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 and has a 360° view, 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?
Show Hint
When determining visibility in geometric setups, always consider the line of sight from each point and whether the combined coverage is complete.
Step 1: Understand the situation.
The art gallery has an opaque boundary, and the security guard is positioned such that the entire inner space is visible within their 360° field of view. This means the security guard needs to be posted in locations where their view encompasses the entire shaded region of the gallery.
Step 2: Analyze the options.
- (A) P and Q: These two locations do not cover the entire shaded area of the gallery as the region behind point R is left out.
- (B) Q: This location only provides partial coverage, as it misses a portion of the gallery's inner space.
- (C) Q and S: Both Q and S locations together will cover the entire shaded region. Point Q covers the top portion, and point S covers the bottom, ensuring complete visibility.
- (D) R and S: These points miss certain areas in the middle of the gallery.
Step 3: Conclusion.
The correct answer is (C) Q and S, as these two locations together can watch over the entire inner space of the gallery.
