This Geometry math Challenge Will Test Your Skills – Can You Solve for R?

Let's find R which is the radius of this circle. Welcome back to my channel. So I'm going to remove this because I don't want this drawing to look somehow. So first thing I will do is to join, get a perpendicular from this point to this point. But before that, let's call this point A, this point B and this one point C. So I'm going to drop a perpendicular from here down. Okay, so this is a perpendicular 90 degrees. Now let's call this Y and this one H. Okay, so now we have triangle A, B, let's call this M, A, B, M. So let's draw that out, triangle A, B, M, this is A, this is B and this is M. This is 90 degrees, this is Y, this is 13 and this is H. Now using the Pythagorean theorem, we have that A, B squared is equal to B, M squared plus M, A squared or A, M squared. All right, therefore A, B is 13, so we have 13 squared is equal to B, M is Y, so we have Y squared plus A, M is H, so we have H squared. So 13 squared is 169, so we have 169 is equal to Y squared plus H squared. So if we make H the subject of the formula, H squared instead the subject of the formula, we have that H squared is 169 minus Y squared. So we are going to write that solver because it's very important. H squared is 169 minus Y squared. If this is the first time you are coming across this channel, please click the subscription button and turn on your notification bell so that you don't miss our videos. We upload our videos all the time, as often as we can, by grace of God. So tune in, you will see something new to watch. Give us a thumbs up. We love what you are doing. We are always in the comments, so drop your comments. I will fully attend to you and I will take your advice or read what you have to say. So we've gotten H squared to be 169 minus Y squared. Okay, so this is A, B, C. All right, so if B, C is 14, then C, M will be 14 minus Y, right? So here is 14 minus Y. So we have another right angle triangle. Our A, M is 90 degrees. C, here is 14 minus Y. Here is H and here is 15. Now using the Pythagorean theorem again, you will have that A, C squared is H squared, okay, is A, M squared plus M, C squared or C, M squared. So A, C squared is 15 squared. A, M is H and C, M is 14 minus Y, all squared. So if we solve this further, if we solve this further, we have 15 squared is 225, H squared plus. Now let's expand this. When you have A minus B squared, it's equal to A squared minus 2AB plus B squared. Therefore, if we expand this that way, we are going to have 225. We are going to have 225 is equal to H squared plus 14 squared. 14 squared is 196. Then minus 2 times 14 times then plus Y squared. All right. Now let's bring everything to one side and leave H squared. So we are going to have 225 minus 196 plus 28Y minus Y squared is equal to H squared. Now 225 minus 196, 15 minus 6, we give us 9. 11 minus 9, we give us 2. So we have 29 plus 28Y minus Y squared is equal to H squared. So now we have this equation and we also have this equation and both of them are in terms of H squared. So I will say recall from here that H squared is also equal to 169 minus Y squared, 169 minus Y squared. That means that we can equate this to be equal to this. So we can say that 29 plus 28Y minus Y squared is equal to 169 minus Y squared. Now minus Y squared minus Y squared, that one will go. So we are left with 28Y is equal to 169. Now let me transfer this 29 to the other side. It becomes minus 29. So 28Y is equal to 9 minus 9 is 0, 6 minus 2 is 4 is equal to 140. Now let's go ahead and find the value of our Y. So to get our Y, it will be 140 divided by 28. So when 4 divides 14, we have 33 remainder 2. If we divide 20, we have 5. If we divide 28, we have 7. Now 7 into 35 is 5. So this implies that Y is equal to 5. So we are going to replace this with Y equals 5. So I'm which is 9. So our Y is equal to 5. Now remember that from here, H squared is 169 minus Y squared. But H squared is 169 minus Y squared. And our Y is 5. So it means that H squared is 169 minus 5 squared. So H squared is 169 minus 25. H squared is 9 minus this is 4, 6 minus this is 4, and we bring down 1. So our H is square root of 144. Our H is 12. It should be plus or minus 12 being that this is length, we are taking only the positive answer. So our H is equal to 12. All right. Now let's also look at this. If we decide to join this, this is 9. So if we decide to join this and then this. Now at C, we have 90 degrees here. Now why did I say that? Remember that this is the center of this circle. Therefore, AN, let's call this N. Therefore, AN is the diameter of this circle. So if AN is the diameter of the circle. Now this circle theorem states that the diameter of a circle sustains an angle of 90 degrees on the circumference of the circle. The diameter of a circle sustains an angle of 90 degrees on the circumference of the circle. So that is why our C is 90 degrees. Now this place is 90 degrees and N is 90 degrees. Now this angle here will also be equal to this angle here. Let me come here. This angle here will also be equal to this angle here. Why? Now remember that when you draw a chord of a circle, S is a chord. Okay. S is a chord. Now the chord of a circle divides the circle into two sectors, right? Into two segments. The major segment and the minor segment. Now angles sustained by the chord of a circle on a segment are equal. All the angles sustained by the chord of a circle on a segment are equal. So you can see this. This is like this. This is the chord. And it sustains angle B at the circumference. This way. This angle B. And it also sustains angle N at the circumference. This angle N. So this and this are equal because both of them are in the same segment of the circle. Therefore they are equal. Now where am I driving at? Where am I driving at? This simply means that this particular side, alright, these two triangles are similar triangles. Both of them are similar. So as a result of that, since this angle is equal to this angle and this angle is equal to this angle as well, it simply means that triangle ABM is similar to triangle ACN. Alright? Both of them are similar triangles. This triangle is similar to this triangle. And that simply means that the ratio of their corresponding sides are equal. Alright. Checking to be sure that this is still on. Now if we say AM divided by AB, right? AM over AB. It should be equal to AC over AN. AC over AN. Now AM is 12. AB is 13. Is equal to AC is 15. AN is 2R. Because this is the radius, this is the radius, this is the diameter. Now if we cross multiply, we have 12 times 2R is 24R. Is equal to 13 times 12 times 15. Let's write this as 13 times 10 plus 5. So we have 24R is equal to 130 plus 45. No, sorry. 5 times 3 is 15. 5 times 1 is 65. Okay. Now 24R is equal to 195. So if we divide both sides by 24, because we are looking for R, we have that R is equal to this. We divide this, we give us 8.8 remainder. I think 3. So if we divide that, it's 1. And we continue like that. So let's assume it's going to give us 8.1 units. Okay. Thank you so much for watching and see you in my next video. Bye.