A Nice geometry math question on polygons and circles| find x

Welcome back to my channel. Is this really possible? Yes, it is possible. Even though the only information given to us is that this is a regular hexagon with side length 5. A regular hexagon means that all its sides are equal. Here is 5, here is 5, here is 5, here is 5, here is 5, and here is 5. And this meets it at its center. These two sides meet at the center of this circle. As I said, please, this is not drawn to scale. And we are asked to find the length x. Please, I would like you to take your time and concentrate so that you understand what I'm doing. Don't get bored before the end of this video. First of all, before we get started, permit me to write down some things. I don't like memorizing things, or I tend to forget a lot. So I have a habit of remembering everything I'm doing in terms of my formulas and everything. So I need to write out my signs 60, 30, and 45. cos 60°, cos 30°, cos 45° So I will do this thing, make a table. This is a sign for 30 degrees, 45 degrees, and 60 degrees. And this is for the course. Now let's derive that. I now have to write 1, 2, 3, over 2, over 2, over 2. Now this is root and this is root. That's it for sign. Of course, I'll just reverse it. As I said, I don't cram much. I have a lot on my mind, so I tend to forget things quickly. So I will repeat this backwards. So, having said that, it simply means that sign 30 is half. So sign 30 is half. And sign 60, because that's all we'll need. Sign 60 is root 3 over 2. Now similarly, cos 30 is this. √3 over 2 and sign 30. And cos 60, sorry, is one half. So they switch places. Now, have I written that I can clean this so that whenever I need it, I can simply pick it straight from this place? As I said, I don't keep some memories. I don't. Alright, now look at what we have on the board. It's going to be a long ride, so let's watch it to the end. First of all, let us draw this diagonal from here to here. Alright, so let's name this A, B, C, D, E, F. Okay, so we have E, D, C. The angle we have here for angle E, D, C. Remember that if you want to find the interior angle of any regular polygon, The formula is (n minus 2) multiplied by 180 divided by n. Okay, the interior angle of any regular polygon. So this is a hexagon, which means we have n equal to 6. So this is equal to 6 minus 2 multiplied by 180 over 6. n means the sides. So we have 4 brackets, 180 divided by 6. So this will give us, 180 divided by 6 is what? 30. And 30 times 4 is 120 degrees. So it means that the internal angle is 120. So here is 120 degrees. Alright, let's clean this up. We don't need so much of this here. Now remember that the sides of this hexagon are equal. So this is 5 and this is 5. It simply means that triangle E, C, D is an isosceles triangle. And if triangle ECD is isosceles, the base angles are equal. Okay, triangle ECD is an isosceles triangle. Yes, so that means the base angle should be 30, and this one should also be 30. Alright, now can we get C, E? Let's find angle C, E. We can find the lengths of C and E using the sine rule. So the sine rule states that A over sine A equals B over sine B equals C over sine C. So let us use this here. So we want to get E, C. So if we say E, D over sine 30, right, that is 5 over sine 30. It should be equal to this point we are looking for, E, C over sine 120. Okay, so if we cross-multiply, EC is 5 times sine 120 divided by sine 30. Now for sine 120, remember that sine (180 minus theta) equals sine theta. Okay, now sin 120 equals sin (180 minus 60). This implies that sin 120° is equal to sin 60°. Okay, and what is sine 60? We're taking it from here, root 3 over 2. So we are going to replace sin 120 with sin 60. Alright, as I said, grab your popcorn because this isn't going to be too long. But it's going to be long, and you're going to enjoy it too. So this is equal to, which implies that E, C is 5 multiplied by, So sin 60 is root 3 over 2. Okay, now divided by sine 30 is half. So E, C is 5 root 3 over 2 times; if I want to multiply, I will take the inverse of half. 2 over 1 So it means that E, C is 5 root 3. So E, C is 5 root 3. Yes, very important. Now, it simply means that each diagonal of this particular hexagon is equal to 5 times the square root of 3. Remember that since this is a regular polygon, the diagonals are equal. So it means that if I connect O, E, O, then E should equal C because they are diagonal points. So it means that O, E is also equal to 5 root 3. So this is 5 root 3. Alright, now if we join O to the intersection point between this circle and this tangent line EB. This is E. Okay, so note that E, B is a tangent. So if I draw a straight line from the center of this circle to this point, let's call it point P. Okay, O, P is the radius of this circle, and a 90-degree angle is formed here. Yes, remember that in a circle with center O, a tangent AB touches the circle at exactly one point. If you draw a straight line from the center to the point of contact between the circle and the tangent, a 90-degree angle is formed. So this is where this 90-degree angle is coming from. Now, also, since we are told that this is a regular hexagon, I don't want you to be confused. Just follow me step by step. Alright, I will explain anything I am getting; I am explaining it. So you don't need to be confused. By the time we have assembled everything, we will be able to obtain E, B, which is X, the value we are asked to find. We are almost close to the solution. Stay there, don't get tired. Alright, so that means if this is a regular hexagon, and FE is 5, then OF is 5 as well, right? Yes, so O, F is the radius of this circle and O, P is also the radius of this circle. So O, P is also 5. I told you I would be explaining everything, each and every thing. So O, P is 5. Now, remember that we already know this internal angle is 120, right? So here is 120. That means it is also 30 degrees here. Okay, now if we call this angle alpha and this angle beta. Now, I want you to look at angle E. I'm going to clean this. Take your time and look at angle E. Everything here, angle E. Remember that it is also equal to the interior angle of this hexagon. Okay, so everything should equal 120, right? Now, angle E is equal to 120 degrees. Now, we have 30 plus alpha plus beta plus 30. Should everything be 120, right? Alpha plus beta plus 30. Everything should be 120. So we have 30 plus 30 is 60. So we have alpha plus beta plus 60 equals 120. Okay, now this 60, if we move it to this side, We have alpha plus beta equals 120 minus 60. So alpha plus beta equals 60. So if you want to improve in terms of alpha, Beta is 60 minus alpha. Alright, so beta is 60 minus alpha, okay? So we'll set that aside and clean this up to write it out very well. Now, if this is the first time you're seeing this amazing channel, Please click the subscription button and turn on your notification bell. So you don't miss our videos, okay? Give us a thumbs up if you're actually enjoying this video. And if you are a returning subscriber, thank you very much. Listen in at your convenience. There must be something new for you to watch, by God's grace. Okay, thank you very much. So this is what we have: beta is 60 minus alpha. Now, this is where the whole thing begins, okay? Let us consider triangle OEP. For triangle OEP, using the Pythagorean theorem, 5 root 3 squared should be equal to 5 squared plus E p squared, okay? So 5 squared is 25, root 3 squared is 3, is equal to 25 plus E times p squared. So what's coming here is going to be 25 multiplied by 3, okay? Then minus 25 equals E p squared. So if we factor out 25, 25 times 3 divided by 25 will leave 3. Minus 25 divided by 25 is 1, is equal to E times p squared. So we have 25 multiplied by 2 equal to E p squared. So if we take the square root of both sides, We have that E p will be the square root of 25 multiplied by 2, okay? So if we solve for that, we have that E_p is 5√2. 5√2 So from E to P is 5 root 2. So we have 5 root 2. Now, we are done. Yes. Trust me when I say we're done. We are close to the answer. Once we have obtained all these things, we are very close to the answer. So we're going to get our cos beta, okay? So cos β is the same as cos(60° − α), right? But remember that cos(A - B) equals cos A cos B plus sin A sin B, okay? cos A cos B + sin A sin B That simply means that cos(60° − α) equals cos 60° cos α plus sin 60° sin α, okay? So this is equal to... Now remember that cos 60 is half. Now what is cos alpha? We need to get that. We also need to find sin α. So let's pause here and get cos alpha and sin alpha. So this is our triangle. Triangle EOP. Let me draw it out for easy understanding. E P O. This is our alpha, right? Yes, this is our alpha. Then E_p is 5 root 2. This is 90 degrees, okay? Not to scale, sorry. This is 90 degrees, as you can see. This is 5. And from here to here, E O or O E is 5 root 3. So from here, let's get our cos α and sin α. So cos alpha equals adjacent... Sorry. Yes, adjacent over hypotenuse, right? Now the adjacent side is 5√2. The hypotenuse is 5√3. So it means that cos α is √2 over √3. Okay? So let us write it here. Cos alpha is root 2 over root 3. Now let us talk about sin alpha. Okay? So sin α is equal to... Sine is opposite over hypotenuse. So the opposite is 5. And the hypotenuse is 5√3. So this, we divide this. We have that sin α is equal to 1 over root 3. Okay? So write it here. Sin α is 1 over root 3. Now we are done. I keep saying that you think I'm joking, right? No, I'm not. So let us go ahead and solve what we have on the board. Alright. Now cos 60 is what? Cos 60 is half. And cos α is root 3 over root 2. Plus sin 60. Sin 60 is root 3 over 2. Multiplied by sin α is 1 over root 3. So this is equal to, that is cos beta. is equal to 1 times root 3, which is root 3 over 2 root 2. Sorry. This is cos alpha. Cos alpha is root 2 over root 3. I wrote something else. This is root 2. This is root 3. So we are going to have 1 times root 2, which is root 2. 2 times root 3 is 2 root 3. Plus root 3 times 1 is root 3. Root 2 times root 3 is sin 60, which is root 3 over 2. Sorry. I keep making mistakes. So this is root 3 over 2 root 3. Because root 3 times 1 is root 3. 2 times root 3 is 2 root 3. Alright. Now this is equal to what? The square root of 3 plus the square root of 2, divided by the shared descent denominator 2 times the square root of 3. Now that is what we get by cos beta. So we have that cos β is equal to the square root of 3 plus the square root of 2, divided by 2 times the square root of 3. Now, if we decide to find cos β without all this hassle. Okay. So I'm going to write this down somewhere. I think we're done with this. Let me write down the answer we received here. So cos β is root 3 plus root 2. I know you can't see this. Divided by 2 root 3. Okay. Alright. Now let me clean this to get the final part of our journey. The final part. Okay. Now let us consider triangle EBC. EBC Triangle EBC So if we draw it out. This is B. This is C. Now this C is 5 root 3, right? EB is the X we are looking for. This is our beta. And BC is not given. So if we consider this triangle and say cos beta. It should be equal to the adjacent side, which is 5√3. Over the hypotenuse, which is X. Okay. So remember that initially we got cos beta. The first one we saw. We obtained cos β as √3 + √2. Divided by 2 root 3. And here we also have cos beta. Equal to 5 times the square root of 3. Divided by X. So you can see that both of them are equal to the same value. So we can say that this is equal to that. So we are going to create that. And use it to obtain our X. Have you handed out your B terms yet? If not, please do so. Thank you very much. So please, you need to do that because I've put a lot of effort into it. And a great deal of research compiled just for you. Okay. Alright. Now we have √3 plus √2. Divided by 2 root 3. Is equal to 5 times the square root of 3. Divided by X. So let me clean all this up. Now we can say that X is equal to 2 root 3. Multiplied by 5 times the square root of 3. Divided by what? √3 + √2 So X will be equal to 2 times 3, which is 6. Root 3 times root 3 is 3. Divided by root 3 plus root 2. So this is equal to 30 divided by the square root of 3 plus the square root of 2. Now we can proceed to rationalize. Or we can simply enter this into the calculator and see what we get. So let me use my calculator and do that. Okay. I went to get my calculator. Now 10 times 3 is 30. Divided by the square root of 3, we get 1.732. Plus root 2 is 1.414. I told you I don't keep track of things. √2 is 1.414. Okay. So this is equal to 30 divided by. 1.414 plus 1.732. So we have 3.146. Now, if we solve this. 30 divided by 3.146 is approximately 9.535. We are going to have 9.54 as the value of our X. Thank you so much for watching. See you in my next video.