The image shows a math problem asking to determine whether the given shapes are polygons and whether they are convex or nonconvex.

The answers provided on the image are:

• a) Polygon and Convex.
• b) Polygon and Convex.
• c) Polygon and Nonconvex.
• d) Polygon and Nonconvex.

Let’s review the definitions to confirm:

1. Polygon: A closed, flat shape with straight sides.
2. Convex polygon: All interior angles are less than 180°, and no line segment between any two points on the boundary will pass outside the shape.
3. Nonconvex (or concave) polygon: At least one interior angle is greater than 180°, and some line segments between points on the boundary will pass outside the shape.

Analysis:

• a) This shape has straight sides and is a closed figure, so it’s a polygon. However, since it has interior angles greater than 180°, it is nonconvex, not convex.
• b) This is a polygon (hexagon) with all interior angles less than 180°, so it is convex.
• c) This shape is also a polygon but has an interior angle greater than 180°, making it nonconvex.
• d) The cross-like shape is a polygon, and since some interior angles are greater than 180°, it is nonconvex.

Thus, the corrected answer for shape (a) should be nonconvex, not convex.

Would you like more details or have any questions? Here are some additional related questions:

1. What is the difference between convex and nonconvex polygons?
2. How can you determine if a polygon is concave or convex based on its angles?
3. What are some real-life applications of convex and nonconvex shapes?
4. Can a polygon have both convex and concave properties?
5. Why