A polygon is convex if, for every pair of points, P and Q belonging to the…
2021
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?
Attempted by 85 students.
Show answer & explanation
Answer: The polygon shaped like an arrow with an inward notch is not convex.
Definition: A polygon is convex if for every pair of points inside the polygon, the line segment joining them lies entirely inside or on the boundary of the polygon.
Why the arrow-shaped polygon is not convex: It has an inward notch (a reflex interior angle greater than 180 degrees). If you pick one point inside the notch and another point on the opposite side of the shape, the straight line segment between them passes outside the polygon. That violates the convexity definition, so the polygon is concave.
Why the triangle, square, and trapezoid are convex: Each of these shapes has no inward dents and all interior angles are less than 180 degrees. Therefore, for any two points inside each shape, the connecting segment stays inside the shape, satisfying the convexity condition.
Quick check students can use: Look for inward notches or interior angles greater than 180 degrees. Alternatively, choose two points that appear on opposite sides of a potential notch; if their connecting line leaves the polygon, the polygon is not convex.
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