Explain the difference between the following pairs of
geometric figures:
a simple closed curve and a nonsimple closed curve
a convex polygon and a nonconvex polygon
A simple closed curve is a curve, such as a circle, that is closed and does not intersect itself.
A simple closed curve is a curve that does not intersect itself. It forms a shape without any holes or self-crossings. A nonsimple closed curve, on the other hand, is a curve that intersects itself, creating loops or repetitions in its path.
To determine if a closed curve is simple or not, you can follow these steps:
1. Trace the curve from start to end, without lifting your pencil.
2. Note if the curve crosses itself at any point or if it forms any loops.
3. If there are no intersections or loops, the curve is simple. If there are intersections or loops, then it is nonsimple.
A convex polygon is a polygon where all its interior angles are less than 180 degrees. It means that if you draw a straight line segment between any two points inside the polygon, the line segment will always stay within the polygon. In simpler terms, a convex polygon does not have any "bumps" or indentations.
On the other hand, a nonconvex polygon has at least one interior angle greater than 180 degrees. This means that if you draw a straight line segment between some of its interior points, the line segment will extend outside the polygon, showing that the polygon has "bumps" or indentations.
To determine if a polygon is convex or nonconvex, you can follow these steps:
1. Select any three consecutive vertices of the polygon.
2. Draw two straight line segments connecting the first vertex to the second vertex, and the second vertex to the third vertex.
3. If both line segments lie entirely inside the polygon, then the polygon is convex. If either line segment extends outside the polygon, then it is nonconvex.
By examining these characteristics, you can differentiate between simple closed curves and nonsimple closed curves, as well as between convex polygons and nonconvex polygons.