2010/11 AHSMC Part 1 Problem 13

Problem 13 of 16

The edges $DA$ , $DB$ and $DC$ of a tetrahedron $ABCD$ are perpendicular to one another. If the length of $DA$ is $1$ cm and the length of each of $DB$ and $DC$ is $2$ cm, the radius, in cm, of the sphere passing through $A$ , $B$ , $C$ and $D$ is

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Solution

The sphere which passes through $A$ , $B$ , $C$ and $D$ also passes through the other four vertices of a $1 \times 2 \times 2$ block having $A$ , $B$ , $C$ and $D$ as four of its vertices. Since the space diagonal of this block is of length $\sqrt{2^2 + 2^2 + 1^2} = 3$ , the radius of the sphere is $\frac{3}{2}$ . The answer is (A).

University of Alberta

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