A Nice geometry math question| solve for the perimeter|equilateral triangle

Hi, this is not drawn to scale. Welcome back to my channel. Today we are given an equilateral triangle. in which three circles of the same size, each with a radius of five units, are inscribed inside the circle So we are asked to find the perimeter of this particular triangle. Okay, first of all, this The triangle is an equilateral triangle, meaning that the sides are equal. Okay, presumably because this is not drawn to scale, so all sides are equal; that means if we can get just one line, okay, one side of this triangle, then we'll multiply whatever we got by three to get the perimeter because the perimeter is assuming this side is x, okay, this side is also x, and this is also x, so the perimeter is equal to x plus x plus x, which is equal to 3x. So the main question now is What is the value of x if we are able to obtain the value of x, then we are done, okay? So how do we proceed? First of all, let me draw a larger triangle here. As I said, this is not drawn to scale. Scale just a freehand sketch, so presumably this is an equilateral triangle we have this. We have this, okay, and we have this, all of them joined together, okay? Yeah, I know what you are. Think please: three circles, each with a radius of five. So from here to here, it's five. from here to here is five as well, now since this is an equilateral triangle, the sum of the angles In a triangle, the sum of the angles is 180 degrees. So, for triangle ABC, the sum of angles A, B, and C is equal to 180 degrees. All right, then it means that if you want to get one of these triangles, it should be 180 divided by three, which will give us 60 degrees. Okay, all right. Now, so if we add triangle A to angle B To angle C, which is angle A plus angle B plus angle C equals 180—I believe you understood what I did. So it means that here is 60 degrees, okay? Here is 60 degrees, and here is 60 degrees as well. Now, let me not write that. Let's bisect this one. If we bisect it, that means one side is 30 degrees. Right, so we have 30 degrees here, all right. Now remember that this is what this is—a tangent. this line here is a tangent, and when you draw a straight line from the center of a circle At the point of contact between a tangent and a circle, you form a right-angled triangle. Okay, so this place is 90 degrees, all right, and because this from here to here is also The radius of this circle, so it means that from here to here is also five. Now let us draw this. Angle this right triangle that we found—if we draw it out, we have this, this, this, okay. This is five, this is 30, this is 60, and we don't know here either; neither do we know here. All right now. Let us try to find the value of x, then the value of y, okay, or the value of y, then the value of x. but just follow along and see what I'm trying to do, all right? Please give us a thumbs up if you Actually enjoying this video, and if this is the first time you've seen this amazing channel Please click the subscription button and turn on your notification bell so that you're always informed. There must be something new for you to watch by God's grace, and yes, let's continue, so from this Draw this triangle. This angle is 90. This is 30. Why did I say this is 60? The sum of angles in a triangle A triangle is 180, you already know that, so 90 plus 30 is 120, so we need 60 to complete it. This angle should be 60 degrees now. In this particular drawing, what I need is the y-coordinate. You are going to So if we say, using the Pythagorean theorem, using the Pythagorean theorem, okay, now you can say that five squared, okay, we cannot use the Pythagorean theorem, yes, except if we have found x, let's Find x first. Now, using sine: sin 30° is equal to the opposite side, which is 5, over the hypotenuse, which is Now, if we cross multiply, we have x times sine 30 equals 5. Now, to get x, we divide both sides by sin 30, so x equals 5 divided by sin 30, but sin 30 is equal to one half, all right. So it means that x is equal to 5 divided by half. Okay, so x is equal to 5 times 2 divided by 1. x is equal to 10, so now we have the value of this side as 10, so we can now find y using the Pythagorean theorem. Okay, so let's go ahead and do that. So, to find y according to the Pythagorean theorem. your hypotenuse squared, which in this case happens to be 10, is going to be equal to this side squared plus this side squared, okay, so we have 10 squared is equal to 5 squared plus y squared Now let's transfer this to this side: 10 squared minus 5 squared equals y squared. Now 10 10 squared is 100, and 5 squared is 25, which is equal to y squared. Now, 100 minus 25 gives us 75. is equal to y squared, so we take the square root of both sides, the square root of both sides. if we do that, we have the square root of 75 equal to the square root of y squared. Now, 75 can can also be expressed as the square root of 25 multiplied by 2 multiplied by 3. Sorry, so we have the square root 25 multiplied by 3 equals y; now we can split this as the square root of 25 multiplied by The square root of 3 is equal to y. The square root of 25 is 5, so we have 5 times the square root of 3 is equal to y. Okay, now we've got everything we need. Yes, you're going to see what I mean by that, so it means that which is from here to here is 5 feet 3; now let's go back to this diagram, if we go back Now look at this diagram—this is what we did, right? This is it, and this is what we cut, okay? to get 30, and from here to here is our y, which we obtained as 5 feet 3 inches; this is 5√3, and from here To here is 5, okay? Now, why did I get from here to here? Remember, from here to here is 5. And from here to here is another 5, so it means that from here to here will give us 5 plus 5. right, which will give us 10, and from here to here is equal to from here to here, which is 5 Root 3, are you seeing anything all right now? We can say that the side of this triangle is 5 root 3 plus 5 root 3 plus 10 okay now remember that a root b plus b plus c root b is equal to a plus c multiplied by the square root of b, okay? So that means we can solve it. This is 5 plus 5 root 3, then plus 10; if we solve further, we have 10 root 3. plus 10 as the length of this triangle; now remember that the perimeter is the sum of the three sides, which is the same as one side multiplied by 3, because this is an equilateral triangle, so the perimeter is equal to 3 multiplied by 10 root 3 plus 10, okay, so the The perimeter is equal to 3 times 10 root 3, which gives us 30 root 3, plus 3 times 10 is plus 30, now. If we solve this, they have 30 in common; we factor out 30, root 3 divided by 30, which gives us √3 plus 30 divided by 30 gives us 1, so this is the answer in simplified form, okay. If I had a calculator, I would have solved this without one. Okay, thank you so much for Watching, see you in the next video