7. It is interesting to look at various features of concave and convex polygons. Ramesh and Suresh have their own observations.
Convex Polygon
Concave Polygon
Ramesh: For a concave polygon, it feels like we are entering a "cave" for at least one of the vern
Suresh: if you stand at any point on the boundary of a convex polygon, you can see all the othe points without any obstruction (as shown in figure). But in case of a concave polygon, there c be some points that are not visible to you due to obstruction by other edges of the same polygon (as shown in figure).
Answer the following questions about convex and concave polygons
i) Can a convex polygon have any of its internal angles as a reflex angle?:
ii) is it possible that a concave polygon has none of its internal angles as a reflex angle?
iii) Can all the sides of a concave polygon be equal?
iv) Can all the angles of a concave polygon be equal?
v) Can a concave polygon be regular?
vi) is it possible that any of the diagonals of a convex polygon lie in the exterior of the polygon?
vii) Is it possible that a convex and concave polygon with same number of sides have a different number of diagonals?
(p.s. I give 90 points for this answer)
Convex Polygon
Concave Polygon
Ramesh: For a concave polygon, it feels like we are entering a "cave" for at least one of the vern
Suresh: if you stand at any point on the boundary of a convex polygon, you can see all the othe points without any obstruction (as shown in figure). But in case of a concave polygon, there c be some points that are not visible to you due to obstruction by other edges of the same polygon (as shown in figure).
Answer the following questions about convex and concave polygons
i) Can a convex polygon have any of its internal angles as a reflex angle?:
ii) is it possible that a concave polygon has none of its internal angles as a reflex angle?
iii) Can all the sides of a concave polygon be equal?
iv) Can all the angles of a concave polygon be equal?
v) Can a concave polygon be regular?
vi) is it possible that any of the diagonals of a convex polygon lie in the exterior of the polygon?
vii) Is it possible that a convex and concave polygon with same number of sides have a different number of diagonals?
(p.s. I give 90 points for this answer)
Answers & Comments
Verified answer
Answer:
Here are the answers to your questions about convex and concave polygons:
i) No, a convex polygon cannot have any of its internal angles as a reflex angle. A reflex angle is an angle that is greater than 180 degrees and less than 360 degrees. A convex polygon is a polygon in which all the interior angles are less than or equal to 180 degrees. Therefore, a convex polygon cannot have a reflex angle as an internal angle.
ii) Yes, it is possible that a concave polygon has none of its internal angles as a reflex angle. A concave polygon is a polygon that has at least one interior angle greater than 180 degrees. However, it is not necessary that all the interior angles of a concave polygon are reflex angles. For example, a polygon with four sides and one interior angle of 270 degrees is a concave polygon, but it has only one reflex angle and three acute angles.
iii) No, all the sides of a concave polygon cannot be equal. If all the sides of a polygon are equal, then the polygon is equilateral. An equilateral polygon is always convex, because the sum of any two sides is greater than the third side. Therefore, a concave polygon cannot be equilateral.
iv) No, all the angles of a concave polygon cannot be equal. If all the angles of a polygon are equal, then the polygon is equiangular. An equiangular polygon is always convex, because the sum of any two interior angles is less than 180 degrees. Therefore, a concave polygon cannot be equiangular.
v) No, a concave polygon cannot be regular. A regular polygon is a polygon that is both equilateral and equiangular. As we have seen in the previous answers, a concave polygon cannot be either equilateral or equiangular. Therefore, a concave polygon cannot be regular.
vi) No, it is not possible that any of the diagonals of a convex polygon lie in the exterior of the polygon. A diagonal of a polygon is a line segment that joins two non-adjacent vertices of the polygon. A convex polygon is a polygon in which no line segment joining any two points in the interior or on the boundary goes outside the boundary. Therefore, any diagonal of a convex polygon must lie entirely inside or on the boundary of the polygon.
vii) No, it is not possible that a convex and concave polygon with same number of sides have a different number of diagonals. The number of diagonals of any n-sided polygon (convex or concave) is given by the formula n(n-3)/2. This formula does not depend on whether the polygon is convex or concave. Therefore, if two polygons have the same number of sides, they must have the same number of diagonals.
Answer:
Here are the answers to your questions about convex and concave polygons:
i) No, a convex polygon cannot have any of its internal angles as a reflex angle. A reflex angle is an angle that is greater than 180 degrees and less than 360 degrees. A convex polygon is a polygon in which all the interior angles are less than or equal to 180 degrees. Therefore, a convex polygon cannot have a reflex angle as an internal angle.
ii) Yes, it is possible that a concave polygon has none of its internal angles as a reflex angle. A concave polygon is a polygon that has at least one interior angle greater than 180 degrees. However, it is not necessary that all the interior angles of a concave polygon are reflex angles. For example, a polygon with four sides and one interior angle of 270 degrees is a concave polygon, but it has only one reflex angle and three acute angles.
iii) No, all the sides of a concave polygon cannot be equal. If all the sides of a polygon are equal, then the polygon is equilateral. An equilateral polygon is always convex, because the sum of any two sides is greater than the third side. Therefore, a concave polygon cannot be equilateral.
iv) No, all the angles of a concave polygon cannot be equal. If all the angles of a polygon are equal, then the polygon is equiangular. An equiangular polygon is always convex, because the sum of any two interior angles is less than 180 degrees. Therefore, a concave polygon cannot be equiangular.
v) No, a concave polygon cannot be regular. A regular polygon is a polygon that is both equilateral and equiangular. As we have seen in the previous answers, a concave polygon cannot be either equilateral or equiangular. Therefore, a concave polygon cannot be regular.
vi) No, it is not possible that any of the diagonals of a convex polygon lie in the exterior of the polygon. A diagonal of a polygon is a line segment that joins two non-adjacent vertices of the polygon. A convex polygon is a polygon in which no line segment joining any two points in the interior or on the boundary goes outside the boundary. Therefore, any diagonal of a convex polygon must lie entirely inside or on the boundary of the polygon.
vii) No, it is not possible that a convex and concave polygon with same number of sides have a different number of diagonals. The number of diagonals of any n-sided polygon (convex or concave) is given by the formula n(n-3)/2. This formula does not depend on whether the polygon is convex or concave. Therefore, if two polygons have the same number of sides, they must have the same number of diagonals.
Sam