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# Calculate the length of the longest diagonal of a rectangle prism

Question:
In the figure show below, ABCD-EFGH is a rectangle prism. The area of ADHE is 1, the area of ABCD is 2 and the area of EFBA is 8. AG is the longest diagonal. What is the length of AG?
Solution:
connect EG
Let x = AE, y = AD and z = AB
then we have
xy = 1 , this is our equation1
yz = 2 , this is our equation2
xz = 8 , this is our equation3
The product of the left side of the equations equals to the product of the right side of the equations
x2y2z2 = 1 × 2 × 8 = 16
Square both side of the equation
xyz = 4, this is our equation4
Substitute equation1 into equation4, we get, 1 × z = 4
z = 4, this is our equation5
Substitute equation2 into equation5, we get, y × 4 = 2
y = 1/2
Substitute equation3 into equation 5, we get, x × 4 = 8
x = 2
In plane EFGH, EG2 = EF2 + FG2 = z2 + y2
In plane AGE, AG2 = AE2 + EG2 = x2 + z2 + y2 = 22 + 42 + (1/2)2 = 4 + 16 + 1/4 = (16+64 + 1)/4 = 81/4
so AG = 9/2 = 4(1/2)
So the length of AG is four and half.

If you are interested in watching the video for the solution of this problem, please click the video talk of the rectangle prism problem.