A Nice geometry math question| can you find the area of the circle

Welcome back to my channel. This is a very interesting multiplication. Very, very interesting. Okay. So today we are given a circle. Please, this is not drawn to scale. All right. So we are given a circle. And if this is the center of the circle, this right angle triangle, the area is 96 and the length from here to here is 20 centimeters. So the equation says we should find the area of this circle. Now, note that we need to, first of all, let's write the formula for area of a circle. Okay. So area of a circle is equal to pi R squared. Now we all know that pi is constant. So the only thing we need to find here is the radius of this circle. So everything we are going to do here is to get the radius of the circle. All right. So, but this is where I want to come in. If we can get the length of the diameter of this circle, we can divide it into two to get the radius. Let's get straight to the point. Okay. First, we need to work with the examiner. Now for the examiner to give us the area of this particular triangle, it means that that is very helpful to us. It's going to help us to get to our destination. So first of all, what is the formula for finding the area of a triangle? Now area of a triangle is half base times height. So area of the triangle is half multiplied by the base multiplied by the height. So area of the triangle is equal to, okay, we are given the area of the triangle as 96 centimeters squared. So we are going to, we are going to write that here. So we have 96 is equal to half multiplied by, now the base is here. Let's call this A, this place B, this place C. Okay. So the base is AB and AB is not given. So write AB and the height is AC and AC is not given. AC. All right. Now first, instead of us use AB and AC, let's call AB X and AC Y. All right. So that simply means that 96 is equal to half multiplied by X multiplied by Y. So this is 96 over 1. If we cross multiply, we have that XY is equal to 96 multiplied by 2. So XY is equal to 2 times 6 is 12, 2 times 9 is 18, plus 1 is 19. So we are going to write this as equation 1 and we keep it aside. Now let's look at this diagram again and see another information given to us and try to work with it. Now Y is length BC given to us Y. Now if you look at this triangle, it is a right angled triangle for a reason. And this CB is the hypotenuse of the triangle. So let's bring in the Pythagorean theorem here. Okay. So in triangle ACB, using the Pythagorean theorem, we can say that CB squared is equal to AC squared plus AB squared. Okay. Now CB is 20. So we have 20 squared is equal to AC is Y, Y squared plus AB is X, X squared. All right. So 20 squared is 400. We have that 400 is equal to X squared plus Y squared. Now let's call this equation 2. Okay. At this point, you can see that we actually stop. We don't, what should we do? Okay. Remember that if we can get the value of X and try to get the remaining parts of this diameter, then we can get the radius we are looking for. And here you can see that we have two unknowns. We have tried the first equation. We still have two unknowns. Try the second equation. We still have two unknowns. That is going to tell you that the only way to find the values of our X and Y is through simultaneous equation, right? So that means we need to form two simultaneous equations, two equations and solve them simultaneously to get the values of our X and Y, and then probably get the remaining parts of the diameter. Now, if you look at this, we have XY and we have X squared plus Y squared. Now an algebra identity comes to my mind. Now the algebra identity that states that X plus Y all squared is equal to X squared plus Y squared plus 2XY. Okay. So if you look at this, we have X squared plus Y squared already, and we have XY as 192. So let's solve this and see where we are heading to. So we have X plus Y all squared is equal to, now X squared plus Y squared is 400. 2 multiplied by XY is 192. So X plus Y all squared is equal to 400 plus, now 2 times 2 is 4. 2 times 9 is 18. 2 times 1 is 2 plus 1 is 3. X plus Y squared is equal to 400 plus 0. Now 400 plus 384 will give us 48 and 7. 784, right? Now let's take square root of both sides. If we do that, we will have X plus Y is equal to 28. Yeah, because square root of 784 is 28. Now please, hit that subscription button, turn on your notification bell, if this is the first time you have seen this amazing channel. Give us a thumbs up if you're actually enjoying what we are doing. Please don't just watch, like, comment, share, okay? Comment, let us know what you feel about what we are doing. Even if you don't like what we are doing, tell us what is it that you don't like about us so that we can improve on those things, okay? And tune in at your convenience. There must be something new for you to watch. All right, enough of the break time. So let's get back to what we are doing. All right, now we have X plus Y is equal to 28. This is one of the equations we are looking for. Let's call this equation 3. Now similarly, okay, we have X squared and Y squared, and we have XY. We can also get another X minus Y squared equals X squared plus Y squared minus 2XY, okay? So if we solve that, we have XY squared is equal to X squared plus Y squared is 400. Two brackets XY is 192. Okay, permit me to clean that. We don't need them anymore, okay? Yeah, we don't. All right, now we have that X minus Y squared is equal to 400 minus 384. So X minus Y squared is equal to 16. So if we take square root of both sides, we will have that X minus Y is equal to 4. Now, let's call this equation 4. So you can see that we can solve equation 4 and equation 3 simultaneously, okay? Now let us add equation 3 to equation 4. So 3 plus 4. So our 3 is X plus Y is equal to 28. This is our equation 3. And equation 4 is X minus Y is equal to 4. This is our equation 4. So if we add them together, we have X plus X is 2X, all right? Plus Y minus Y is 0, is equal to 28 plus 4 is 32. Now let's divide both sides by 2. Divided by 2, divided by 2. So we have that X is equal to 16. Okay, X equals 16. So let's replace AB with 16, okay? All right. Now, permit me to clean everything. I need enough space to work with, okay? All right. Now, we are going to leave this part because I will still need them. Okay, so let's from equation 3 or from equation 4, any one. So let's say from 3, we have that X plus Y is equal to 28, okay? Now our X is 16. So we have 16 plus Y is equal to 28. So Y is equal to 28 minus 16. So Y is equal to 12. So we can replace Y with 12, okay? Now at this point, what do we do from here? Okay, yes, there is a theorem that is going to save us at this point. Now, what is that theorem? The code-code power theorem. Now, what does this theorem state? The theorem states that in a circle, if you have codes, two codes, say code A, B, C, D, that intersect at point B. This code-code theorem states that Cp multiplied by Pd is equal to Ap multiplied by Pb. Yeah, that is what the code-code theorem is saying. Now, how do we apply it here to get the remaining parts of our diameter? We are just going to extrapolate our lines so that we have two codes that meet each other at a point. So this is all we are going to do. So if we increase Ac down and extrapolate Ap to this point, okay? So let's call this place B. That means they meet at point A, okay? So remember that this is a diameter. So it divides this code into two equal parts, right? So if Ac is 12, then from A to this point is also 12, okay? Sorry about that. I'm trying to draw a straight line. Alright. So I'm trying to say that from this point A to this point is also 12, okay? So let's call Pa the remaining part. Let's call it x. Alright. Now, using the code-code theorem, you can see that x multiplied by 16 is equal to 12 multiplied by 12, okay? So we have 16x is equal to 144. So to get our x, x is equal to 144 divided by 16. So if we solve further, when 4 divides this, we have 4. Divide this, we have 3, 3 remain. That 2 divides into 4, we have 6. And when 4 divides into 6, we have 9. So it means that x is equal to 9. x is equal to 9. So we can replace here with 9. Now, note that Bb is what's the diameter of this circle, alright? Now, diameter is equal to half of the radius. Diameter is what? Radius times 2, sorry. Radius multiplied by 2. So if you are looking for the radius, it means that the radius will be equal to the diameter divided by 2, okay? Alright. Now, let's get the radius. Therefore, the radius will be, the diameter is Bb, that is 9 plus 16 divided by 2. So our radius is 25 divided by 2. So our radius is equal to 12.5 units, okay? Now, the question says we should find the area. At this point, I believe you already know what the answer is, right? So the area then is pi r squared. So the area of the circle is pi multiplied by 12.5 squared. So the area is pi multiplied by 156.25, alright? And that is the answer in centimeter squared in terms of pi. Thank you so much for watching. See you in my next video.