**Joseph Malkevitch: Practice 1: Polygons**

Prepared by:

Joseph Malkevitch

Department of Mathematics

York College (CUNY)

Jamaica, New York 11451

email:

__malkevitch@york.cuny.edu__

web page:

__http://york.cuny.edu/~malk/__

__Geometric Structures
__

Practice Questions

1.

a. Which of the diagrams below represent polygons?

b. Which if any of the diagrams is convex?

c. How many sides does each figure which is a polygon have?

2.

a. Draw a convex 5-gon.

b. Draw a non-convex 5-gon.

c. Draw a self-intersecting hexagon.

d. Draw a convex 7-gon all of whose sides have equal length but for which not all the angles are the same size.

3.

a.

A diagonal of a (planar) polygon is a line segment which joins two vertices of the polygon and lies completely in the interior of the polygon.

How many different diagonals does the polygon above have?

b. Can one subdivide the interior of the polygon above into triangles by drawing diagonals? If so, how diagonals did you need and how many triangles did you get?

c. How many sides does the above polygon have?

d. Draw a convex polygon with the same number of sides as the polygon above.

e. Answer parts (a) and (b) of this question for the polygon you draw in (d).

f. An *ear* of a plane simple polygon is a vertex e with the property that the line segment joining the endpoints of edges that meet at e is a diagonal of the polygon.

How many ears does the polygon above have? Can you find an infinite family of polygons where each polygon in the family has exactly 2 ears?

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