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A triangular pyramid ABCD has vertices at A(3, 0, 1), B(-1, 4, 1), C(5, 2, 3), and D(0, -5, 4). Let G be the centroid of triangle BCD. Four quantities are listed below along with their possible values: (I) Length of vector AG (II) Area of triangle ABC (III) Length of perpendicular from vertex D onto the opposite face (IV) Area of triangle ABD Possible values: (P) 14/sqrt(6), (Q) 4/sqrt(6), (R) 4*sqrt(6), (S) sqrt(51)/3, (T) 2*sqrt(82), (U) sqrt(82) Which of the following is the ONLY incorrect combination?

1. IV -> T
2. II -> R
3. III -> P
4. II -> P

Correct answer: II -> P

Solution

Computing the area of triangle ABC gives 4*sqrt(6), which matches (R), not (P)=14/sqrt(6). Therefore the incorrect matching is II -> P.

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