A Nice geometry math question| find the radius of the smaller circle

Welcome back to my channel, please. This diagram, just to manage it, this is a circle, okay, and this square is inscribed inside this circle, and we have another smaller circle inscribed inside this square. Now the question says that if the shaded area is 20, okay, that is the area of the shaded portion is 20 units squared, we are asked to find x, and this x happens to be the area of the smaller circle, all right? So first, like I always say, let us read the mind of the examiner. For the fact that the examiner gave us the area of the shaded portion shows that this area is going to play a vital role for us to get the value of x. So first of all, the area of the shaded portion is equal to what if we intend to get the area of the shaded portion, what is the formula that we are supposed to use, okay? Now area of the shaded portion, before I said that, let us assign the radius of the shaded portion, the radius of the bigger circle, sorry. Now if x is the radius of the smaller circle, now from here to here is going to give us the radius of the bigger circle, right? This bigger circle, its radius is R, so we say let R be equal to radius of the bigger circle, okay? So R is the radius of the bigger circle. Now for us to get the area of the shaded portion, okay, area of the shaded portion is equal to area of the big circle, right? It will be equal to area of the big circle minus area of the square, right? Okay, so for us to get the area of the shaded portion, which is what we are given, it should be area of this big circle minus the area of the square. So the first thing is to get the formulas for these two areas. Now area of a circle is equal to, okay let me now repeat this, just know that the first thing I am going to write is that of the big circle and the other one is that of the small circle. So the shaded area is equal to, now area of the big circle now is what, pi R squared. Remember that R is the radius of the bigger circle. Now minus area of the square, remember that the area of the square is length squared. Now we have not actually defined our length, okay? So the length of the circle of the square, remember that from here to here is x, that means from here to here is also x. This side is also x, right? Yes, it's also equal to the radius of the small circle. Alright, if this place is x, that means from here to here should be x and from here to here should be x as well. Here should also be x, alright? So it means that the length of this square is 2x, okay? X plus x, which is 2x. So that means that if we say length squared, but L, let me write it here. The length of this particular square is equal to 2x, okay? Alright, so we can now write this as area of the shaded portion, which is 20, is equal to pi multiplied by R squared multiplied by, then the length is 2x, right? 2x squared. So if we solve this further, we have pi R squared minus 2 squared will give us 4 and x squared will give us x squared. But notice that with this, we have two variables here. We have the radius of the bigger circle and we have the radius of the smaller circle. So we cannot get whatever we are looking for here if we can't find one of these variables. So let's go back to this diagram. Let's see what we can get as the value of our R in terms of x, so that we can substitute it here to get the value of our x. 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 amazing videos, alright? We upload our videos every day. So in your convenience, there must be something for you to watch by the spiritual grace of God, and then give us a thumbs up if you actually enjoyed this video, thank you. So, permit me to erase everything here, and then let's concentrate on this diagram. So the important thing we have formed here, the important equation we formed here, is that 20 is equal to pi R squared minus 4 x squared. So I'm going to erase everything. Okay, now back to what we are doing. Like I said, let's get R in terms of x. So if we draw out this triangle, let's call this, this is center O, and this is A, B, okay? So triangle A, O, B, is what? Right angle triangle. Alright, so let's draw it out. We have x, we have x, please this is not drawn to scale, and this is our R, okay? Now, this is 90 degrees, right? Why did I say so? Remember that this line drawn from the center to the circumference of this circle is our radius, right? The radius of the smaller circle, which is this x. And whenever a straight line that touches a circle at just one point is called a tangent. So this line here is a tangent. Let me draw out and let me explain what I'm trying to do here. If this is the center of a circle, and you have a tangent, that is a line that touches this circle externally at one point, if you draw your radius, okay, from the center to that point of contact between the tangent and the circle, we are going to form a right angle. So that is exactly why I can confidently say that this place is right angle. That is where this right angle is coming from. Okay, so let's continue. Now, by Pythagorean theorem, by Pythagorean theorem, we can see that R squared is equal to x squared plus x squared, right? R squared is equal to 2x squared. So we can get our R, or we can leave it like that, yes, since what we want to substitute into this place is R squared. So we have R squared to be equal to 2x squared. So let's now copy this out. Remember, recall that 20 is equal to pi R squared minus 4x squared. So we can write 20 is equal to pi 2x squared. That is our R squared is 2x squared minus 4x squared. Okay, let me just clear everything, clear everything. So now we have 20 is equal to pi 2 pi x squared, 2 pi x squared minus 4x squared. Now let's factorize, okay? So we have 20 is equal to, let's factorize x squared, x squared bracket, 2 pi x squared divided by x squared will give us 2 pi minus 4x squared divided by x squared will give us 4. So we have 20, okay, let's divide both sides, since we are looking for the value of x, let's divide both sides of this equation by 2 pi minus 4. So we have 20 divided by 2 pi minus 4 is equal to x squared divided by 2 pi minus 4. So x squared multiplied by 2 pi minus 4 divided by 2 pi minus 4. Alright, now this will cancel this. So we have that x squared is equal to 20 divided by 2 pi minus 4. Alright, so now we have x squared is equal to, now 20 divided by 2 pi minus 4 will give us what? Let me get my calculator, okay, I'm back. So let's use this as my calculator. Now we want to solve 2 pi minus 4, okay, because I don't want to leave my answer in terms of pi. So we have 2 pi minus 4, 2 times pi minus 4. This will give us 2.2832, approximately, 2.2832. So we have that x squared is equal to, now let's do 20 divided by what we just got, okay. Have you given us a term? So if not, please do that, 20 divided by 2.2832. We have approximately 8.7596, okay. Now we take square root, square root of both sides. So we have square root of x squared is equal to the square root of 8.7596. So this, we remove this, and we have x is equal to, let's get the approximate value of our x. So it's going to be square root of 8.7596, which is approximately 2.96 units, okay. So that is x, which is what we are asked to find. Thanks so much for watching. If you find this video interesting, don't forget to give us a thumbs up, and then see us in our next video. Bye!