A Nice geometry math question|find the radius of the circle.

Welcome back to my channel, my amazing friends. So this is a continuation of a video we have. I've solved this before using a different method, and I promise to solve it using another method, okay. Now this method right now, um, is going to be, um, we are going to make use of so many theorems. Now, first of all, this is the center of the circle, and this is a chord. If I go from here to Here is 13, even though this is not drawn to scale. Okay, this is not drawn to scale. If I draw a a straight line from this center to meet this chord divides the chord into two equal parts and A right angle is formed; this is what I mean in a circle, okay? If this is the center of a circle. You have a chord, that's correct, if you draw a straight line from the center of the circle to the chord. it forms a 90-degree angle with the chord and divides the chord into two equal parts, so if this is a The meeting point is O, and this is B. According to this theorem, AO, that is, from here to here. is equal to op from here to here, okay? So if we apply this theorem here, we are going to see that if we draw a straight line, this is 13; let me write it here. If we draw a straight line Line from the center of the circle to this point—okay, not straight, okay? Let me try. again, if we draw a straight line from here to the center of the circle, all right, now this straight The line is not a straight line; this straight line divides this circle, this chord, into two equal parts. so it means that this side is 13 over 2, this side is also 13 over 2, okay, that is the The first theorem that we want to make use of now: let us reason, if from here to here is 18 and from here to here is 13; if we should drop something like this, it means that from here to here should be 18 minus 13, which will give us 5, right? Yes, so it means that from here to here is 5, okay. Now remember, if we extrapolate this, and call this h(x), can we obtain The value of x. Yes, we can get the value of x. Remember that this is also a chord—the chord. we have just formed it, and if we draw a straight line from the center to this point, it should divide Now we have from here to here is 13 over 2, right? Plus 5, so it. means that 13 over 2 plus 5 should be equal to now from here to here is 13 over 2 plus x, right so we have 13 over 2 plus x; this should be equal, using the angle bisector theorem, the chord bisector theorem Alright, now let us move this plus 13 over 2 to the other side of the equation; it becomes minus 13 over 2, so we have 13 over 2 plus 5 minus 13 over 2 equals x, so 13 over 2 minus 13 over 2 equals x Over 2 we give us 0, so we now have 5 equals x, so it means that this side will also be equal to So we have here to be 5 as well, all right? So we have 5, 13 over 2, 13 over 2, and 5. Okay, so where Am I driving at the question? I'll say that we should find the radius of this circle, okay now. If I should draw a line from here to here, if I should draw a line from here to here, let me clean it up. We already know that here is 13 over 2 and here is 13 over 2. Okay, if I draw a line from here to Here, then, I have two chords like this: I have this chord drawn from here to here, and another from Here to here right now, using this chord, now we can see that from here to here is 5, right? Yes, exactly. and from here to here is 18, from here to here is 18 according to the diagram, okay, because from Here is 39, here is 5, so everything is 18. All right, now here we have 6, and let us find the Value of this side, okay, so now I want to explain the chord power theorem according to the chord. Power theorem: it states that when two chords intersect each other, then the products of the lengths of the segments of each chord are equal. Now, if we have chord A as a b and the second chord as c d, okay, and they meet at this point zero. According to the chord-chord power theorem, it states that AO multiplied by OB. is equal to OC multiplied by OD, all right? Yes. Now, if we apply that here, remember that from here Here is a chord from here to here; this is another chord, which is what I have drawn, and they meet. each other at this location, so if we apply the chord chord theorem, that means that 6 times 4 times x times 6 times x should be equal to 5 times 18 times 5 times 18, okay? So we have 6x is equal to 90, do you understand? So, in order to get our x, we have to divide both sides of this equation. Divided by 6, divided by 6, we have that x equals 6. We divide this to give us 1, 1. Remember, 3 divides 30 to give us 10, so it means that x is what 15, so this side is 15, all right. Alright, so we don't need this anymore, and if we now decide to join from here to this point From here to here, we already know that this side is 6 plus 15, and we already know that here is 13. and from here to here is diameter 2r, right? This place is 2r. Now let us draw the right angle that If this is the first time you're encountering this lovely channel, please give us a thumbs up. Subscribe to the channel by clicking the notification bell so you don't miss our amazing videos. Alright, then, tune in at your convenience—there must be something new for you to watch. Thank you very much. Okay, now look at what we have. Let me draw the right angle formed. and this is 15, oh, this is 15 plus 6, which will give us 21, so here is 21, here is 2r, and here is 13 from here to here, okay. Now remember, we are going to solve this using the Pythagoras theorem. Now, according to Pythagoras, Pythagoras states that in a right-angled triangle, the side facing the The right angle is where our radius is the hypotenuse, okay, and we have the other sides as a and b. Alright, now Pythagoras says that h squared is equal to a squared plus b squared. h squared equals a squared plus b squared, okay, so that means we can write this As a theorem, we have 2r squared equals 13 squared plus 21 squared. Okay, so this is 4r squared. is equal to 169 plus 21 squared is 21 times 21 1 times 1 is 1 1 times 2 is 2 2 times 1 is 2 2 times 2 is 4 2 is 4, so 4, 4, 1. Okay, so we have 4, 4, 1, 4, 4, 1. All right, now 4r squared is equal to 169 plus 4, 4, 1. 1 plus 9 is 10, 4 plus 6 is 10 plus 1 is 11, 4 plus 1 is 5 plus 1 is 6, so let us divide both sides of divide this equation by 4; if we do that, we have 4r squared divided by 4 and 16 divided by 4, so if We do that, we will have 4; we divide it by itself, and we have r squared equal to 4; we divide 6 to Divide 21 to give us 5, 5 remainder 1. Divide 10 to give us 2, 2 remainder 2. Right then, add 0. in front of the last one to give us 5, so r squared is equal to 152.5, so to get the value of r which What we are asked to find is r. The square root of r squared is equal to the square root of 152.5. Okay, now you can see this, and we have r equal to the square root of 152.5. Okay, this gives us the square root of 152.5 gives us 12.35, okay, approximately 12.35, just like we got Using the other methods, okay, thanks so much for watching, see you in my next video, bye.