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An art gallery engages a security guard to ensure that the items displayed are protected. The diagram below represents the plan of the gallery where the boundary walls are opaque. The location the security guard posted is identified such that all the inner space (shaded region in the plan) of the gallery is within the line of sight of the security guard. If the security guard does not move around the posted location, which one of the following correctly represents the set of ALL possible locations among the locations P, Q, R, and S, where the security guard can be posted to watch over the entire inner space of the gallery? The gallery floor plan is a chevron (inverted-V / arrowhead) shape pointing downward with shaded interior. Points are placed in and around the shape: P is near the top-left, Q is near the top-right inside the notch, R is on the left side, and S is near the top-right area. A. P and Q B. Q C. Q and S D. R and S

GATE 2022 · General Aptitude · Patterns In Two Dimensions · medium

Answer: The set of all valid guard positions is Q and S — option C.

  1. Analyse the gallery shape: The gallery is a non-convex chevron (inverted arrowhead). The inward notch at the top creates concavities on the left and right sides. Any guard position in the top-left or top-right exterior cannot see the opposite arm of the chevron.
  2. Test position P (top-left area): From P on the upper-left, the right wing of the chevron lies behind the central dividing wall. At least one interior region is hidden. P is INVALID.
  3. Test position Q (inner notch): Q is placed at or near the inner apex of the chevron where both wings radiate outward. Every point in the shaded interior can be reached by a straight line from Q without crossing a wall. Q is VALID.
  4. Test position R (left side): From R on the left side wall, the right arm of the chevron interior is hidden behind the angled walls. R is INVALID.
  5. Test position S (top-right area): From S near the top-right, the guard has an open line of sight to the entire interior including the left wing. S is VALID.
  6. Compile the answer: Valid positions: Q and S. Invalid: P and R. Answer = C.