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This problem trains for: SAT-I, AMC-8, GMAT, AMC-10.

In a rectangular box with dimensions 3, 4, 12 what is the area of the shaded triangle?

Z and Y are the centers of the faces ADD'A' and BCC'B', respectively. O is the point where the diagonal of the box intersects the segment ZY.

BC=3, BB'=4 and AB=12.

The triangle ZDO belongs to the plane A'DCB'. This is a figure of the rectangle A'DCB'.

The side A'D is the diagonal of the face: AA'D'D. It has length (Pythagoras):

The segment ZD is half of this length, since Z is the center of the face ADD'A':

The diagonal of the box with sides a, b, c is: