A Nice geometry math question| find the length AB

This is a very interesting one, yes. We are not giving anything; we are only giving the radius. of this small circle, and we are asked to find ab. Now this is not drawn to scale, and these two circles are equal, so if they are equal, that means their radii are equal as well, and this is The center of this large circle, all right. Now, how do we solve this? Welcome back to our channel. Alright, let's do this. Okay, first, let us call this the radius of this circle x. Okay, so if we call it x, you'll see why I need to find the value of x in order to get my a and b. All right, now remember, if I decide to join from this center to this center, so let me draw a Straight line, so I'll draw a straight line. Please, wow, I tried. So if that is a straight line Note that from here to here is also equal to x, remember that this is not drawn to scale. so this is also equal to x, which is the radius of this circle as well, so that is why it's Now from here to here is the radius of this smaller circle, which is 10, so we have 10 here. Now from here to here, from here to here is also x, right? We have x now. If I draw a tangent... from the center of this circle to pass through this particular center as well, right? So let me Alright, now remember the circle theorem: in a circle with center at zero and a tangent to the circle a b okay if you draw a straight line from the center of the circle to the point of contact A right angle of 90 degrees is formed between the circle and the tangent, which is exactly what we have here. Okay, because this is a tangent line and it touches the circle at this point of contact. The radius at the point of contact meets here, so that means we have 90 degrees here, okay now Remember that this is the center of the circle, so it means that the radius of this yellow circle is the diameter of this smaller circle, which is 2x, so the radius from here to here Okay, so 2x is the radius of the yellow circle. Now we are told that from here to here is 10 and from here to here is 2x, so it means that from here now, from here to this point will be 2x minus 10, okay, so here is 2x minus 10. Now permit me to draw the right angle that we just Found if this is the first time you have seen us, please click the subscription button and turn on your notification bell so that you don't miss our amazing videos, all right? We upload our videos As much as we can by the grace of God, so tune in—you have to see something new to watch. depending on how frequently you log in and give us a thumbs up if you're actually enjoying this video thank you very much. If you're my returning subscriber, just know that I seriously appreciate it. Thank you for your support. Now, this is what we have: so this is x, from here to here is x. From here to here is x plus 10, and from here to here is 2x minus 10. Now this is a right angle, so you can see that using the Pythagorean theorem. Using the Pythagorean theorem, we can see that (x + 10) squared is equal to x squared. plus 2x minus 10 squared; now we are no longer allowed to claim that we don't need it anymore, so a plus b all squared is equal to a squared plus 2ab plus b squared. a minus b all squared. is equal to a squared minus 2ab plus b squared, so due to space constraints, I'm going to simplify that. then let us solve; so this is going to be x squared plus 2 multiplied by x multiplied by 10 plus 10 squared, which is equal to x squared plus 2x squared, right, multiplied by 2 2x multiplied by 10, then plus 10 squared. Okay, this is going to be plus 2x squared. minus sorry a squared minus 2ab, so it's going to be minus 2 times bracket 2x multiplied by 10 plus 10 squared, okay, so we have x squared plus 20x plus 100 is equal to x squared plus 4x squared minus 2 times 2 times 10 is 40x plus 10 squared is 100 now let us cancel what we Have the same, okay, let's cancel out. So when you look at this, x squared is here, x squared is here. Um, what else is 100? Is 100 there? So we have 20x equals 4x squared minus 40x. Okay, now let's move this to the other side: we have 20x plus 40x equals 4x squared. So 20x plus 40x gives us 60x, which is equal to 4x squared. Now let's divide both sides By 4, if we do that, we'll have so this shows we are going to get two values for our x. Let's see. the one that is right; so if we divide both sides by 4, we have 60x divided by 4 is equal to 4x squared divided by 4; now when 4 divides this, we have 15, and when 4 divides itself, we have 1 So we have 15x equal to x squared. Now let's move this to the other side. or let's move this to the other side so we have 15x minus x squared equals zero Let's factorize: x times 15x divided by x is 15, x squared divided by x is x, is equal to zero. so x is equal to zero or 15 minus x is equal to zero, but one thing is for sure Our x cannot be equal to zero because it already has a value, which we can see. so we are left with this one, okay? 15 minus x is equal to zero. So if we move minus x to this side, We have that x equals 15. Now the question states that we should find AB, the length of this line. Remember that this circle is equal to this circle, so their radii are equal as well. So it means that from here to here is x, from here to here is x, from here to here✍️ is another x. And from here to here is another x, so it simply means that AB is equal to 4 multiplied by x. Alright, so a times b is equal to 4 multiplied by 15, which is equal to 60 units. Thank you so much for watching