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Consider a slightly more general situation, where the line segment S of the statue is being viewed from a point on a line V positioned generally (rather than a line perpendicular to the line segment as given in the problem). To determine the best vantage point on V from which to view S, note that if the circle determined by the end-points of the segment S and the current vantage point is such that the line V moves cuts the circle, then any point on the corresponding chord will have a larger viewing angle (follows readily using the constancy of angles at the circumference standing on a given chord, S in this case). Therefore, at the best viewing point, the circle must be tangent to V.
In the given case, where V is perpendicular to S (extended), the distance of the viewer to S extended can be shown by elementary means to be given by the formula above.