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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?

How to solve a solid rectangle prism problem?

Solution:

connect EG
find the length of the longest diagonal in a rectangle prism.
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.