A Geometry math question| find area of the shaded portion

This is not easy, but it is very easy. It's not just tricky. Welcome back to My channel. We are asked to find the value of the area of the shaded region. I forgot my microphone. I just had to put it on. All right, now, as I was saying, we are asked to find the area of the shaded portion. Now this is a quarter circle, and this is a quarter circle as well. This is not drawn to scale, and this is a rectangle. So if this is a rectangle. Then this is a right angle, right? This is also a right angle, but these are not important. Yeah. They are not important for now. First of all, let us call this the length of the rectangle, right. So we call this a, b, c, d. Okay, so a d is equal to l. We stand for the length of the rectangle. So that means this is also equal to c b, because if this is a rectangle. then this side is equal to the other side. All right, now let us call the radius of this smaller quadrant, small r, because this is smaller, right? Let's call it r. So if we call the meeting point M. So it means that a M is R. Now if we call this point, if this point, then it means that a e is also equal to r. So a e is equal to a m, which is equal to r, which is the radius of the quadrant. Okay, now similarly, let us denote the radius of this larger semi-quadrant as capital r. So if we do that, that means if we call this point point B. Okay, so it means that BP is also equal to capital R. Now, where am I heading? As I said, this is easy but tricky. First of All we note is that d a is equal to l. All right, so it means that small r e a is equal to d a minus d e right. So e a, which is small r, is equal to d a. Remember that d a or a d is capital l. minus d e, which is 16. Okay, so that is our small r. Now, also capital r. Okay, to get capital r We are going to solve for p b, right? Now, c b, sorry, p b, which is what we are looking for, right? That is, capital R is equal to C B minus C P. Okay, now P compat R. I'm going to explain. What I'm trying to do is, PB is capital R, and this is equal to CB, which is also equal to capital L, right? minus c p, which is 14. So if we call this equation one and this equation two, I want us to note That small r plus capital R is also equal to 18 because AB is also equal to DC. Okay, because both of them are opposite sides of a rectangle, so they are equal. That means that a b, which is r plus capital r, is equal to d c, which is 18. Now let's call this Equation Three. Please hit that subscription button. sonar notification bell if this is the first time you have seen this lovely channel. Give Give us a thumbs up if you love what we're doing. Like, comment, and share—it means a lot to us. and then please tune in at your convenience. By the grace of God, yes, by His grace alone there There will always be something new for you to watch on this channel. Thank you. Very much. Now look at what we have here. Remember that r plus capital R is equal to 18. From equation three we have that r plus capital R is equal to 18. Now remember that your r is equal so we can replace a small r with l minus 16. So length minus 16 plus and our capital r where is it? Here is l minus 14. So we have l minus 14, and this is equal to 18. Now if we Combining like terms, we have l plus l, and then we have minus 16 minus 14, which equals 18. Now l plus l is 12 minus 16 minus 14 is minus 30, which is equal to 18. Now if we add 30 to both sides Add 30 to both sides; we will have 2l minus 30 plus 30 equal to 18. Oh, sorry, we didn't get it. 30 through. Yeah, minus 18 minus 14, yeah, so I'm correct. So 18 plus 30. All right, so we have That 2l because minus this plus this is zero, so we have 2l is equal to 18 plus 30, which is 48. Dividing both sides by two, we have 2l divided by two equal to 48 divided by two. So two we divide two we have l is equal to 24. All right now remember oh we have cleaned that Part. Remember that small r equals l minus 16, so r equals l minus 16. That's it. means that small r is equal to 24 minus 16. So small r is equal to 8. All right, so we're going Replace this with 8 and this with 8. All right, now similarly, capital R is equal to l minus 14. That means capital R is equal to, we have our L as 24, right? 24 minus 14. That means capital R. is equal to 10. Now why am I getting these values? Remember that we are asked to find the area of the shaded region. Now, the area of the shaded region is the same thing as area of this quarter circle. All right plus area of this quarter circle. Now area of a circle is pi r squared. All right, so but being that this is a quarter circle, the area The area of a quarter circle will be pi r squared divided by four. Now we have two quarter circles. For the first one, the area of the shaded portion will be equal to the area of the first. quadrant which is pi multiplied by small r which is eight right squared divided by four plus the area of this larger quarter circle, which is pi multiplied by the capital R, which is 10 squared divided by four. So the area of the shaded portion will be equal to pi times x squared is 64 divided by four plus pi times 10 squared is 100 divided by four. Now the area of the shaded portion is equal to when four divides this, we have one more remainder two. Dividing 24, we get six. So what we have here is 16π plus when four divides 100, we get 25. So what we have is 25. Therefore, the area of the shaded region is equal to 41π square units. Yes, thank you so much for watching, and see you in my next video.