A polygon is convex if, for every pair of points P and Q belonging to the polygon, the entire line segment PQ belongs to the polygon. Which of the following shapes is always convex?
Options:
(A) Regular hexagon
(B) Star-shaped decagon
(C) Crescent polygon
(D) Arrowhead quadrangle
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Official Solution & Concept Analysis (GATE BT 2021 Q.2)**
**1. Governing Concept**:
- **Topic**: Geometry & Convexity of Polygons
- By definition, a polygon is **convex** if, for every pair of points $P$ and $Q$ lying inside or on the boundary of the polygon, the line segment $PQ$ lies entirely inside or on the polygon.
- If any internal angle is greater than $180^\circ$ (a reflex angle), there exists at least one pair of points whose connecting segment passes outside the polygon. Such a polygon is **concave (non-convex)**.
- Figure (A) displays a concave polygon with an inward dent/re-entrant corner where the segment connecting two internal points exits the figure.
**2. Option Analysis**:
- **Option (A) is CORRECT**: Depicts a concave polygon whose line segment between points across the indent falls outside the boundary.
- **Option (B) is INCORRECT**: Convex regular polygon.
- **Option (C) is INCORRECT**: Convex regular polygon.
- **Option (D) is INCORRECT**: Convex rectangle.
**3. Final Answer**:
- **Correct Choice**: **A**
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