Question

A polygon is convex if, for every pair of points inside the polygon, the line segment joining them lies completely inside or on the polygon. Which one of the following is NOT a convex polygon? 

• A. A
• B. B
• C. C
• D. D

Hint:

Any polygon with a 'dent' or inward angle greater than \(180^\circ\) is automatically non-convex.

Answer 1

The Correct Option is A

A polygon is said to be convex when every interior angle is less than \(180^\circ\), and for any two points chosen within the polygon, the straight line segment connecting them remains entirely inside the polygon.

Step 1: Understand convexity.
Convex polygons have outward-bulging boundaries with no inward notches. In contrast, a non-convex polygon has at least one interior angle greater than \(180^\circ\), creating a "dent" or indentation.

Step 2: Evaluate each option.
- Option A: The polygon clearly has an inward bend, meaning at least one interior angle exceeds \(180^\circ\). This violates the convexity rule.
- Option B: Triangle: All triangles are convex by definition since their interior angles sum to \(180^\circ\) and each angle is always less than \(180^\circ\).
- Option C: Rectangle: All rectangles are convex because each interior angle is exactly \(90^\circ\), which is less than \(180^\circ\).
- Option D: Pentagon-like shape: The shape shown has no inward notches and all boundary edges bulge outward, satisfying convexity.

Step 3: Conclusion.
Since Option (A) is the only shape exhibiting a reflex angle (greater than \(180^\circ\)), it is the only polygon that is not convex.