If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# 2003 AIME II problem 4 (part 2)

## Video transcript

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 centre of that face which is also one of the vertices of the smaller tetrahedron and when we figure out that coordinate then we could 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 at this point right over here let's get the coordinates of this thing and to do that we just have to take the average of the x y&z coordinates of the fate of the vertices of the face now for X you can see hopefully maybe you see that this is sitting on the Y z axis actually let me draw the z court the z axis here just for just for visualization so Z could go up like that and then this would be the negative 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 you could just average the X 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 0 plus 0 plus square root of 3 over 3 square root of 3 over 3 all of that over 3 and then you can average the Z points let me do it in a different color average the Z 0 plus 0 0 plus 0 plus 2 square roots of 2 over the square root of 3 all of that over all of that over 3 now this first coordinate right over here clearly just 0 second coordinate over here square is going to be the square root of this is the square root of 3 over 3 so it's 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 to two square root of two over the square root of three divided by three so it's going to be two times the square root of two over three times the square root of three this right here are the coordinates for the center of this face this coordinate right over here now using that we can now find the distance between that point and that point we already know that this points coordinates we already know the center of the base we already know that right over here so let's figure out let's figure out this distance or this distance the length of this 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 zero this has an x value of zero so zero minus zero squared plus the difference in Y values square root of three over nine minus square root of 3 over 3 minus square root of 3 over 3 just we have a common denominator that's the same thing as three square roots of three over nine I just wanted to rewrite it this way so that I have a common denominator squared plus plus two square roots of two it's getting messy because I'm plus two square roots of 2 over 3 square roots of three - this guy is Z coordinate well this guy was on the XY plane so it's minus zero minus zero 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 of the small tetrahedron squared is going to be equal to well this is just going to be zero this right here is negative two 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 right negative 2 times square root of 3 squared is going to be 4 times 3 over 9 squared over 81 over 81 so we did that part right over here right yeah we had negative 2 times square root of 3 over 9 you square that you get this over here so that turns into that and then this term right over here this is just going - 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 over 9 times 3 or another way of saying it this is going to be equal to let's see what the numerator denominator are divisible by 3 so this is the same thing as 4 over 27 so this first term is 4 over 27 plus we have over here 4 times 2 is 8 over 27 which is equal 2 which is the same thing as 12 over 27 you divide the numerator and the denominator by 3 this is equal to 4 4 over 9 so the length of the small side squared the length of a 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 or 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 the 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 it is 2/3 now before we even cube it we can divide the numerator and the denominator by 2 actually well yeah we can take the new divide the numerator and the denominator by 2 so this becomes 1/2 1/3 which is + 1 / 1/3 is the same thing as 3 so 3 cubed is 27 this is or I can write it as 27 - 1 since we want to write it as a ratio so when you simplify all this you get 20 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 big tetrahedron - the volume of the small so that ratio is M over N this is equal to M over N M and n are relatively prime positive integers these are definitely relatively prime their only common factor is one find M plus n so 27 plus 1 is 28 so our answer is 28