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

Welcome back to my channel. Today we are asked to find the area of this circle. Now this is not drawn to scale, and this circle and this is a semicircle. Both of them are inscribed within this semicircle, and the length of the hair to this point is 18. So if we call this A, let's call this meeting point 0, and call this B. This is actually the center of this semicircle, okay So let's call this 0, so from here to here AB is 18, okay Alright, now if we call the center of this particular place C Now let us consider this larger semicircle and this smaller semicircle, okay So I'm going to draw another diagram that I'm going to use to explain something. So if this is our larger semicircle, okay, and this is the center And this is our smaller semicircle, oh pardon my drawing. Okay, now you can see that if we complete this, it forms a semicircle. And the other semicircle is inscribed within it And there is this theorem that states that when two semicircles are joined to each other Or connect to each other internally or externally, okay If you join the centers, their centers to the point of contact You're going to form a straight circle, okay So this is internal; both of them are joined internally. Now, if each were to be external, let's say they touch each other like this. If this is the center and this is the center If you join the center and the point of contact to the other center They should all be on a straight line, okay That is what this particular theorem is stating. Now let us consider that particular theorem, okay So this is our semicircle, and this is the center, okay. So another circle is inscribed here, alright And this is the center of this circle Using this particular theorem, it simply means that Remember that from here to here is also the radius of this semicircle, alright And from this center to this point, if we call this point P This is our point A, our point C already So this is our C and this is point O, alright PO is the radius of this semicircle. And remember that AB is the diameter of this semicircle, alright And AB is 18, so it simply means that The radius PO should be 18 divided by 2. Which is 9, okay Since PO is the radius of this circle, okay It's the radius. This is for those who actually find my accent difficult to understand. So because PO is also the radius of this circle It should be equal to half of the diameter of this semicircle. Which is 18 So it means that from here on out to this point Okay, will give us 9 Now PC is the radius of this semicircle. Let's call it R, okay So R is the radius of this semicircle. So it means that OC will be equal to 9 minus R. So it's going to be 9 minus R Okay, alright Having said that We are going to write this Remember that this is the point of contact I don't know if this is small. But let's try to understand what I'm trying to do This is the point of contact Between the circle and this particular diameter, okay So if we draw a straight line to this point of contact And this diameter Then a 90-degree angle is formed here Because this diameter is tangent to this circle at this point Okay So because of that That simply means that Remember that in a circle That has a tangent AB If this is the center of the circle If you draw a straight line from the center to the point of contact A 90-degree angle is formed. That's why this place is at 90 degrees. Now let's call this place P. Okay, we already have P. Let's call it D. So do Let's call DO X Okay So if DO is X Then this triangle Remember that this point is C So in triangle CDO Alright We can say that Using the Pythagorean theorem We can say that This is what we have I want to draw it so that you can see what I have there So this is 9 minus R This place Remember that from here to here is also the radius of this smaller circle. So we have a small r here And we have X here Which is our DO Okay And this point is our C So using the Pythagorean theorem Using the Pythagorean theorem, we have That 9 minus R All squared Is equal to R squared Plus x squared So if we solve again, we have Remember That A minus B All squared Is equal to A squared -2AB Plus B squared So That means that we have 9 minus R All squared will be Let's not repeat it. Let's just go ahead and solve it. Alright So 9 squared -2 × 9 × R Plus R squared Then plus R squared plus X Sorry is equal to R squared plus X squared Now we remove R squared, right? Because if this comes to this point, we have Minus R squared So we definitely remove this. Now So now we have 9 squared is 81. Minus 2 times 9 is 18 times R is 18 R Is equal to X squared So this simply means that 18R Will be equal to 81 minus X squared Okay Because if this crosses to this side Then this will come to this side We have 18R equal to 81 minus X squared. Now this is very important, so we are going to write it down somewhere. That 18R is 81 minus X squared Alright Now Let us go back again And Try to get this particular diagram So if we now consider This particular circle and this semicircle Okay So if we consider this circle and this semicircle Is this the center right? This is the semicircle. This is the They are supposed to touch somewhere. Okay And this is the circle So if we consider both of them Is this the center of this circle right? And this is the center of this particular circle Using the same theorem that I explained If we join the two centers With the point of contact We're going to form a straight line, right? So let's do that So this is what we have Okay Now remember from here to here Is 9 OB is 9 So it means that Remember that from here to here is the radius of this smaller circle. Which we call small r Okay From here to here will be the radius of this large circle. Of this semicircle, sorry What will be 9 divided by 2? Now remember that we have already explained Why is it 90 degrees at this point? Okay Now if you look at this This place is our X. Okay That is what we call it here That is between D and O Is our X And from here to here is the radius. which is also 9 over 2 So you can see that this triangle CDB What? A right-angled triangle Yes It is a right-angled triangle. So using the Pythagorean theorem Also We can say that R plus 9 over 2 R-squared Should be equal to CD squared Remember that our CD is small. So it should be R squared plus DB squared What is X plus 9 over 2 squared Now remember That A plus B squared Is equal to A squared Plus 2AB Plus B squared Okay So it simply means that this will be R-squared Plus 2 multiplied by R multiplied by 9 over 2 Plus nine over two Squared Okay Is equal to R squared Plus this will be x squared Plus 2 multiplied by 9 over 2 Multiplied by X Then plus 9 over 2 Squared So we have R squared Plus We removed this 2 right here. So we have 9R Plus 9 squared is 81 2 squared is 4 Is equal to R-squared Plus x squared Besides this, we removed this; we have 9X Besides this, we give ourselves 81 over 4 So if you look at these 2 Sides of the equation You can see that we can remove R squared and R squared And we can also remove 81 over 4 and 81 over 4 So we are now left With just 9R is equal to x squared Plus 9X So I'm going to write it here Because it is very important to us So we have 9R Is equal to X squared Plus 9X If this is the first time you have seen us Please click the subscription button Turn on the notification bell So you don't miss our videos Give us a thumbs up If you love what we are doing And please tell us There must be something new for you to watch By God's grace Thank you very much. So I'm going to clean everything now Because at this point I think I've gotten everything I need To find the value Of the area of this circle Yes, you are going to see Pretty soon Let's go ahead and get that. So these two Particular These two equations Are enough for me So first of all This is equation 1 And this is equation 2 Let's multiply equation 2 by 2. So that I will have 18R as well On the left-hand side So if I do that, I will now have 18R equals 81 minus X squared That is for 1 And for equation 2 It will now be 18R Equals 2x squared Plus 18X Perfect Now you can see That 18R is equal to this And 18R is equal to To this as well So we can say that this is equal to this. Right? So we can write that 81 Minus x squared Is equal to 2X squared Plus 18X So Now we can say 81 Minus x squared Let's move everything over to this side. -2x squared -18X Is equal to Zero We have 81 Minus x squared -2x squared We give ourselves minus 3X squared -18X Is equal to zero Alright Now we can decide. To write this first Because this is already in the form of A quadratic equation So we have minus 3X squared -18X Plus 81 Is equal to zero Now at this point We need to remove this negative sign So we need to Multiply both sides Minus 1 If we do that We will have 3x squared Plus 18X -81 Is equal to zero Now let's Divide both sides Dividing both sides By 3 So we have If we do that We will have 3x squared Divided by 3 will give us X squared 18 plus 18 divided by 3 will give us 6 And minus 81 divided by 3 Will give us minus 27 Is equal to zero So at this point I know you are like What am I actually doing? Why am I searching for all of this? Alright Now At this point I have even cleaned the equations. I actually still need them To obtain the value of R But no problem, when we get there I will pause the video and get the equation. So let's continue Now at this point We have a quadratic equation before us. And we need 1 and 27, 3 and 9 That when we add them together We get 6. 1 and 27, 3 and 9 That when we add them together we get 6 And that should be 9 And 3 Now because this is positive This will be 9 minus 3 OK Now there is something I need you to always remember When you are solving quadratic equations And you will notice that The equation of your x squared is 1. And your equation is factorable All you need to do is To open two brackets Set them equal. To be equal to 0 Now we have X squared So what we should have is X and X X times X is x squared Then plug in the roots you obtained This is plus 9 And minus 3 That's all You don't need to stress yourself too much. I'm sorry. I believe you heard the voice of my baby. So I actually had to take care of you. Alright, let's continue. Now this will be x plus 9 Is equal to 0 And x minus 3 Is equal to 0 So X equals -9 And X is equal to 3 Now we cannot take X equals Minus 9 because definitely Remember our X from our diagram This is Our Right So this point is our X. From here to here So X is actually a length that we can see. But the truth is A length cannot be a negative number So this is not taken Actually, we now have Only one value of X That X is equal to 3 Now this is where we need to get one of those Equations we found Equation 1 and Equation 2 Substitute our X to get our R Ok, I'm back Now these are the two equations we found And we got our X To be equal to 3, right So let's substitute it here To get our R Note that once we get our R We have gotten the area, remember So we have 18R Using equation 1 You can also use equation 2, but I feel this is Easier for me Okay, since I just have X squared here In fact, both of them are easier. Use any one So 18R Is 81 Negative 3 squared So 18R Is 81 minus 9? 18R Is 72 So if we divide both sides By 18 We have that R is Equal to 4 Okay, yeah, because 4 times 8 is 32 divided by 4 times 1 So our R is 4. Now we haven't gotten the radius. Four of us in the circle That means that the area Of the circle So the area of the circle Will be equal to π arrow squared right Because it is a full circle It's pi times squared So it should be equal to Pi multiplied by 4 squared equals 16. Which will give us 16π Square unit And that is the area of the circle I hope you enjoyed this video Please give me a thumbs up, like, comment, and share. Thank you for joining and See you in my next video