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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
- x
^{2}y^{2}z^{2}= 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, EG
^{2}= EF^{2}+ FG^{2}= z^{2}+ y^{2} - In plane AGE, AG
^{2}= AE^{2}+ EG^{2}= x^{2}+ z^{2}+ y^{2}= 2^{2}+ 4^{2}+ (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.