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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.