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

Let's find the radius of this circle. Welcome back to my channel. First of all, let us draw a line from here to this part of the circle, a straight line. So if we should draw a straight line to this point and a perpendicular line to this line Now let's call this A, B, C, D and E. Alright, now from A to E is equal to 4 units. A to E is equal to 4 units. Remember we are asked to find the radius of the circle. So A, E is 4 units and C, D is also equal to 4 units. They are equal. This side is equal to this side. So if we should draw a straight line from the center across and call this point, this particular point, point N. It means that C, N is equal to half of 4, right? So C, N should be equal to N, D which is equal to 4 divided by 2 which is equal to 2 units. So it means that here is 2 and here is 2 as well. Now because this is the center of the circle, B, N must be equal to N, let's call this place F. So B, N must be equal to N, F. So B, N must be equal to N, F. So B, C is 2, 2 plus 2 is 4. So B, N is 4 which is equal to 4. Now because we have N, F to be equal to 4 and N, D is just equal to 2. So D, F should be equal to N, F minus N, D. So D, F will be equal to N, F is 4 minus N, D is 2. So it simply means that D, F is equal to 2. So this particular point now, D, F is also equal to 2. That is all I want to prove here. So in order to avoid having a lot of this thing here, permit me to clean this up. Okay, all right. So D, F here is also equal to 2. Now remember this is the center of this circle. If we should extrapolate A, C to complete it, okay, let's call this place G. Now you notice that A, G is what? A chord. A, G is a chord and also B, F is a chord, right? B, F is a chord. All right, now using chord-chord theorem, using chord-chord theorem, we can say that, okay, let me use the other side. So using chord-chord theorem, using chord-chord theorem, we can say that A, C times C, G, right? A, C multiplied by C, G is equal to B, C multiplied by C, F. B, C multiplied by C, F using chord-chord theorem. All right, the theorem that states that when two chords meet at a point, if this is A, B, and this is C, D, and let's say this is O. This chord-chord theorem is saying that C, O multiplied by O, D is equal to A, O multiplied by O, B. So that is exactly what I applied here. All right, click the subscription button and turn on the notification bell if this is the first time you have seen us. All right, give us a thumbs up if you love what we are doing. Always say that you're confident. There must be something new for you to watch. All right, so now applying the chord-chord theorem, A, C. Our A, C is six. So six multiplied by C, G is unknown. We don't know the value of C, G. Multiplied by C, G is equal to B, C. B, C is just two. And C, F is, from here to here is four. So C, F is four plus two, which is what? Six. So we divide both sides by six. This, we remove this, and we have that C, G is equal to two units. So here also is two. All right, okay. We're actually very close to our answer. Very, very close. Okay, now when you look at this, when you look at the diagram, let us join the center of this circle to this point, okay? If we do that, then we have formed a right angle triangle. So this is the center. Let me draw the right angle triangle that we formed. So this is A, right angle. This is E, and this is G. A, E is four. A, G is six plus two. This six plus this two. And E, G is what we don't know. All right, so let's call E, G, X. Now using the Pythagorean theorem, we know that X squared should be equal to four squared plus six plus two is eight squared, right? So X squared should be equal to 16 plus 64. X squared should be equal to 80. So that means that X is equal to square root of what? 80. That should be our X. Now what is square root of 80? If we should simplify this further, we are going to have 80. Let's do prime factorization. We have two times 40. We have two times 20. We have two times 10, and we have two times five. So using prime factorization, we take one two out of this. We take another one two out of this. So now we have two times two, which is four. Four root five, right? So it means that our X is equal to four root five. That is our X. All right, now why do we need to find our X? Why do we need to find our X? The truth remains that finding our X is very important. Remember that X is the same thing as E, G. So it means that E, G is equal to four root five. Now we are asked to find the radius of this circle. Remember that the radius, that E, G is the diameter, right? It is the diameter because it's a straight line that pass through the center of the circle. And radius is equal to diameter divided by two. So it simply means that our radius is four root five divided by two, which will give us two root five units. And that is our answer. Thank you for watching, and I wish to see you in my next video.