A Nice geometry math question| find the area.

I was amazed when I saw this question, and I decided to share it with you, my lovely. Subscribers, welcome back to my channel. Today we are asked to find the area of this portion. Okay, we are asked to find the area of this place. Now the first thing is to try to understand what We are about to do okay, so that is the first thing we need to do: first of all, analyze the diagram. Okay, so if we do that, the area of this place now—this place, if we divide it into two, starting from the center like this, you will notice that from here to this place, okay, everything here this is a sector of a circle, right? All right, and this particular place we call this O. then AOC is a triangle, it is a triangle, all right. Now, if that is so, that means that the area of the The entire area should be the area of this triangle AOC plus the area of this region. BOC this sector of a circle, okay? So now the big question is, how do we get the area of this? triangle, and how do we get the area of this sector? Now, the first thing we need to know is to remember the formulas that we can use to get the area of these places now if you have a triangle ABC; if you have triangle ABC, ABC, okay; now this is the small a, this is the small b And this is the small c now area of this triangle, okay, is equal to half AB. If theta is c, and we have c, and we know two sides of this Triangle. Okay, then we can find the area of this triangle now. If you come to what we have here. We know this place, and we know this place, okay? Now let me explain. Remember that the length... of this diameter is from here to here is four right now; if AB is equal to four, okay, then AO is equal to four divided by two, which is equal to two, right? So it means that here is two now if you Look at this place closely; you notice that this place is half of the diameter, so it is the the radius of the circle and then the line you draw from the center of the circle to the circumference is also the radius, which means this place is also two, so you can see that we know two sides of this triangle, so if we can get our theta here, if we can get five here, okay, then we can get the area of this triangle, now coming to this one, this side is a sector of a circle, okay, now the area for a sector is theta over 360 times pi r squared. Now, if you look at this diagram, we know our radius is already given, pi is a constant, so the only obstacle to finding the area of this sector Is this our theta? So you can see that everything we are going to do in this particular video to try to find the value of phi and the value of theta; once you are able to do that, we are going to find the area of this place and the area of this place, then add them together, are you ready? Now, let's Go ahead and do that, so everything I'm doing right now is going toward these two explanations, these two. Things I said, now I'm going to clean this because I'm not ready to use them right now Okay, so back to the main diagram. This is the center of the circle. Now, first of all, let us Talk about our triangle ACD. Okay, triangle ACD. For triangle ACD, all right, since the two sides are equal, okay? So this is C, and this is D, and this is 30 degrees because AD So since AD is equal to DC, this is what an isosceles triangle is, okay? This is an isosceles triangle, and there is a theorem regarding isosceles triangles: the base angles are equal, therefore the base angles are equal. Okay, the base angles are equal, so if the base angles are equal, which means angle A is also 30 degrees. Okay, and now we can find angle D. Now remember The sum of the angles in a triangle is 180, so that means 30 plus 30, that is this plus this. plus angle d is equal to 180 degrees, okay. Now 30 plus 30 is 60, so you have 60 plus d is equal to 180 degrees; now we take this to this side, so we have d is equal to 180 minus 60, so d is equal to 120, all right, so this angle here is 120 degrees. Please click the subscription button and turn on your If this is the first time you've seen this amazing channel, give us a thumbs up if you Actually enjoyed this video. Always check us at your convenient time—there must be something new. to watch by the grace of God, okay, all right. Now let us continue. So now that we have found this Let us complete this triangle, okay? Now remember that this is 30 degrees. as well, now if we complete this triangle, you notice that this is the angle formed semicircle, yes, according to the circle theorem, if this is the center of the circle and this is Okay, this is 90 degrees; if you form another one here, it's 90 degrees, so that means this place Is 90 degrees okay? Now you're going to understand what I'm doing. All I'm trying to do now is to get one of the angles we are just expecting either our five or our theta, okay? All right, now if you Look at this: this is 90 and this is 30, and you notice that this is a cyclic quadrilateral. right yes because the vertices are touching the sides of the circle now exactly what is if you and you have an inscribed quadrilateral as long as the vertices touch If the circumference passes through all these vertices, then we have a cyclic quadrilateral. So ABCD is a cyclic quadrilateral, okay? So ABCD is a cyclic quadrilateral. Okay, now what is cyclic, and why did I even bring it up? Remember that opposite angles? Opposite angles of cyclic quadrilaterals are supplementary. I'm trying to write this down because some of you complain that you don't really understand. What I say, you don't understand my language, my accent, okay? So, opposite angles of a cyclic quadrilateral... Quadrilaterals are supplementary, which means that when you add them together, you get 180 degrees. Now let me go back to that. What am I doing? So if you look at this, okay, so this is 90. Now 90. plus 30 for angle C, so C is 90 plus 30, which is 120, right? 120 degrees, and this is the angle that Okay, so we have 30; we can get this because when we add all of this and this, we get 180. Okay, so we can say 30 plus 90, 30 plus 90 plus this 30 plus, let's call this x, okay, let's call it We almost have the answer; we are very close to the answer. Okay, so we have 30 plus 90, which is 120. 120 plus 30 is 150, so we have 150 plus x is equal to 180 degrees. Okay, now let's transfer this to on this side, so we have x is 180, now 150, we cross and become minus 150, okay, all right, so our x is equal to 30 degrees, so that means this place is also at 30 degrees now. Why, why am I bothering myself? Remember that we need the value of this; now I can drop this again. Let's connect this and this. Again, now remember that we need the angle that will be here; let's call it theta, okay, we need theta, and we have already obtained that—remember that, according to the circle theorem the angle at the center is equal to two times the angle at the circumference, okay? Now, this is it if you have A circle—this is a circle—and you have a chord AB, and this is the center of the circle, okay? This chord subtends an angle at the circumference called theta, and you sustain another angle here. called phi, okay, then two theta is equal to five, okay, so the angle at the center is two times angle at the circumference now where is it coming from remember that line BC is a chord Okay, so this line BC is a chord, and it subtends an angle θ at the center. and it maintains another angle at the circumference, let me come here So this is our chord, because it subtends an angle at the center, theta, and it subtends an angle at the circumference 30, okay, so using this particular theorem, now using this theorem we can say that two multiplied by 30 degrees equals theta, so it means that 60 degrees equals to theta, right? Yeah, so we're going to replace this with 60 degrees. Now, again, remember that We need this angle, which is now very easy to get, okay, very easy, because remember that angle. The sum of angles on a straight line is 180, so if this is x and this is y, then y plus x is equal to 180. Okay, now you can see that here it will be 180 minus 60, which is equal to 120 degrees, okay. So we have 120 degrees. All right, now we are very, very close to our answer. Now remember what I said: the area of ABC equals the area of triangle AOC. a o c plus area of triangle o c o b, okay, c o b or b o c. All right, now let's go ahead and find. So the area of triangle ABC should be equal to twice the area of triangle AOC. Sine c okay, now recall what we said—this is the radius of the circle. So from here to here is also the radius, so we have two of them as well. Okay, now that means this is half a b sine c plus, for this we have theta over 360. times pi r squared, just like I explained, COB or BOC will give us a sector, so it's going to be theta over 360 times pi r squared, so this is equal to half times, now this is our a and this Our b and both lines are radii of the circle, which are 2 and 2, so we have 2 times 2. then sin 120, okay, sin 120 degrees, which is this, then plus now the theta of the sector is 60 so we have 60 divided by 360 times pi times the radius squared, which is 2 squared, okay, so this is times 4 times 9, so 120 is root 3 over 2, okay, plus now here we have this, we divide it by itself Now 6 divided by itself is 1, and 1 divided by 6 gives us 6, so we have 1 over 6 times pi times 4 over 1. All right, now let us go ahead and solve this; it is equal to half. Okay, let's go ahead and divide. We divide this to give us 2; we divide this to give us 1, so we are left, we are left with root 3. Then here 2, we divide this to give us 2 divided to give us 3, so we have 2 times pi. divided by 3, and this is what we have: square meters, square centimeters, or centimeters squared or square meters, so I don't know the units, but this is the value of our area Thank you so much for watching, see you in my next video.