I don't even know how to solve this

追加: 2026-05-10

[00:00:02]Okay, everyone. So, this is probably one of the sickest problem-solving questions I've ever seen come up in a GCSE UK maths paper. All right? And of course, Malik has sent it to me. Thank you so much, Malik.

[00:00:15]And we've got both Malik and Kwame here on the mic, and they're going to help me solve this question. Okay? And it's worth four marks.

[00:00:22]Now, I love questions that involve geometry and problem-solving because at the end of it, the answer is so satisfying. And once you see the answer, you're like, "Oh my god, that's so simple." But of course, to get there, you need some mad brain skills. And of course, that's why we've got Kwame and Malik here on the mic. Welcome, boys.

[00:00:42]How are you doing?

[00:00:43]Good, thanks, sir. I'm good, sir. Good stuff, guys. Good stuff.

[00:00:47]Now, I'll read the question. I'll have a go, and then you boys just interrupt me at any point. And when you do interrupt me, make sure you clarify clearly what you're saying, and you know, you have to understand people are listening to you, and they want to understand what you're saying. So, you have to be clear with your explanations.

[00:01:04]Okay, boys?

[00:01:05]Yes, sir. Yes, sir. All right. Now, question 24.

[00:01:09]The diagram shows a circle and two semicircles inside the square.

[00:01:14]Okay?

[00:01:15]The circle touches the square and both the semicircles as shown in the diagram.

[00:01:21]So, both the semicircles and the circle is touching the square. Okay, cool.

[00:01:27]Of course, they're touching the square.

[00:01:28]They're in the in the middle of the square. Well, they're inside the square.

[00:01:33]Now, and I think it means the the perimeter of the square, right? Yes, sir. Yeah.

[00:01:37]Now, the length of each side of the square is 12 cm. Okay, cool. Let's write this down.

[00:01:43]You know, you know, I love these questions cuz it looks so simple.

[00:01:47]And it is, but it requires brain power.

[00:01:52]So, there we go.

[00:01:54]We've got that side there. And cuz it's a square, every side is 12, isn't it?

[00:01:58]So, this blue is just going to represent 12 cm. I'm not going to write it down.

[00:02:03]Uh but you guys understand that the blue represents 12 cm on each side, right?

[00:02:08]Yes, sir. Okay, cool.

[00:02:11]And then they tell us, right?

[00:02:14]Sorry, I I want to do that properly.

[00:02:17]And then they tell us And and and students like even Dilara can understand this, to be honest.

[00:02:26]Cuz it requires basic geometry but a lot of problem-solving.

[00:02:30]Okay?

[00:02:31]Now, the length of each side of the square is 12 cm and the radius of each semicircle is 3 cm. Okay, cool.

[00:02:38]All right?

[00:02:41]No, I only mentioned Dilara cuz she's a new student and um you know Do you know what I mean?

[00:02:47]Uh I'm I I want to formally retract my statement and say even students like Dilara can understand this cuz they're amazing and they're super smart.

[00:02:59]Okay? Um I do apologize. Dilara's a new student and she's just started studying the GCSE curriculum, so that's what I meant.

[00:03:06]Okay?

[00:03:09]Anyway, let's continue. Uh boys, are you with me?

[00:03:12]Yes, sir. So, if we if we go back to this question, I'm just going to draw the the uh radiuses in, right? The radii in.

[00:03:20]So, if if we look here uh from here that's roughly the center, right?

[00:03:27]Yes, sir. To here is 3 cm, correct? Yes, sir.

[00:03:31]>> And it's the same on this side. So, if I just get the center here and draw it here, that's also three.

[00:03:42]Yes, sir. And therefore, that means that the gap must be six, right?

[00:03:48]Yes, sir. Yes, sir.

[00:03:50]Uh you boys agree with me, yeah? Yes, The gap must be six. Okay, cool. I don't know how that's going to help us, but I've just I've just got it down, okay?

[00:04:05]Oh.

[00:04:06]>> [sighs] >> Oh, okay. Okay, so um Kwame before we started this recording, I know you were talking about a triangle, right?

[00:04:18]Yes, sir.

[00:04:19]And if I start forming a triangle um is this the triangle you're talking about?

[00:04:26]Yes, sir.

[00:04:28]And and therefore, you know this part of the triangle um cuz it's a radius of that smaller circle, that must be three.

[00:04:36]Correct, sir.

[00:04:37]And so we just have to Okay. Okay. Okay.

[00:04:41]All right.

[00:04:42]I'm not sure where I'm going with this, but I'm just By the way, I was not thinking of a triangle until Kwame mentioned it off off recording off the off the recording.

[00:04:53]Um let me just put these down in black uh just so we're all clear.

[00:04:59]Uh 1 second, everybody. I just want to highlight You know, cuz this is a radius A radius to the circumference A radius is a line going from the center to the circumference of a circle, right?

[00:05:09]Yes, sir. So that's three as well, and that's three. So I'll tell you my general understanding of this question.

[00:05:15]We know that the area of the square is 12 * 12, right? Yes, sir.

[00:05:21]Which is 24 cm squared.

[00:05:24]And all now all we need to do is find out the areas of three circles, two semicircles, one full circle. And then we just have to minus it from 24. That's the plan, right?

[00:05:36]Yes, sir.

[00:05:36]Um wait, no, sir. 12 * 12 is not 24, it's 144.

[00:05:43]This is what I mean, man. This is what This is on the recording as well.

[00:05:51]I can't even stop the [clears throat] recording. I can't even pause the video.

[00:05:54]It's not working.

[00:05:55]>> [laughter] >> It's not I'm trying to pause the video.

[00:05:59]It's not working.

[00:06:01]Okay, there we go. What was that delay, man?

[00:06:05]Oh, man.

[00:06:09]Uh thank you, Kwami. I do apologize. 12 * 12 is not 24.

[00:06:13]Uh it's 148.

[00:06:15]And that's the one that everyone knows.

[00:06:18]>> Sorry? Oh, so 100 Okay, the length of each side of the square is 12 cm and the radius of each semicircle is 3 cm. I'm going to start off with the the area of the semicircles, okay, boys?

[00:06:38]So, it'll be pi r squared. So, I'm allowed the calculator. We're allowed the calculator.

[00:06:44]So, it's pi r squared.

[00:06:48]So, it's pi * 3 squared, which is 9. So, it's 9 pi. Can I just keep it in pi for now?

[00:06:55]Yes, sir.

[00:06:56]I would divide it by two because it's a semicircle, but we've got two of them.

[00:07:00]So, can I just leave it as 9 pi for both the circles? Yes, sir.

[00:07:03]Okay, cool.

[00:07:05]Uh 9 pi.

[00:07:07]Uh you know, I'm really uh I'm sh- I'm ashamed of myself here because 12 * 12 is like the first thing the first square that you learn that you're really excited about. Like, you know, in school like everybody goes around when we were kids. What's 12 * 12? What's 12 * 12?

[00:07:20]So, I'm I'm sorry. I knew it was 144. It's just So, it was just a mistake. So, 9 pi, okay?

[00:07:29]Is the area of the the two semicircles.

[00:07:33]Now, we need to find out the radius of this larger circle, which is that gap there, isn't it?

[00:07:41]Right?

[00:07:43]Yes, sir.

[00:07:44]Which is that gap there. We need to find out cuz as soon as we know that length, we can the question is easy from there, isn't it?

[00:07:53]But I I I I don't know what to do right now, so I'm going to let you boys take it away.

[00:08:00]Okay.

[00:08:02]Well, what we could do, I guess, is we could start using it as we could just name the radius of the larger circle X.

[00:08:11]Okay, cool.

[00:08:14]And then we could start like working out the area. Say, no, not area. Um we could work out the try and work out like a length of the triangle. So.

[00:08:26]The length of the triangle, the whole length?

[00:08:29]Oh, it'll be 3 + X.

[00:08:31]Yes, sir. Is that what you mean?

[00:08:33]Yes, sir. Is that a What kind of triangle is that, by the way? We know that the base is six and the two lengths are 3 + X. It's isosceles, sir.

[00:08:42]Is it a 100% isosceles triangle? Yes, sir.

[00:08:47]Uh could you guys quickly geometrically um explain yourselves? How's that isosceles?

[00:08:53]Well, it's because it's from the center of the two semi-circles.

[00:08:57]If you're going up >> of the Uh yes.

[00:09:01]>> And and both of it meets at the same point. Yes, sir.

[00:09:05]Uh what theorem is that?

[00:09:07]Uh So, if if the both um uh both lines are meeting at the same point and they come off the semi-circles, right?

[00:09:19]Yes, sir. I just want you to just geometrically um uh just explain yourselves. Does that make sense?

[00:09:28]How is that a isosceles triangle?

[00:09:30]Well, we can name the other side of the radius um of the [snorts] large circle X as well.

[00:09:37]>> Yes.

[00:09:37]>> yes, we can. Well, very well done. We can call that X as well. Very well done.

[00:09:41]So, we know we know 100% um that these two lengths are the same, right?

[00:09:47]Yes, sir.

[00:09:48]We know that 100%. I totally agree with you. Because the other radius will also be X, wouldn't it?

[00:09:53]And we know that the radia the radii of the semi-circles are three each. So, we know that part of the triangle is definitely the same length.

[00:10:03]Yes, sir.

[00:10:04]So, therefore, does it mean that the base is also the same length?

[00:10:07]Yes, sir.

[00:10:09]Is that >> Ah, wait. No. No, that's not where it is. That's what I That's That's what I'm looking for. Do you get it? Yes, since there's two sides equal, we can say for now that it's a um isosceles triangle.

[00:10:22]Yes, of course. Of course, that's because that's what makes an isosceles triangle, right? Yes, sir. Right? I'm sorry, guys. You know, the entire time I was thinking of an equilateral triangle for some reason.

[00:10:32]An isosceles triangle is both sides are equal, right?

[00:10:35]Yeah, sorry, guys. It's an iso- Sorry.

[00:10:37]Sorry, go ahead, Malik.

[00:10:38]Yeah, I was saying yes, sir.

[00:10:40]Yeah. So, so yeah, it's an isosceles triangle. There we go. Yeah. There we go. That That we've proved why it's an isosceles triangle. I promise, guys, I was thinking of an equilateral triangle for some reason. Okay. So, cool. So, therefore, that um we know that the the lengths here are 3 + X, 3 + X, and the base is 6. What are we going to do with that information?

[00:11:01]Um Well, look. I think what we could do, right? Cuz essentially, we just want X, don't we?

[00:11:13]Yes, sir. Yes, sir.

[00:11:14]>> Right?

[00:11:17]Could we uh draw a line in the middle?

[00:11:20]Yes, sir.

[00:11:21]Yes, sir.

[00:11:21]>> Just to get a right-angle triangle. Or two right-angle triangles, yes, sir.

[00:11:25]Or two right-angle triangles.

[00:11:27]Yes, sir.

[00:11:28]So, I'm just going to draw this. I don't know what I'm doing. I'm just drawing it cuz I know I have two right angle triangles.

[00:11:33]Yeah.

[00:11:34]And therefore, this will be free and that will be free, right? Yes, sir. Yes, sir.

[00:11:40]Can we not do some sort of um uh socatoa or something? Yeah, well, we could work out the perpendicular height.

[00:11:49]Oh, of course, we could use Pythagoras, can't we? Yes, sir. Pythagoras, yeah.

[00:11:54]But then you don't only get the perpendicular height in terms of that.

[00:11:57]>> In terms of X, which I don't think is a bad thing right now.

[00:12:00]Yeah.

[00:12:02]But I could be wrong.

[00:12:03]Yeah.

[00:12:06]What?

[00:12:09]Okay, everybody, both my whole class and Kwame and Malik, uh we have been proper thinking about this and we haven't been able to get it.

[00:12:16]Uh this is the closest that we've that we've come.

[00:12:19]And of course, we've got the model answers with us, but we'd rather not look at them and we're both We're all of us are super tired. My brain's gone to mush and I can't think anymore.

[00:12:29]So, I'd rather just upload this video and then later, we'll come back to this fresh with fresh brains and we'll have another go.

[00:12:36]Do you got Do you boys agree?

[00:12:38]Yes, sir.

[00:12:39]Are you sure?

[00:12:40]Yes, sir.

[00:12:42]Cuz I feel like it's it's come to a point where I can't think anymore.

[00:12:47]Do you know what I mean?

[00:12:49]Yes, sir.

[00:12:50]Uh and by the way, we just want to give a big uh clap to Layla.

[00:12:54]Uh she was the one that told us to use coordinates from the very beginning, actually.

[00:13:00]Uh so, thank you very much, Layla.

[00:13:02]Um All right, then. Thank you very much, everybody. I hope this I hope this video has been of benefit. I'm just going to end it here.

[00:13:08]Thanks, guys. Bye.