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 5 of 5: Take the ratio

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V_{other}=V-V_{ABWXYZ}=\frac{3k+1}{(k+1)^3}V\\quad\Longrightarrow\\quad V_{ABWXYZ}:V_{other}=k^2(k+3):(3k+1).

Detailed analysis

Subtracting from the total volume gives the other part. Cancelling the common factor $V/(k+1)^3$ yields the required ratio $k^2(k+3):(3k+1)$ ; the lengths $a,b$ , distance $d$ , and angle $\theta$ do not affect it.