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in Three-dimensional geometry by (36.3k points)

A parallelopiped T has one of its face as ABCD. The face opposite to ABCD is A'B'C'D'. The parallelopiped T is compressed to another parallelopiped S with the face ABCD remaining the same but A'B'C'D' changes to A''B''C''D''. If the volume of S is 90% of T, show that the locus of A'' is a plane.

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Best answer

Let AA' = a, AB = b, AD = d

Volume of the parallelopiped T,

V = a.(b × c)

Let AA'' = r

Let V1, the volume of the new box = r.(b × c)

It is given that V1 = (90/100)V2 = (9/10)V2.

Thus, V1 = (r – 0.9a).(b × c) 

r lies on a plane passing through 0.9a, and perpendicular to (b × c). 

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