB plus B squared. So moving forward, we have that this can be written as 6 squared which is 36 plus 2AB, 2 times 6 times 2, which is 2 times 12 is 24. So we have 24x plus 2x squared, which is 4x squared plus 6 squared is 36. 9 squared, 2 times 6 is 12. 12 times 9 is 108. So we have 108 x plus 9x squared is 81x squared is equal to 13 squared is 169. Sorry, this is just x squared, right? 13x squared. Now x squared is x squared. Now, collecting like terms, we have 36 plus 36, which will give us 72. So we have 72. 24 plus 108 will give us 132, right? 132x. All right. Then 4x squared plus 81x is 85x squared is equal to 169x squared. Now, if we transfer everything to the other side of the equation, transferring everything to the other side of the equation, we will have 169x squared minus 85x squared minus 132x minus 72. Now, everything is equal to zero. Permit me to clean this side of the board. If you have not subscribed to our channel, I think this is the right time for you to do that so that you don't forget again. Give us a thumbs up, like, comment. Tell us how you feel. It's important for us, okay? We need the feedback. Now, 9 minus 5 is 4. 16 minus 8 is 8. So we have 84x squared minus 132x minus 72 is equal to zero. Now, dividing through by 12. Let's divide through by 12, which is the HCF, okay? Dividing through by 12, we have 84 divided by 12 will give us 7. 132 divided by 12 will give us 11. And 72 divided by 12 will give us 6. All right. Now, we are going to find the value of x using the almighty formula according to my students or the quadratic equation formula, okay? Now, this formula states that x should be equal to minus b plus or minus square root of b squared minus 4ac. Everything divided by 2a. All right. Now, here, a is 7. b is negative 11. c is negative 6. And we are going to substitute these values into this place, okay? So moving forward, we are going to do that. So we have minus. Minus 11 should be positive 11. Plus or minus, that is our x, should be equal to. So x is equal to 11 plus or minus square root of b squared, that is 11 squared, which is 1 to 1, right? Minus 4 multiplied by a, which is 7, multiplied by c, which is negative 6. Now, everything divided by 2a, that is 2 times 7, which is 14. This needs to fill the ink in this. All right. Thank you for staying behind. Now, continuing, we have x is equal to 11 plus or minus the square root of 121. Minus minus becomes plus. 4 times 7 is 28. 28 times 6. Let's see. 6 times 8 is 48. 6 times 3 is 12. Plus 4 is 16. So we have 168 divided by 14, okay? So moving forward, moving forward, we have x is equal to 11 plus or minus. Now, 121 plus 168 will give us 289, right? So we have square root of 289 divided by 14, okay? Therefore, x is equal to 11 plus or minus square root of 289 is 17, divided by 14. So we have x should be equal to 11 plus 17 divided by 14. And x should also be equal to 11 minus 17 divided by 14. I'm going to forfeit this because this is going to give us a negative number already. And x in this situation is a length. And since it is a length, it's not supposed to be a negative number, okay? So let's focus on this. We have that x is equal to 7 plus 1 is 8. 1 plus 1 is 2, divided by 14. So this implies that our x is equal to 2, finally. And an amazing question. So since we have x to be 2, we can now find the value of the area of shaded portion, which is very easy to find. So area of the shaded portion, which is As, will be equal to 33 multiplied by 2, which is the value of our x, plus 18. So we have 66 plus 18, right? And this will give us 6 plus 8 is 14. We carry 1. 6 plus 1 is 7 plus 1 is 8. So it's going to be 84 units squared, the area of the shaded portion. Thank you for watching and see you in my next video.