This Geometry Problem Tricks Many Students! Can You See How To Solve It?

追加: 2026-05-28

[00:00:00]If you did pretty well in geometry, you ought to be able to solve this problem.

[00:00:05]Now, of course, you may not remember everything, but let's see if you can actually figure out the solution to this geometry problem. Okay, so what do we have here? Well, we have a circle and then of course we have a triangle inside this circle. But this part of the triangle is the diameter of the circle.

[00:00:24]In other words, it is the width of the circle. It's going through the center right here. Now this triangle, this side of the triangle is 12 and this side right here is 7. And the question is, we want to know the radius of the circle.

[00:00:40]Okay, so what is the radius of this uh circle given this information? Well, we have a multiplechoice question here. And let's take a look at our answers. So, A is 4.12.

[00:00:52]B is 4.73.

[00:00:55]C is 6.94 and D is 8.61.

[00:01:00]There's a lot of people that can solve this problem, but they're probably stuck on one part of the problem. Or they're kind of looking at it and they're saying, "All right, there's something about this problem that I it looks it appears to be something." Okay, and I'll kind of leave that, you know, really general here in a second. this appears to be. Matter of fact, I'll just tell you right now, some of you might be saying, "Hey, Mr. D2 math, man, this looks like a right triangle, but uh you know, I'm not quite sure it is." Well, if that is what you were kind of looking at and you were saying, "Boy, if this was a right triangle, I could figure this out, but I'm just not sure I can prove or establish that that uh that is a right triangle." Well, you were on you were definitely on the right track, and that is going to be the strategy here.

[00:01:44]I'm going to show you why this is a right triangle. Now, if this is a right triangle, which it is, and we're looking for the radius, which is this part right here, that distance, well, what we can do here is find the hypotenuse of this right triangle because we have the two sides out of the three sides in a right triangle. And we can use what? Well, you might be thinking the Pythagorean theorem. A square + b^2 is equal to c^2.

[00:02:10]So, this coupled with uh this being a right triangle. Well, we can figure out the radius. But why is this a right triangle? Well, I'll show you that in just one second. And of course, I'll finish up the rest of the math that uh we need to do to get the right answer.

[00:02:24]But uh for those of you that still have to take math exams and let's suppose you came across a problem like this and be like, "Hey, uh I have no idea what to do, Mr. YouTube Math Man." Uh well, listen, one thing you don't want to do is ever leave a multiplechoice question blank. Matter of fact, never leave any math question blank with very few exceptions. Uh for those of you that may have to take like the SAT or ACT exam, you may get a slight well not may, you'll definitely get a uh some sort of penalty. Usually it's like a quarter point or something. I I can't remember off the top of my head, but you'll get some sort of uh negative uh point value for wrong answers. Okay, but for the vast majority of situations, always at least take a guess. Okay, that's a longwinded way of saying, hey, you know, just take a guess. So, you know, you might be looking at this, well, this is 12, this is seven. Well, this, you know, appears to be kind of like a right triangle. And if this is a right triangle, well, this has to be the longest side. So, you know, it have to be longer than 12. But remember, we're looking for the radius. Okay. So, the radius is what? That's half the diameter. Okay. So, if I kind of doubled this, let's just call this instead of 4.1, let's just kind of think of this as four. If I double that, that's eight.

[00:03:40]So, that would mean that the diameter is eight. If the radius is four, well, that's not going to make sense because if this looks like a right triangle, this should be greater. Okay? So, in other words, these two answers would be better guesses. Okay? So, again, for those of you that are professionals in guessing, which I was way back in the good old days, I used to remember I could complete a math test in like five minutes. I was so good because I would be like, "All right, my lucky letter today is B." And I would just circle everything B. Of course, I would not pass the exam. But anyways, let's get back to the problem. And remember, the key here is to uh figure out u what this angle is. And this is a right angle, which of course is going to make solving this problem very easy. But why is it a right triangle? Well, let's go ahead and get into that right now. Okay. So, let's take a look at the problem. I'm going to give you a little bit more detail here.

[00:04:34]Let's put some points on um uh the vertices of this triangle. So this is the diameter. So anytime you see a like a circle figure and you see a dot in the center, well that indicates the center.

[00:04:46]Okay? So that is the center of the circle. And if you see a line going through from edge to edge, that's called technically a chord. So the longest chord in a circle is what we call the diameter. Okay? So if you see a line and it's not explained, that notation is indicating that yes indeed that is the diameter. And remember, we're looking for the radius. Okay? So the key here is figuring out this angle. So what is this angle? Well, there's a few observations here that we want to consider. Okay. And that is uh here. Okay. Uh the diameter is basically chopping this circle in two. So in other words, this is like a semicircle and this is like another semicircle, right? So two semicircles, we have a complete circle. So, a diameter of a circle will obviously create two semicircles. And that's going to be important. Okay? You're going to see why in just one second. Because really what we're talking about here is something called inscribed angles. Let's go back to this figure here and see why we are dealing with an inscribed angle.

[00:05:52]So, let's just forget about the diameter here for a second and just focus in on this part right here. Okay? The edges of the um triangle. So, let me just draw this out. this way. So these two sides, the 12 and the seven really could look like this. Okay? So we have like 12 here and seven here. Okay? So here's 12 and here's seven. But let's just look at this simple figure. So what we have is an inscribed angle, right? This is an inscribed angle inside of a circle. And uh we can uh figure a few things out about inscribed angles if we know a simple relationship. So let's go and take a look at that relationship. And you can see it right here. So an inscribed angle in a circle is the following. Okay. So what's the relationship? You might be saying, "Hey, Mr. YouTube Math Man, it looks like the arc formed by this triangle. The angle is 1/2 the arc arc formed by the triangle." All right. So hopefully I didn't mess that up too bad, but you can just kind of see it better than I can even say it, right? So we have an inscribed angle. Now, by the way, this angle is inscribed, meaning the uh vertex is on the edge of the circle, okay? It's kind of coming out like this and forming an arc. Well, the angle is half of the ark, okay? Or the arc is double the angle. This is absolutely a formula or relationship that you need to know in geometry, but it's pretty straightforward. Okay? Now, we're going to use this fact here to figure out this angle. So I kind of gave you a few uh clues here but let's just go ahead and formalize this uh formula right now.

[00:07:30]Okay. So here if I have uh angle inscribed angle A B C all right so this is point A this is point B and this is point C. So the measure of angle A B C which is this angle right here is equal to 1/2 the measure that little M means measure uh 1/2 the measure of arc AC. So from A to C, uh what we're talking about is an arc. Okay? So in other words, we're talking about the degree measure from here to here. Remember, a complete rotation one lap around the circle is going to be 360°.

[00:08:06]Okay, so hopefully this makes sense. But this would be the formula that you would need to know if you were taking some sort of geometry course. But it looks pretty fancy, right? I mean, you look you kind of take a look at this, but really it's means something very simple, right? just half of the ark is what the inscribed angle is going to be equal to.

[00:08:23]Okay, so now let's go back to our problem and now let's think about this, right? So we have an inscribed angle.

[00:08:30]Now you might you may not be thinking about it in in those terms and that's why you have to be very careful when you study a figure in geometry because some of this other information might kind of um you know confuse you. But here these two parts of the triangle do form a arc from here to here. Okay. Now because this is the diameter this is a semicircle. So how many degrees is it from here to here? Again we have a semicircle. And if you said hey Mr. DJ Math man I think that's 180°. Well you would be absolutely right. Okay. So a semicircle is halfway around a circle or 180°. So if we look at this, we have our inscribed angle like so. Here's the diameter, but our arc is from here to here, which is what? 180°. So if this is the ark, what is this angle? Well, it's half. Okay, we just looked at the formula. So that makes u uh this angle 1/2 of 180°, which of course is 90°, which is a right triangle. And we are very happy about that because yay, we get to use the Pythagorean theorem to solve the problem. Okay, so remember the problem is we want to determine the radius of the circle. Well, before we get uh to the radius, let's just figure out the hypotenuse of this right triangle, which of course is the di the diameter. And 1/2 of the diameter is going to be the radius. So we need to solve for x, which is the di the diameter. So we have to uh break out the Pythagorean theorem. Now remember uh a^2 + b^2 is equal to c^2. Let me just kind of show you this real quick. So here is our right triangle. The longest side of uh a right triangle is c which is called the hypotenuse. And it's always going to be opposite of that 90° angle. So this is c. So don't confuse this with a or b.

[00:10:26]these two sides where here seven and 12 uh is uh one could be seven or I'm sorry one could be a one could be b it doesn't make a difference but c is always the hypotenuse. All right so let's go ahead and just plug in the numbers. So a^2 + b^2 is going to be what? Well it's going to be uh oh let me do this consistently here 12^2 + uh 12^2 + 7^2 is equal to c ^2 but here we have the variable x. So we'll just have as x^2. All right. So here is the equation that we need to solve. So let's go ahead and do that number crunching right now. Okay. So 12^2ar is 12 * 12 which of course is 144. 7^2 7 * 7 which is 49. 144 + 49 is 193. And that's going to be equal to x^2. So now I have a lovely basic quadratic equation. So all I have to do is take uh the square root of both sides. So I'm going to take the square roo of 193. the<unk> of x^2. The<unk> of x^2 is x. So our answer x is going to be the<unk> of 193, which is approximately 13.89.

[00:11:36]So I really should have put this um as an option in the multiplechoice uh uh part of this problem because a lot of you may have selected this, right? You did all this great work, you were super excited and like, oh, here's the answer.

[00:11:50]And uh no, that's wrong. That's the diameter. Remember the question is what is the radius? Well, the radius is 1/2 of the diameter. So, we're just going to take this number, divide it by two, and we're going to get approximately 6.94.

[00:12:04]All right, so hopefully you enjoyed this little problem. And again, the key here, you know, I think for most people, uh, was determining that this, in fact, was a, uh, right angle. Okay. Now, let me give you a little bit of a tip here. If you uh let's say you are taking a math exam and you just can't uh come to the absolute conclusion that you are dealing with a right triangle. Sometimes it's not a bad strategy. Okay, if you have to do this, you know, I think it's um you know, a good thing is you're you're like, well, I got to finish this question. I only have a couple minutes to try to solve it. So, I'm just going to assume here for a second that is a right triangle and then do the math. And if you find the right answer, well, you're probably uh right. That's going to be far better than just, you know, doing some random guessing. And of course, we could talk about the other things as well in terms of like, well, if this does look like a right triangle, these two answers definitely don't make sense being that this is a diameter, right? This has to be greater than 12.

[00:13:04]You know, if this is again the we're talking about the radius, right? So, what's the radius? So, the radius has to be greater than six. Okay? Cuz 6 and 6 is 12. So, this can't be um 12, right?

[00:13:16]This can't be 12 and this can't be 12.

[00:13:18]has to be greater than 12 for another reason called the triangle inequality.

[00:13:22]All right, I have to stop myself here because I can just ramble on and on because I love teaching math that much.

[00:13:29]But hopefully you got something out of this video. And if that's the case, don't forget to like and subscribe. And with that being said, I definitely wish you all the best in your math adventures. Thank you for your time and have a great day.