In context of the adjoining figure, answer the following questions: (i) Is ABCDEFG a polygon? (ii) How many sides does it have? (iii) How many vertices does it have? (iv) Are adjacent sides? (v) Is a diagonal of the polygon? (vi) Are diagonals of the polygon? (vii) Is point P in the interior of the polygon? (viii) Is point A in the exterior of the polygon?

Question

KnowledgeBoat · Accepted Answer

(i) The figure ABCDEFG is a simple closed curve made up entirely of line segments. Therefore, it satisfies the definition of a polygon. ∴ Yes, ABCDEFG is a polygon. (ii) The sides of the polygon ABCDEFG are . So, the polygon has 7 sides. ∴ The polygon ABCDEFG has 7 sides. (iii) The vertices of the polygon ABCDEFG are A, B, C, D, E, F and G. So, the polygon has 7 vertices. ∴ The polygon ABCDEFG has 7 vertices. (iv) Two sides of a polygon with a common end point are called adjacent sides. The side has end points A and B, and the side (same as ) has end points F and E. They do not share any common end point, so they are not adjacent sides. ∴ No, are not adjacent sides. (v) The diagonals of a polygon are the line segments formed by joining non-adjacent vertices. The segment (same as ) joins the vertices G and F, which are adjacent vertices (they are the end points of the same side ). So, is a side of the polygon, not a diagonal. ∴ No, is not a diagonal of the polygon; it is a side of the polygon. (vi) Let us check each line segment: - In : vertex A is adjacent to B and G, but not to C. So, A and C are non-adjacent vertices. - In : vertex A is adjacent to B and G, but not to D. So, A and D are non-adjacent vertices. - In : vertex A is adjacent to B and G, but not to E. So, A and E are non-adjacent vertices. Since each of these segments joins two non-adjacent vertices, they are all diagonals of the polygon. ∴ Yes, are all diagonals of the polygon. (vii) Looking at the figure, point P does not lie inside the boundary of polygon ABCDEFG. ∴ No, point P is not in the interior of the polygon. (viii) Point A is a vertex of the polygon ABCDEFG, so it lies on the boundary of the polygon, not in its exterior. ∴ No, point A is not in the exterior of the polygon; it lies on its boundary.

Mathematics

Geometrical Concepts

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