Problem 3

Given tetrahedron

ABCD

, let opposite edges

AB

and

CD

have lengths

a

and

b

. Their skew-line distance is

d

and their angle is

\theta

. A plane parallel to both

AB

and

CD

divides the tetrahedron into two solids. The ratio of the plane's distances from

AB

and

CD

k

. Compute the ratio of the volumes of the two solids.

Step 3 of 5: Compute the prism-like solid

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V_{WXPBYZ}=\frac{3k^2}{(k+1)^3}V.

Detailed analysis

The base triangle $BYZ$ is similar to $BCD$ with area scale $k^2/(k+1)^2$ . Its prism height is $1/(k+1)$ of the altitude from $A$ to $BCD$ . Since $V=Bh/3$ , this gives $V_{WXPBYZ}=3k^2V/(k+1)^3$ .