A Nice geometry math question| solve for the area of the small square.

Welcome back to my channel. Today we have an amazing geometry problem to solve. We are told to find the area of this smaller square if the area of this bigger square is 64. Now friends, let's Try to understand what the examiner is saying or where the examiner is coming from. Why would the examiner give us the area of its square? All right, now it shows that this is going to help us find the area of the smaller square. So we are going to start from that. That is the First clue. Since you are given the area of a square, what is the area of a square? Okay now Remember that the area of a square is equal to the length squared. Now let's let l be equal to the length. Of what? The larger square, and let a be equal to the length of the smaller square. All right, now how do we find our l? As I said, remember that area is equal to length squared. That is it for the large square. So our area is already given as 64. 64 is equal to length squared. Now take the square root of both sides. If we do that, we have the square root of 64. is equal to the square root of l squared. Okay, now remember that the square root of 64 will give us plus or minus 8 is equal to this. We remove this and we have l. But since we are talking about length, Length cannot be negative. Therefore, our l is positive 8. So we have the length of the square. The largest square is 8. That means from here to here is 8. Okay, from here to here is another 8. Right. Yeah, okay. Now I want you to note that there is a reason why we are given this semicircle. It shows that it will also help us find the area of this smaller circle. Now when you look at this semicircle, the only important thing about a circle in an equation It always lies within its radius. So let's find the radius of this circle. Okay, now if you look at if we call this the center. Okay, this is the center of the circle. Now that means that from here To here is 4, and from here to here is 4, right? This is the center of the circle from here to here. is 4 and from here to here is 4 because the whole length is 8. Now remember that any line you draw from the center of the circle to its circumference is the radius of the circle. So now let us draw this line from the center of the circle to touch this point which is also on the circumference and which is the radius of the circle. Now let us draw this right-angled triangle that we just formed. So let me clean this. I don't think it's important for now. We've obtained the length we need. So let's Draw this right-angled triangle. So if we do that, this is the radius. Okay, now this point is 4. Remember that this is the side length of the larger square, which is also 8. Now let us find the value. of our radius. Remember that using the Pythagorean theorem by the Pythagorean theorem By the Pythagorean theorem, R squared, which is the hypotenuse, is equal to d squared. plus d squared. So it's going to be equal to 4 squared plus 8 squared. So R squared is equal to 16 plus 64. R squared equals 80. Alright, so we're going to set this aside. Let's write somewhere that R squared is equal to 80. Now if you look at this diagram again you are going to notice that if we draw a line from here to touch this place, this will also be another radius because this line is coming from the center of the circle to its circumference. So it is also the radius. Now let's recall that we let b denote the length of the smaller circle. So it means that this place is a right and this place is a as well. Now let us try to draw out this particular right angle. So we have obtained R squared to be 80. So I'm going to clean this up as Well, we don't need it anymore. Okay, please give us a thumbs up if you're actually enjoying this. Watch the video and click the subscription button, and turn on the notification bell if this is your first time. You've seen this amazing channel and this amazing face. Okay, thank you so much. Now Tuning in at your convenience, there must be something for you to watch by the grace of God. Now let's look at what we have; let's draw this right angle out. Now let's draw it out. So if we draw it out, we have something like this. Okay, so from here to here will be 4 plus 8. From here to here is our a, and this point is our R. So, using the Pythagorean theorem as well. We have that R squared equals 4 plus a squared plus a squared. Okay, now remember that is equal to a squared plus 2ab plus b squared. So we are going to have R squared is equal to 4 squared plus 2 multiplied by 4 multiplied by a plus a squared plus a squared. R squared is equal to 16 plus 2 times 4, which is 8, so we have 8a plus a squared plus a squared, which is 2a squared. Now recall that R squared equals 80. So we can write 80 equals 16 plus 8a plus 2a. squared. Now let's transfer this to this side. If we do that, we have 80 minus 16. is equal to 8a plus 2a squared. So we have 80 minus 16, which gives us 64. So we have 64. is equal to 8a plus 2a squared. Now if you look at this, you can see that we can divide through by 2 to reduce or simplify the equation. Divide both sides by 2. So 64 divided by 2 32 divided by 8 is equal to 4. 2 divided by 2 is 1. Now let us rearrange the equation. Let's rearrange. If we do that, we have a squared. plus 4a minus 32 equals 0. Now let's factorize. Because the equation is a squared Yes, we can factorize it this way. A times a will give us a squared. Now we need two factors. of minus 32, which when added together give plus 4. So let's start with 2 and 16. There is nothing we can do to 2 and 16 to get 4. Now let's talk about 4 and 8. Yes. Because if you say 8 minus 4 you get 4, and 8 multiplied by minus 4 is minus 32. So what we're going to do now is take this as it is and put it here. So we have plus 8 and minus 4 equal to 0. Now remember that if you say that a b is equal to 0 this implies that either a is equal to 0 or b is equal to 0 or both are zero. Now that means that you can say a plus 8 equals 0, a minus 4 equals 0. Now if we transfer this to this We have a is equal to minus 8. If we transfer this to the other side, we have a is equal to to plus 4. But you cannot have two lengths for this. So how do we know which one is correct? and the one that isn't? The first thing is that we are looking for length, and length cannot be Negative. So this is not our length. The length is 4. Now the equation says we should find the area. Now remember that area is equal to length squared. So the area of the smaller circle Remember that a is the length of the smaller circle, so it should be a squared. So this will give us 4 squared, which is equal to 16 square units. Thank you so much for watching. See you in my next video. Bye.