2003 AIME II problem 4 (part 2)

Where we left off in the last video, we were able to figure out the coordinates of all of the vertices for this face right here of the tetrahedron. And so now we can use those coordinates to figure out the coordinates of the center of that face, which is also one of the vertices of the smaller tetrahedron. When we figure out that coordinate, then we can figure out the distance between that point and that point, or between this point and this point. And that will give us the length of one of the sides of the small tetrahedron. And then we can compare that to the length of the large tetrahedron and get this ratio right over here. So let's get the coordinates of that point or this point right over here. Let's get the coordinates of this thing. To do that, we just have to take the average of the x, y, and z-coordinates the vertices of the face. Now for x, you can see, hopefully maybe you see that this is sitting on the yz axis. Actually, let me draw the z-axis here just for visualization. So z could go up like that, and then this would be the negative z direction would be going down like that. So this is clearly lying on the zy plane. But even if you didn't see that, you could just average the x points here. So we have negative 1 plus 1 plus 0 over 3, which is going to be 0 over 3. And then you could average the y points. 0 plus 0 plus square root of 3 over 3, all of that over 3. And then you can average the z points. Let me do this in a different color. Average the z's. 0 plus 0 plus 2 square roots of 2 over the square root of 3, all of that over 3. Now this first coordinate right over here, clearly just 0. Second coordinate over here is going to be the square root of-- this is the square root of 3 over 3. This was hard to read. Square root of 3 over 3 divided by 3. So it's the square root of 3 over 9. And then we have 2 times the square root of 2 over the square root of 3 divided by 3. So it's going to be 2 times the square root of 2 over 3 times the square root of 3. This right here are the coordinates for the center of this space face, this coordinate right over here. Now using that, we can now find the distance between that point and that point. We already know this point's coordinates. We already know the center of the base. We already know that right over here. So let's figure out this distance or this distance, the length of a side of the smaller tetrahedron. So let's write it. Let me do it over here. The length of the side of a smaller tetrahedron squared is going to be our difference in x value squared. So this has an x value of 0. This has an x value of 0. So 0 minus 0 squared plus the difference in y values. Square root of 3 over 9 minus square root of 3 over 3 minus-- Square root of 3 over 3, just so we have a common denominator, that's the same thing as 3 square roots of 3 over 9. I just wanted to rewrite it this way so that I have a common denominator, squared. Plus 2 square roots of 2. It's getting messy because I'm-- plus 2 square roots of 2 over 3 square roots of 3 minus this guy's z-coordinate. Well, this guy was on the xy plane, so it's minus 0, minus 0 squared. So this is going to be equal to-- let me write it here. Let me scroll to the left a little bit so I have some space. So we have a length of the small tetrahedron squared is going to be equal to-- well, this is just going to be 0. This right here is negative 2 times the square root of 3 over 9 squared. So negative 2 times the square root of 3 over 9 squared is 4 times 3. Negative 2 times square root of 3 squared is going to be 4 times 3 over 9 squared, over 81. So we did that part right over here. We had negative 2 times square root of 3 over 9. Square that, you get this over here. So that turns into that. And then this term right over here, this is just going to be 2 root 2 over 3 root 3 squared. So it's going to be plus-- so this is going to be 4 times 2 over 9 times 3. Or another way of saying it, this is going to be equal to-- let's see. Well, the numerator and denominator are divisible by 3. So this is the same thing as 4 over 27. So this first term is 4 over 27 plus what we have over here, 4 times 2 is 8 over 27, which is equal to, which is the same thing as 12 over 27. You divide the numerator and denominator by 3. This is equal to 4 over 9. So the length of the small side squared is equal to 4 over 9. You take the principal root on both sides, positive square root. We only care about positive values because we're talking about distances. So the length of one side is going to be equal to the square root of that, which is going to be 2 over 3. So this distance right here, this length right here is going to be 2/3, or this length right here is going to be 2/3. So what's the ratio of a length of a big side to the length of a small side? So the length of a big side we already figured out the. Length of a big side is 2. This thing is 2. What's the length of a small side? Well, based on all the math we just did, it is 2/3. Now before we even cube it, we can divide the numerator and the denominator by 2. Actually, we can divide the numerator and the denominator by 2, so this becomes 1/2 1/3. And 1 over 1/3 is the same thing as 3. So 3 cubed is 27. Or I could write it as 27 to 1, since we want to write it as a ratio. So when you simplify all this, you really will just get 27. But if you want to express it as a ratio, it is 27 to 1, the volume of the big tetrahedron to the volume of the small. So that ratio is m over n. This is equal to m over n where m and n are relatively prime positive integers. These are definitely relatively prime. Their only common factor is 1. Find m plus n? So 27 plus 1 is 28, so our answer is 28.