A polygon is convex if, for every pair of points, P and Q belonging to the polygon, the line segment PQ lies completely inside or on the polygon. Which one of the following is NOT a convex polygon?

Question

Prepp Team · Accepted Answer

Understanding Convex Polygons

A polygon is defined as convex if, for any two points P and Q chosen within or on the boundary of the polygon, the entire line segment PQ remains inside or on the boundary of the polygon. Visually, this means the polygon has no "dents" pointing inwards, and all its interior angles are less than 180 degrees.

A polygon that does not meet this condition is called a non-convex or concave polygon. It will have at least one interior angle greater than 180 degrees, or there will be at least one pair of points (P, Q) inside such that the line segment PQ goes outside the polygon.

Analyzing Polygon Shapes

We need to examine each provided image to see which polygon violates the definition of a convex polygon.

Option 1: This image displays a star-shaped polygon. If you select two points on opposite points of the star (e.g., on two different 'rays'), the line segment connecting them will pass outside the polygon. This violates the definition.
Option 2: This image shows a regular hexagon. All interior angles are equal (120 degrees), and it's easy to see that any line segment connecting two points within it will stay inside. It is a convex polygon.
Option 3: This image shows a rectangle. All interior angles are 90 degrees. It is a convex polygon.
Option 4: This image shows a standard pentagon. All its interior angles appear to be less than 180 degrees, and it seems to satisfy the line segment condition. It is a convex polygon.