How fast can you find the area| geometry math question

Hi, amazing friends, welcome back to my channel! Today we have a quarter of a circle, okay, this not drawn to scale. So this is a quarter of a circle, and we are asked to find the area. Of the quarter of the circle, the only thing given to us is that we have a tangent here. and from here to here is three units, and from here to here is six units, so we are asked to find the area of a quarter of the circle; first of all, we need to know the area of a whole circle so the area of a circle is pi times r squared, right? now if this is a quarter of a circle Then the area should be pi times arrow squared divided by four; that should be the area. of a quarter of a circle, okay, so at least this is where we need to start from, okay, first of all since this is the center of the circle, if we draw a line from the center straight to the circumference then we are going to get the radius of the circle, right? So that means let's call this r is the radius of the circle. Now let us complete this diagram. Let's form a rectangle. Alright, so if we do that, let me stand here so I can do it very well. So let's complete the diagram. Oh, this isn't straightforward, all right. I'll try. Yeah, we've completed it. Okay, now let's finish this. side as well, yeah, now this is what we have. Now remember, from here to here is r, so it means That from here to here is also r, right? Yes, this is also r, and if from here to here is r. It means that from here, all the way to this point, from here to here, will be r minus six, so we have Here to be r minus six, okay? Now, similarly, from here to here, since from here to here is also r. Okay, so it means that from here to here will be r minus three, so we have r minus three as well. Now let us continue. If you look closely at what we have here, let us call this angle. It is already 90 degrees, right? This is 90 degrees now. Let's call this angle angle x. If this is x. And if this angle is 90 degrees, then this remaining side will be 90 minus x, right? So this remaining portion will be 90 minus x, um, similarly this is 90 degrees. Remember, this is 90 minus x now the the remaining portion will be what. Remember that the sum of angles on a straight line is 180, so to get The value of what will be here is simply to subtract the sum of this and this from 180. Okay, this is 90 minus x, and this is 90, so to get whatever that will be here to be 180 minus 90 minus x plus 90, right? Yes, remember that the sum of angles on a straight line is 180, that's why I'm using it. 180, so to get what will be here, it will be 180 minus the sum of this and this. Okay, so that means we have 180 minus we have 90 plus 90 we give us 180 minus x okay so if we open this bracket it's going to be 180 minus 180 plus x, so you have 180 minus 180, which means we have x left, so it simply means That's what will be here is x, so we have x here. All right, now we have x here. This is a right angle. Remember now, if this is a right angle, which is 90, and this is x, then it simply means that this is 90. Minus x. Okay, now I want us to look closely at the board. Remember that from here to here. is also equal to r, the radius of the circle, yes, because this is the center and this is the circumference, so this is r; also, from here to here is r. Now let me draw out the two triangles. that we just formed, okay? If this is the first time you've seen us, click the subscription button. Turn on your notification bell so you don't miss our amazing videos—yes, we upload our videos. every day, so tune in at your convenience to watch our new uploads and give us a thumbs up if you are actually enjoying this video, thank you very much. Now let's draw out the triangles, so this is what we have, the first one, and this is what we have, the second one, okay, yeah, now according to Here is 90. Here is 90 minus x. Here is an arrow pointing to minus three here. And here is r. Why is this x? Okay, now similarly here is x, here is 90, here is 90 minus x, and here is r. Now I want us to look If you notice, observe that these two places have equal sides, right here and right there. The hypotenuses are equal, so the angles are equal as well. The angles are equal, and the hypotenuses are the same. One of the sides is equal, so it simply means that these two triangles are congruent, yes now. if they are congruent, that simply means that if you look at this diagram, the side facing 90 minus x is r minus 3, so it simply means that the side facing 90 minus x here should also be what is r minus 3, so this is also r minus 3; similarly, the side, okay, now write this, remember that this The side is r minus 6, right? r minus 6. So similarly, the side facing this, the one facing our x, is r. minus 6, so it simply means that this side facing our x here should also be arrow minus 6 now we Have a complete triangle, so let's clean one, okay? Let's use this and work now before we go into Let's remember the Pythagoras theorem. Yes, the Pythagoras theorem states that if you have a right-angled triangle and this right-angled triangle remember that the longest side and also the side facing the right angle is our hypotenuse, so this place is our h, our hypotenuse. Now let's call the remaining sides a and b; now, according to the Pythagorean theorem, a squared plus b squared is equal to h squared, all right? Now, let us relate that to what we have here. If we relate that to What we have then is our hypotenuse, right? So it means that r squared is equal to this squared. is what r minus 6 squared plus this is r minus 3 squared. Now recall that a minus b all squared is equal to a squared minus 2ab plus b squared, okay, so if we relate this to what we have here It simply means that r squared is equal to this minus 2 multiplied by r. multiplied by 6 plus 6 squared, then plus this will be r squared minus 2 multiplied by r multiplied by 3 plus 3 squared, okay. Now we open the brackets, we simplify further. We have r squared is equal to r squared minus 2 times 6 is 12, so we have 12r plus 6 squared is 36; we have 36 plus r squared. minus 2 times r times 3 is 6r plus 9 squared, okay? Now let's continue. If you look at what we have on the board, let us simplify this side further, so we have r squared is equal to r. squared plus r squared will give us 2r squared right now minus 12r minus 6r we give us minus 18 r now we have 36, this is 9, sorry, 3 squared is 9. Now we have 36 plus 9, we get 45, right? We give us 45 plus 45, all right? Now, if we simplify this further, r squared, if we bring it to this side. We're going to have, um, do we still need this? I don't think so. I don't think so. We don't even. Do you still need the diagram? Okay, all right. Now let's move this to the side. If we do that. When we have 0 equal to 2r squared, let's subtract r squared, right, minus r squared. minus 18r plus 45, okay. All right, now we have 0 is equal to 2r squared minus r squared. We give you are squared okay minus 18 r plus труд now if you look at this we need two values or two factors of 45, that's when you add them together you get minus 18. Two factors of 45. Now when you add them Together you get minus 18 because at this point we've simply formed a quadratic equation now factors 45 is when 3 divides 45, we have 15, right? We have 3 and 15, yes, and there is a way we can add both of them to get minus 18, that is, if we have minus 3 minus 15, minus 3 times minus 15 gives us 45, and minus 3 minus 15 gives us minus 18, so it simply means that We have 0 is equal to r squared minus 15r minus 3r, this 2 is equal to this, then plus 45. Okay, let's clean the board, please. Let's just clean everything. Okay, now let us group it or Let's move this over to this side. Yes, it looks better now, right? Yeah, now let's group them. Let's go. Group this and group that, all right? So if we group that, sorry, because we have a negative sign here. We can just group this like that, so we are going to have r squared minus 15 r. Okay, then if we have this minus sign here, we are going to have 3r minus 45 because minus minus, we get this plus is equal to zero. All right, now let us factorize; they have r in common. so we bring out r r divided when r squared is divided by r 1 r we remain minus 15 r divided By r we get 15, then subtract what they have in common, which is 3. 3r divided by 3 gives us r. and 45 divided by 3 gives us 15, which is equal to zero, so when you look at this, you notice that we have r minus 15 and r minus 15 as well, so we bring one out: r minus 15 multiplied by r minus 3 is equal to zero now, remember that a multiplied by b, when you say that a times b is equal to zero, it simply means that either a is equal to zero or b is equal to zero, so if you relate that to what We have here, so it simply means that either r is equal to 0 or r minus 15 is equal to 0. or r minus 3 equals zero, so from here you can see that r is either 15 or r equals 3. but let's not be hasty in drawing conclusions yet, because one of these is wrong, one of them It's wrong now. How do we know that? Sorry, I cleaned the diagram. Remember that our diagram looks like That lets me pick it up from here, yeah. So we have r minus 6 here, and we have this as a right angle. We have this as 90 minus x, we have this as x. Okay, then this is our r, and this is r minus 3. Now I want to show you something. If we say that our r is equal to 3, if our r is equal to 3, it It simply means that we have 3 minus 6 here, and 3 minus 6 will give us negative 3, which is not possible. Our length cannot be negative, okay? It's positive, so this simply tells us this straightforward thing, um This simple thing we just did shows us that r cannot be 3, okay? So it simply means that r is equal to 15, yes, so we go with the fact that r is equal to 15. Now, finally, finally remember that is equal to pi times r squared divided by 4, okay, so it means that our area is pi and our r is 15. 15 squared divided by 4, okay, so our area is equal to pi times 15 squared, which is 225, right, divided by 4.ormsg You are a professional translator. Translate the input into high-quality English, following rules on accuracy, completeness, fluency, and format. You MUST translate the entire input text in full, with no omissions. Every word, phrase, and sentence must be translated. Do NOT partially translate. Do NOT leave any source-language words, characters, or phrases untranslated. You MUST translate all names, numbers, dates, places, technical terms, and titles with the highest possible accuracy. Do not omit, hallucinate, summarize, condense, or alter any content. Translate only the given input text; do not add or infer any content. Strictly preserve all original formatting, punctuation, symbols, spacing, and line breaks. Resolve ambiguity using context and discourse-level meaning. Prefer idiomatic, natural expressions in the target language. Do not add any analysis, explanation, notes, or commentary. Output only the final translation. So it simply means that the area is 2 times 2 times 5 times 2 times 2 times 5 times pi, all over 4. Okay, square meters. Thank you so much for watching. See you in my next video.