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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.