A Nice geometry math question|Solve for the area of the circle

You're welcome back to my channel if you're a returning subscriber, thank you so much, and if This is your first time joining us—trust me, you're going to enjoy this. Thank you so much. For joining, so first I'm going to apologize as usual, this is my diagram, okay? Now this is not drawn to scale, and this is a right-angled triangle, okay, and these circles are equal; they are of the the same magnitude in everything, so it simply means that their radii are the same, okay, all right now First of all, since this is a right-angled triangle, let's find the value of BC, okay? Let's Now find the value of bc. First, we are going to obtain it using the Pythagorean theorem, yes. It says if you have a right-angled triangle, okay, now let's assume this is a right-angled triangle. The side facing the right angle is our hypotenuse. Let's call the remaining sides a and b now. According to Pythagoras, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The hypotenuse The square of the hypotenuse is equal to the sum of the squares of the other two sides. Now let's relate that to what we have on the board. Okay, now remember that this is a right angle, which means that BC squared should be equal to ab squared plus ac squared, right? ab squared plus ac squared. Now let's find the value. so BC squared equals AB squared, which is five, so we have five squared plus We have 12 squared, okay. Now, BC squared is equal to 25 plus 144, okay. Now, BC squared is equal to 25 plus 144 is 169, 5 plus 4 gives us 9, 2 plus 4 gives us 6, leaving 1, so it gives us 169. now let's take the square root of both sides, so we have the square root of bc squared is equal to square root square root of 169 if this is the first time you've seen us, please hit the subscription button Turn on your notification bell so you don't miss our videos, okay? Tuning in at your convenience. There must be something new to watch. Give us a thumbs up if you're actually enjoying this video. and if you're a returning viewer, if you've been my subscriber for a while, thank you so much, I appreciate your support; now in this way we cancel the square root and we are left with bc is equal the square root of 169 gives us 13, so this simply means that BC is equal to 13. Now, why are we doing Let's write 13 here, and then let's go back. Let's go back and understand what we are looking at. For because the question states that we should find the area of these three circles Remember, they are the same; they are the same size, okay, and the area of a circle is equal to pi. arrow squared pi arrow squared, and if you look at this formula, our pi is constant—it's approximately 3.142, but the radius is the key point here—once you're able to determine the radius of Any of this circle, we will use it to find the area, then multiply it by three to get our answer. So that's what all this is about. What do we do to find the radius of any of these circles? Now let's start by finding the area of this entire triangle, okay? You're going to see why now. the area of a triangle is equal to half the base times the height, so this implies that the area is equal to half times, now the base is 12 times the height is 5, okay, now we have that the area is equal to this, we divide this to give us 6 times 5, which gives us 30, so the area is 30 square meters or 30 units squared because the meter is not this thing, this is not Given that we are not told whether this is centimeters, meters, or anything else, okay, so 30 square units. Now let us return to this diagram. Remember that this line, this line. touches this circle at this point, and if we draw a straight line from the center of the circle to the angle formed at the point of contact between this straight line and this circle is 90 degrees to our circle theorem, okay, because this line is a tangent, then this line from the center to this The place is our radius, right? That is our radius. So we have a 90-degree angle from there as well. If we drop a straight line from here to the point of contact between this and this circle, which is also the Tangent—this is supposed to be a tangent, okay? So this place will also give us another 90 degrees. Now let's look at this—remember that from here to here is our radius, right? So from here to here. We give ourselves another radius, which we make it 2r, right from here to here, 3r from here to here. 4r from here to here gives us 5r, so the length from this place to this place We give us 5R, okay? So let's draw that straight line. Let's draw the straight line, so this straight The line is 5R. Okay, if we call this meeting point D, this point D, don't worry, I'm still coming now. Let us join from this place to this place from here to here from here to here let's all join the vertices of this triangle to D, okay, so we join from here to here, join from here to here I hope I get this straight. Let me try. Let me draw this one—something easier, somehow easier. From here to here, okay. Now from here to here seems difficult for me, but I will try. I knew it, but this is a straight line—please, I don't want to draw any further, okay? And this is our Meeting point D. Please, this line is a straight line from B to D, so that we can proceed, alright? Okay, now we have three triangles here: we have triangle BDC, triangle BDA, and triangle ADC. three triangles; now, if we add the area of these three triangles together, we are going to get the area of this entire triangle ABC, which is equal to 30, okay, so it simply means that 30 or area of triangle ABC is equal to the area of triangle BDC, so the area of triangle BDC plus the area of triangle b d a right b d a plus area of triangle a d c area of triangle a d c so area of triangle a b c It's already 30 square meters, okay? So we have 30 equal to, remember, the area of a triangle. is half base times height, so for triangle BDC we have half times the base; the base of this Triangle BDC is 13. You can see why we have to find the value of our hypotenuse. Okay, now the uh The base of triangle BDC is 13, so we have 13 times the height. Now, the height is this, right? this place and remember that this place is our radius, so we have a radius r plus half times b d a, we have triangle BDA; the base of triangle BDA is five, okay? This five, and the height of triangle BDA is five R right five R plus hope you're enjoying this video okay if you are please keep watching The end is going to blow you away, trust me on that. All right, but back to what we're doing, all right. Now for a d c a d c, the length is going to be half the base, from here to here, which is 12. So times 12 times the height is from here to here, which is also our radius, right? Okay, now we have 30 equals half times 13r, that is, 13 times r plus half. 5 times 5r gives us 25r, plus half times 12 times r gives us 12r. Okay, now we have 30 is equal to 13 divided by 2, which gives us 6.5, so we have 6.5r plus 25r divided by 2 when 2 divided by 2 we have 1, 5 divided by 2 we have 2 with remainder 1, 10 divided by 2 gives us 5, so this Is 12.5r plus 12r divided by 2 we get 6r, okay. Now we have 30 equals 6.5 plus 12.5. 12 plus 6 is 18. 5.5 is 1 now 18 plus 1 is 19, right? So we have 19 r plus 6 r, which will give 25r, so we have 25r. To get the value of r, we divide both sides of this equation by 25. so permit me to claim this, permit me to claim this, because the most important thing we are What I'm looking for is this r. Once you're able to get it, we are true. Okay, so we have 30 is equal to 25. So we divide this by 25 and this by 25, okay? Now 25 we divide itself to give us our 1 times. r is equal to 30 divided by 25. Now let's reduce this to the lowest term. divide this by 5, divide this by 5, so we have r is 6 over 5, okay, which is equal to 1.2, that is the the radius of this circle; each of them is the radius of the circle. Now you can see that we have gotten What we wanted, right? Yes, the radius is 1.2. Now remember that the area of the circle is pi. arrow squared times 3 now pi arrow squared for 1 multiplied by 3 for 3 of them okay so we have Now we cut our radius to 1.2, right? 1.2 squared times 3, so we have A is equal to 3 times pi. Does 3 pi times 1.2 squared mean 12 times 12, and then we move the decimal point two places to the right? and 12 times 12 is 144; shifting the decimal point two places, we have 1.44. Now the area is equal to now I don't want to leave my answer in terms of pi, okay, so permit me to use my calculator so that I will be faster, so we have 3 times pi times 1.44, and it will give me approximately 13.57 square meters, amazing, right? If you watch to this end Thank you so much, I really appreciate you. Thank you so much for watching, see you in my next video.