A Nice geometry math question| solve for the radius of the circle.

Hello, wonderful friends! You're welcome back to my channel. Yeah, so today we are asked to find the radius of this circle, please; this is not drawn to scale, so this is a circle from here to here is 13 from here to here is 18 from here to here is 6 and we are asked to find the radius where do Yeah, now look at what we have here. The first thing we need to do is how do we bring it in? Radius of this circle into the picture—how, how do we bring it in? Now, let us first of all do this. Using the Pythagorean theorem, I'm going to solve this using two methods, so grab your popcorn and get comfortable. Okay, because this is going to be a bit long, but you're going to enjoy it—trust me. Now, let's also Do this using the first method. The first method is this: I'm going to apply the Pythagorean theorem twice. To achieve our goal, the first thing is this: now we have from here to here is 13, so if we Look at this; it means that if we draw a straight line from here to the center of this circle Okay, remember that this is a chord, okay? This is a chord, and a line from the center of the circle. The straight line from the center of the circle drawn to meet a point on a chord bisects the chord. That means if this is a circle and this is the center of the circle and you have a chord, okay If you draw a straight line from the center, then this point will be equal to that point. Alright, that is the circle theorem. So, having said that, since the entire thing is 13. if we draw a line from the center to meet this chord at this point, it simply means that from Here to here will be equal to from here to here. Now, 13 divided by 2 will give us the value of what? We have here, and 13 divided by 2 gives us the value of what you have here; that means this place will Be 6.5, okay, and this place will be 6.5 as well. Having said that, let us go back to the diagram. Okay, now if I draw a straight line from this center to this place, something like this, this supposed to come from the center, okay? Now this line is our radius. All right, then from here to Here is 6.5; let me draw it out. We draw this, we draw this, and we draw this. Okay, all right. Now from here to here is 6.5; this is 90 degrees—remember that 90 degrees is formed. when this straight line touches this chord, okay, so this angle is 90 degrees. Now let us call from Here to here H, from here to here H, and this place is our radius, right? This is our radius. Now we are going to leave this; let us bring out another right-angled triangle. Now we have this, we have this. Okay, this is 90 degrees. Now this point is 18. So if we join this From here to here, we have another right angle, okay? So let's draw that like this, like this. Like this, and from here to here is our radius, right? Yes, this is a right angle, right angle now. From here to here is what we don't know. Remember, from here to here is 18. Yeah, let me Write it so that from here to here is 18, but from here to here is 6.5, and from here to here is 6.5. According to our diagram, to get the value from here to here, it should be 18 minus 6.5. Okay, and 18 minus 6.5 gives us 11.5, so that means the distance from here to here is 11.5. All right, now remember that from here to here is our h, from here to here is our h, but from here to here is 6, okay? If from here to here is h, and from here to here is 6, it means that this remaining The portion should be h minus 6, right? Yeah, so it's going to be h minus 6. So we have h minus 6 here. All right, yes, now let us remember the Pythagoras theorem, yes, now the Pythagoras theorem. states that if you have a right-angled triangle, the side facing the right angle is our hypotenuse, hypotenuse, so this is our h now the remaining parts, if we call the remaining the parts of the triangle or the remaining sides a and b, then according to Pythagoras, it states that a squared plus b squared is equal to h squared a squared plus b squared is equal to h squared Now we are going to apply this theorem to the two triangles we have formed. If you look at we can say arrow squared, okay, or let us start from this, okay, arrow squared, we have arrow squared will be equal to h minus 6 squared plus h minus 6 squared plus 11.5 squared plus 11.5 squared. Let's Call this equation one. Similarly, let's move on to this. If you look at this, we have arrow squared is also equal to 6.5 squared; let's write this h squared first since this in terms of h plus 6.5 squared, and this is equation two; if we say equation one minus Equation two: what do we get now? Arrow squared minus arrow squared is already zero, right? So we have zero equals h minus 6 squared plus 11.5 squared minus h squared. plus 6.5 squared, okay, so let's open the bracket. I think it's time to clean this up. You don't really need it anymore, right? If you're actually enjoying this video, please give us a thumbs up because That's a way of letting the algorithm know that yes, this particular video is truly interesting. Alright, so here we have zero equals h minus 6 squared plus 11.5 squared. minus h squared is minus h squared minus h squared minus plus 6.5 squared is minus 6.5 squared. Now, when we have (a minus b) squared, this is equal to a squared. minus 2ab plus b squared a squared minus 2ab plus b squared okay so that means that a minus b squared in this case is (h minus 6) squared, okay, so let's expand the bracket like that, so zero is equal to h squared minus 2 times 6 times h, which is 12h, plus 6 squared. Okay, then plus 11.5 squared minus h squared minus 6.5 squared. Now we have h squared. minus h squared, that one is there, that one has gone, okay, so we have zero is equal to minus 12 h Okay, then we have plus 6 squared plus 11.5 squared minus 6.5 squared. Okay, so let us transfer. to this side, so we add 12h to both sides of the equation, so we have 0 plus 12h is equal to minus 12 h plus 12 h plus 6 squared plus 11.5 squared minus 6.5 squared, so we have that 12 h equals minus 12h plus 12h gives us zero, so we have 6 squared plus 11.5 squared. minus 6.5 squared, all right? Let me excuse me—let me clean part of this. Okay, let me clean the front so that we can continue, okay? So this is what we have here. Let me write it out: that 12 h is equal to 6 squared plus 11.5 squared minus 6.5 squared. Now remember that a squared minus b squared is equal to a minus b multiplied by a plus b, okay, then this will be 12 h is equal to 6 squared plus, so this is 11.5 minus 6.5 multiplied by 11.5 plus 6.5, all right, so we 12 h is equal to 6 squared, which is 36, plus now 11.5 minus 6.5, which gives us 5, right? So we have 5. multiplied by 11.5 plus 6.5 gives us 18, so we have 12h equals 36 plus 5 times 8 is 40 5 times 1 is 5, plus this is 9, so it means that 12 h is equal to 0 plus 6 is 6, 9 plus 3 is 12. So to get our h, our h is 126 divided by 12, okay? Are you there? So you now have the h is equal to 12; divide this, we have 1, 1, and divide 6 is 0. point divide 60 is 5, so it means that h is equal to 10.5, okay, but we are looking for the radius of this circle, so what are we doing? What are we going to use the answer for now? First of all, oh, I've already cleaned that up if we go Back to our diagram, let me recreate what I did—remember that's using this right-angled triangle. Okay, from here to here is our radius. This is a right angle. From here to here is 6.5, and here. Is our h okay, and from the particular theorem we had, r squared is equal to 6.5 squared plus h squared, okay, so I'm going to leave the value of our h, and then I will Write this out now. Recall, recall that r squared is equal to 6.5 squared plus h squared. Okay, um. If by the end of this method I notice that the video is already too long, then I'm going to cut it. We'll stick with this first method, and we'll make a different video for the second method, okay? right now r squared should be equal to 6.5 squared plus, now our height is 10.5, okay, so it's going to be 10.5 squared as well, so r squared is equal to. Now I'm going to make use of my Okay, so let's use our calculator. So in our calculator, we have 6.5 squared. 6.5 squared is 42.25. 42.25 plus 10.5 squared. 10.5 squared is 110.25 as well. Okay, so r squared. is equal to 42.25 plus 110.25. 42.25 plus 110.25 will give us 152.5. Okay, now the take the square root of both sides; if we take the square root of both sides, okay, then we are going to have this we can see this we have r is equal to then let's take the square root of 152.5 which will give about 12.35 meters, or centimeters—whatever the unit is; okay, now let me check The video—if it's already long, then we're cutting it here, okay? Oh, this is already 15 minutes long. And we're going to continue this video, the second method, in a different video, all right. Thank you so much, and I hope to see you watch that next video. Bye.