A Nice geometry math question| find the area of the blue square.

Hi, wonderful people! We have an amazing and interesting geometry question to solve. So today we are asked to find the area of this small square if the area of this big square is 25. Now, please, this is not drawn to scale, but here this large square is inscribed inside this Big circle. Now let's go ahead and find it. First, we are asked to find the area of this smaller one. Let us call the side of this smaller square x, because once we are able to find x which happens to be the side of this square, then we square it to get our area, remember that the The area of a square is length squared. Okay, so we are told that the area of this big one is 25. so 25 is equal to length squared; now let us take the square root of both sides: the square root of 25 is the square root of length squared, so when this cancels that, we have that the length The square root of this large square is the square root of 25, which is 5. Okay, so that simply means that this is 5. And this is 5 as well, because this is a square. All right, now I want us to take this as the center. of this square, okay, note that the center of this square will coincide with the center of This circle as well, all right. So if this is the center of the square, now let us take the diagonal. of the square, let's assume this is the diagonal; this is a freehand sketch, but let's assume this is the diagonal of the square now this place is what we have a right-angled triangle this is 5 This is 5. This is not drawn to scale, and the diagonal, let's call it d, so we need to find the length of the diagonal; now remember that we can do this using the Pythagorean theorem and according to the Pythagorean theorem, 5 squared plus 5 squared will give us the diameter squared or the diagonal squared, okay. Now 5 squared is 25, plus 25 is equal to d squared. So we have 50 is equal to d. Squared. Now let's take the square root of both sides. The square root of 50 is equal to d squared square. root of d squared, so if we solve further, we can see that d is equal to the square root of 50. Now we 50 can be expressed as 25 times 2, so d is the square root of 25 multiplied by 2, so d is the square root of 25. multiplied by the square root of 2, so d is 5 root 2, which means that the entire diagonal is 5 root 2 okay? All right, now I want you to do something: if the entire diagonal is 5 root Then half of it will be 5 root 2 divided by 2, so here it will be 5 root 2 divided by 2, and remember that half of it is the radius of this circle, okay, so that gives us that the radius of this The particular circle is equal to 5 times the square root of 2 divided by 2. All right, we are going to keep that in mind. and then let us draw a straight line that will connect the center to this point, the bottom left Of this small square, okay, I'm using freehand sketch, so I may not exactly get it. Exactly, but just know that I'm trying to connect this to that place, okay? Okay, so we're done, all right. and you can see that this length from here to here is also the radius of this circle, so it means that this point is also 5 times the square root of 2 divided by 2, but we are not finished yet if this is the center From this square, let us draw a line connecting the center of this large square to the center of the small square. As usual, I'm going to use a freehand sketch. Okay, in case it doesn't join well, all right now. We are going to notice that this place is a straight line and this is 90 degrees, right, so We have a right-angled triangle. Now let me draw the right-angled triangle. If this is the first time you have seen us, please click the subscription button and turn on the notification bell so that you don't miss our amazing videos, all right? Give us a thumbs up if You're actually enjoying this video until it's convenient for you, there must be something new. For you to watch, by God's grace. Okay, thank you so much. Now I will draw this triangle out. That's the radius of the circle, as I explained, which is 5 times the square root of 2 divided by 2. then remember that this passes through the center, so this is half of x, right? So from here It should be x over 2 here. From here to here has two lengths; the first length is from here to here. Half of this is right, and that happens to be 5 over 2, so here is 5 over 2. and this remaining part is equal to x, the side of the rectangle so the entire length should be 5 over 2 plus x. Now, using the Pythagorean theorem, we can see that 5 over 2 plus x all squared plus x over 2 all squared is equal to 5 root 2 over 2 All squared, okay, so we're going to carefully expand these brackets to obtain a linear equation. Now, a plus b raised to the power of 2 is equal to a squared plus 2ab plus b squared. So if we open this bracket now, the first one is going to be 5 over 2 squared. plus 2 multiplied by 5 over 2 multiplied by x plus x squared, then here we are going to have plus x squared divided by 2 squared equals so this will give us 5 root 2 squared divided by 2 squared; now, if we expand this, it will be 5 squared, which is 25. divided by 2 squared, which is 4, plus here too we divide by 2, we are left with just 5x Okay then, plus x squared plus x squared over 4; for this is equal to 5 squared, which is 25. multiplied by root 2 squared is 2; divided by 2 squared is 4. Here, let's continue. Have you given? Give a thumbs up if you haven't already, and if you're actually enjoying this video, please do like it. Comment and share, okay, so that we can reach more people who really need it, especially students. that are preparing for various types of examinations, thank you so much. So here we have 25 over 4 plus 5x plus x squared plus x squared over 4 equals 25 times 2, which is 50 over 4 All right, now let's multiply through by 4 in order to clear the fractions. Okay, so 25 over 4. Let me write it by multiplying by 4. Okay, let me clean all of this, so 25 over 4 multiplied 4 times 25 is 100 plus 5x times 4 is 20x plus x squared times 4 is 4x squared plus x squared over 4 multiplied by 4 is x squared is equal to 50 over 4 multiplied by 4 Is 50 all right? Now here we have, let's transfer this to the other side. If we do that, we have 20x. plus 4x squared plus x squared is 5x squared equals 15; now plus 25, we move it to the other side to become minus 25, so we have 20x plus 5x squared is equal to 25. Now let us divide through by 5. If we divide divide both sides by 5, you will have 4x plus x squared equal to 5. So if you look at This is a quadratic equation, so let us rearrange it. Well, okay, so if we rearrange it. We will have x squared first, followed by plus 4x, and this will come here to become minus. Now, since the square of x is 1, we can do this easily—we need two factors of minus 5, but when you add them together you get plus 4; two factors: 5 and minus 1. Okay, all right. so we open two brackets equal to 0, x and x, because x times x is x squared, then it's supposed to be um 5 and minus 1, yeah, so x plus 5 is equal to 0 and x minus 1 is equal to 0, x is equal to If we move to the other side, we get minus 5, and x equals if minus 1 crosses over, it becomes Plus 1, then but x cannot be negative. Yes, x cannot be negative, so we have x equal to 1. therefore now we need to find the area of the square which is 1 squared, which is equal to Thank you so much for watching, bye