A Nice Geometry math question|Circle

Let's solve this using the second method. Welcome back to my channel. We've already solved this using a method, okay? I posted it earlier so you can check it to see the method I used to obtain the value of our CD. Okay, this is D. Alright, now let us use another method. So, the second method. The second method. Let me clean this. I actually need all the space I can get. I can get it, okay? So, let me clean this up so I can extend my space and make the most of it. So, the second method. Now, remember that in a circle with center O, alright? And you have a tangent AB. If you draw a straight line from the center of this circle to the point of contact between this tangent and the circle, You're going to form a 90-degree angle, okay? Alright, now if you look at this diagram, you will notice that we have a tangent. This CD is tangent to the circle at this point, with center O and center A, okay? So, this is 90 degrees. Likewise, we have this here, another tangent, and a straight line from the center of the circle to its point of contact. So, this is also 90 degrees. Now, we have an angle here and another angle here, vertically opposite. This shows that the two angles are equal. So, if we call this particular point point E, you'll believe me that triangles ACE and BDE are similar, right? They are similar because they have two angles of 90 degrees each and a common angle that they share. Now, if they are similar, then remember that the corresponding sides of similar triangles are proportional. The ratio of their corresponding sides is equal. Now, look at this. ACE corresponds to BDE, right? It corresponds to BD. Then, AE corresponds to EB, right? AE, here is our A, corresponds to EB. Then also, CE corresponds to ED. Now, where am I heading with this? This simply means that if I say AC over BD, it should be equal to AE over EB, right? This should be equal to CE divided by ED. Now, we need to find the value of CD. This simply means that if we can obtain the value of CE and the value of ED. If we add the two together, we will obtain the value of CD. Okay? So, everything we are doing now is that we are trying to find the values of CE and ED. And once we have two of them, we add them together to get the value of CD. So, CD is equal to CE plus ED. So, let's leave it at that. Then, let's continue. Now, what is our AC? Our AC is 2 and our BD is 5. Now, AE, if we say EB, that is from here to here. If we call that X, if we call this X. Remember that BA, which is AB, is 10. So, it means AE should be 10 minus this X, right? 10 minus X That is for AE. So, let's clean this up to avoid confusion. Now, let's write AE as 10 minus X divided by EB, which is X. Now, let's solve this and get the value of our X. Alright? If this is the first time you are seeing this amazing channel, Click the subscription button and turn on the notification bell. so you don't miss our amazing videos. Yes. Give us a thumbs up if you're actually enjoying this video. If you are a returning viewer, a returning subscriber, Thank you so very much. Now, you can tune in at your convenience. There must be something new for you to watch. By the grace of God. Okay? Alright. Let's continue. Now, let's cross multiply. Cross multiply. So, if we do that, we have 5 multiplied by 10 minus X. is equal to 2 multiplied by X. Now, we have... Let's open the bracket. Five times ten is fifty. Five times minus X is minus 5X, which is equal to 2X. Now, let's transfer this to this side. 50 is equal to 2X plus 5X. 50 is equal to 7X. Now, let's divide both sides by 7. So, we have 50 divided by 7 is equal to 7X divided by 7. So, this will divide itself. We have X is equal to 50 over 7. Now, that is the value of our X. X is equal to 50 over 7. Now, why do we need the X? Because we need this side. This AB. This AE and this BD. We are going to see why. So, our X is now equal to 50 over 7. That means that BE is 50 over 7. So, let me explain this. So, X is equal to 50 over 7. Now, AE will be 10 minus X, right? So, let's clean the board and solve AE. So, AE equals 10 minus X. So, AE equals 10 minus 50 divided by 7. So, this is over 1. Now, AE is equal to... Let's make the LCM 7, right? When 1 divides 7, we get 7. 7 times 10 is 70. Minus... When 7 divides 7, we get 1. 1 times 50 is 50. So, what is AE? 20 over 7 Yeah. So, let's write that. AE 20 over 7. Now, why am I doing this? We are very close to our solution. Remember that our intention is to obtain CE and ED. Then we add them together to get the CD that was requested from us. Alright. Now, let's draw triangle ACD. So, if we do that, this is what we have. Let me clean this so that I have enough space. So, let's draw triangle ACD. So, this is A. This is C. And this is E, right? Now, from here to here is 2. AE is 20 over 7. This is a right angle. And we need CE, right? By the Pythagorean Theorem. Pythagorean Theorem Now, this is our hypotenuse. So, the square of the hypotenuse, (20 over 7) squared, should be equal to AC squared, which is 2 squared, plus CE squared. Now, let's bring this to this side. If we do that, we have 20 over 7 squared, minus 2 squared, equals CE squared. If we solve for that, 20 squared is 400. 7 squared is 49. Negative 4 over 1, right? Is equal to CE squared. Now, the LCM of 49 and 1 is 49. 49 divided by 49 is 1. 1 times 400 is 400. 1 divides 49, giving us 49. 196 Let me switch to my calculator so we can do this more quickly. So, 49 times 4 is 196. Is equal to CE squared. Now, 400 minus 196. We have 204 over 49 equal to CE squared. So, let's take the square root of both sides. So, the square root of 204 over 49 gives us CE, right? So, the square root of 204 is not a perfect square. So, let's try to figure out what it's going to give us. So, 204, let's simplify. If 2 divides this, we have 1, 0, and 2. When 2 divides this, we get 51. So, you can see that this is the same as when we pick 1, 2. This is the same thing as the square root. That means 204... Let me clean part of this. Okay. Alright. So, it simply means that we can write this as square root of... Now, 204 can be written as 4 times 51. 4 multiplied by 51 equals 204. Divided by the square root of 49. Is equal to CE. So, this can be written as the square root of 4. multiplied by the square root of 51 Divided by the square root of 49. Is equal to CE. So, the square root of 4 is 2. So, what we have is 2 root 51. Divided by... And the square root of 49 is 7. Okay. And this is our CE. So, now we have obtained our CE 2 times the square root of 51 divided by 7. Okay. Now, let's get our ED. Remember that once we have that, we add everything together. And our answer is out. Okay. Now, let's draw triangle EDB. If we do that, we're going to have this. This is E. This is D. This is the right angle. And this is B. So, from here to here is our X. Remember, from EB is our X. And our X is 50 over 7. So, here is 50 over 7. And BD is 5. So, ED is what we are looking for. Now, using the Pythagorean theorem as well, We are going to have... Let me write this down here. 2 times the square root of 51 over 7. Okay. Now, using the Pythagorean theorem, we will say 50 over 7 squared is equal to 5 squared plus DE squared. Now, 50 squared is 2500. Seven squared is 49. minus 25, sorry, is equal to 25 plus DE squared. Alright, let's transfer this to this side. We have 2500 divided by 49. Minus 25 is equal to DE squared. Now, let us solve this. The LCM of 49 and 1 is 49, right? So, now 49 divided by itself, we have 1. 1 times 2500 is 2500 minus, well, 1 divides 49, we have 49. Then 49 times 25. Let me use my calculator. We are going to have 1225 is equal to DE squared. Now, let's subtract 1225 plus from 25. We have 1275 divided by 49. is equal to DE squared. So, in order for us to obtain DE, that means that the square root of 25.999999999999996 is equal to our DE, okay? Now, we are going to solve this to find out what it will give us. So, we have 1275 divided by 5. Let's start with 5 first, okay? We are going to have 255. Now, let's use 5 again. We are going to have 51. So, you can see that 1275 can be written as The square root of 25 times 51, okay? So, I'll now write this as what? Square root of 25 multiplied by square root of 51 which will give us the square root of 1275 divided by the square root of 49, okay? Yeah. And this is equal to DE. Now, the square root of 25 gives us 5. The square root of 25 gives us 5. Let me clean this up. We don't need it anymore. Sorry, this is root 51. Alright. The square root of 25 will give us 5. So, we have 5 root 51. Divided by 7 equals DE. Alright. So, this is what we have. 2 times the square root of 51 divided by 7 equals DE. Now, remember that CD is CE What's this, alright? CD is equal to CE plus ED. So, it means that CD is equal to CE is $ \frac{2\sqrt{51}}{7} $ plus ED or DE is 5 times the square root of 51 divided by 7 völlig So, CD is equal to They have the same denominator, right? So, we choose one. Now, we have 2 root 51. plus 5 root 51 So, CD will be equal to This award is of a similar type. So, we just add the award. Their coefficient 2 plus 5 is 7. So, it's going to be 7 root 51, right? Divided by 7. Now, if you solve this, You can see that CD is equal to the square root of 51. If you press this on your calculator, You are going to have an approximate value. of 7.14 units That is precisely what we obtained using the other method. Okay. Thank you so much for watching. See you in my next video. Goodbye.