A Nice geometry math question| solve for perimeter of the hexagon

You're welcome back to my channel, and I'm excited to have you here. Thank you so much for joining. We are asked to find the perimeter of this regular hexagon. Please note that this drawing is not to scale. drawn to scale? Excuse me, yes, it's not drawn to scale, so but this is supposed to be a regular hexagon meaning that this is a polygon with six equal sides, so these sides are equal and then we have a triangle inscribed inside this large triangle, and we are told that the The perimeter of the triangle is 12√3, okay, and we are asked to find the perimeter of the regular Hexagon. First of all, let's read the mind of the examiner: why did the examiner give us the perimeter? Of the triangle, first what is perimeter? Perimeter of a plane shape means the distance around it. around the shape, okay, so it means if we add the length of the triangle, all the sides, that gives Use the perimeter; first, we are told that this is a regular hexagon. Now let us relate the regular hexagon and the triangle first; if this is a regular hexagon, then it means that these diagonals are equal, okay? And if these three diagonals are equal, then it means that we have what? equilateral triangle, so let's call this triangle ABC. Okay, so triangle ABC is an equilateral triangle. Triangle. Okay, so if this is an equilateral triangle, it means that the three sides are equal. and the three angles are equal as well. Now we are told that the perimeter of the triangle is 12 root So it means that one side of triangle ABC will be equal to 12 times the square root of 3, divided by 3, okay. so it's going to be 12 root 3 divided by 3, which will give us 4 root 3. Okay, so we have found the the size or the length of one side of this equilateral triangle; now let us look at what we can deduce From this, so let us draw this out; one part of this means that this place, the length is What is b c? It is 2 times the square root of 3. Right, this is 4 times the square root of 3. Now let us draw that out. Now this is 4 times the square root of 3. Okay, then we need to know the interior angle of a regular hexagon, that is, each of these angles. What are they equal to? Okay, now there is a formula for the interior angle of any regular hexagon. so if this is the first time you have seen us, please click the subscription button and turn on your notification bell so you don't miss our videos—we upload videos as often as possible Okay, so tune in at your convenience—there was something new for you to watch by God's grace. and give us a thumbs up if you're actually enjoying this video, so like I said, um, interior the interior angle of a regular hexagon, which is a regular polygon with n = 6 sides, is equal to n minus 2 multiplied by 180 divided by n, okay, this is the formula, now we are talking about a hexagon meaning that our side is equal to 6, so n is equal to 6. Okay, so let's go ahead and create Oh, this isn't going to be enough. Okay, I'll find a way. All right, now let's try. Find the interior angle of this particular hexagon, which is the angle that will be located here. one of them, so we have n minus 2, that is 6 minus 2, multiplied by 180 divided by 6 so it's going to be equal to 4 multiplied by 180 divided by 6; if we solve that, we will have When 2 divides itself, okay. Let's use this: when 6 divides itself, 1. If we divide 18, we have 3. Divide this: we have 0, so we have 4 multiplied by 30; we have 120 degrees, so it simply means that this angle is equal to 120 degrees, okay. Now let us analyze this carefully, remember that This side and this side are equal; yes, this is a regular hexagon, so we have two equal sides. of this triangle, meaning that this triangle is an isosceles triangle, and there is this theorem about an isosceles triangle that states that if you drop a straight line from the vertex On this side of the triangle, okay, that particular line drops—for example, let me draw it like this. if this is an isosceles triangle, and we drop a straight line from the vertex to the the other side of the triangle; now we are going to notice that this particular line will bisect Divide this angle into two equal parts, form a 90-degree angle here as well, and bisect this length too. Okay, so if we do that here, that means you're going to have this, so this will now be 60 degrees. This will be 60 degrees; 90 degrees is formed here, and it's going to bisect this line into so one part will now be 4 times the square root of 3 divided by 2, and this equals 2 times the square root of 3, so it means that This side is 2 root 3, and this side is 2 root 3. Okay, now where am I taking us? Let us take part of this particular triangle now, okay? So I'm just going to take part of it. That is part of this drawing; if I do that, this is 2 root 3, so this place is 2 root 3, right? The angle is 60 degrees, 60 degrees; this is the side we don't know, and this is 90 degrees now. Where am I driving? Permit me to clean this wall and fill in this ink—it's becoming faint, okay? And I'll be right back, okay? Now let me redraw this. Okay, this is what we have. Yeah, it's now very Clear, so this is 2 root 3. Remember, we have 90 degrees, this is 60 degrees, and this side is What we are looking for on this particular side, and if we get it, that means we have obtained one. side of this particular example, remember that the perimeter will be this one side multiplied By six, okay. All right, so first of all, I want you to note that this side can be obtained in two ways. Now the first method is this: if this is 90 and this is 60, that means this is 30 degrees. Okay, yes, because the sum of the angles in a triangle is 180 degrees, so this one is 30 degrees now, um. We can remember that there is a relationship between the sides of a 30-60-90 triangle. A triangle, the opposite is our x, okay? So let me draw this using another triangle. If this is 30 And this is 90, and this is 60, the opposite is x, the hypotenuse is 2x, and the other side is 2. Is x times root 3 okay? So if we compare these two now, it means that 2 times root 3 is equal to x times root 3. So we have 2√3 = x√3. Now let's divide both sides by √3; we get that 2 is equal to x, okay? So if 2 is equal to x, we already have 2 root 3 here, which means that the hypotenuse is 2x, that is 2 times 2, right? So we should get 4. So that means we have found This is to be 4 now. Alternatively, another alternative method is this. Another alternative method is this. This side is the hypotenuse of this triangle, and this side is the opposite side relative to the 60-degree angle. So the opposite of the hypotenuse is sine; if we can say that sine 60 equals the opposite, which is 2 root 3. over hypotenuse, which is x, let's call it x. Okay, so if we cross multiply, we have that x is equal to 2 root 3 divided by sine 60, now x is equal to 2 root 3 divided in sort form, sin 60 is root 3 over 2, so we can say that x is equal to 2 root 3 multiplied by we take the inverse of this if we decide to change this dimension to multiplication so it becomes 2 over root 3; now root 3 we cancel root 3 and we have that x is 4, 2 times 2, so we still get the same answer. Now we are done with finding this side, so that means that this is 4, okay, therefore the perimeter of this hexagon is 4 multiplied by 6, which is Equal to 24 units. Okay, thanks so much for watching.