Prove that if two diagonals of a rectangular parallelepiped are perpendicular, then its dimensions are equivalent to the sides of a right triangle and vice versa

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- April 6th 2010, 05:18 PMmathwizard325proof for rectangular parallelepiped
Prove that if two diagonals of a rectangular parallelepiped are perpendicular, then its dimensions are equivalent to the sides of a right triangle and vice versa

- April 6th 2010, 10:21 PMSoroban
Hello, mathwizard325!

I've proved it one way so far . . . a vector proof.

Quote:

Prove that if two diagonals of a rectangular parallelepiped are perpendicular,

then its dimensions are equivalent to the sides of a right triangle and vice versa.

The "box" has dimensions: . in the direction,

. . in the direction, in the direction.

Orient it in the first octant with vertex at the origin.

The graph looks like this:

Code:`T | W`

(0,0,c)* - - - - - * (0,b,c)

/| /|

/ | / |

U / | / |

(a,0,c) * - - - - - * V |

| | | | S

| P * - - - | - * (0,b,0)

| / | /

| / | /

Q |/ |/

(a,0,0) * - - - - - * R

/

One diagonal is: .

. . . . Another is: .

Since

Therefore: .