Geometry

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Below is a worksheet I have to do for my geometry class. I am not sure about the answers I have given and I have no clue about the ones not answered. Any help will be much appreciated and thank you in advance.

A. Determine the number of triangulations for a hexagon.
5


B. How would the number of triangulations change if you ignored the vertices’ distinctness? If you remove the labels A, B, C, D, … from the vertices, and say two triangulations are identical if one can be transformed into the other via rotation or reflection, this may change the number of ways a polygon can be triangulated. In other words, determine the number of triangulations of a quadrilateral, pentagon, and hexagon if the vertices are no longer distinct.

If the vertices are no longer distinct the number of triangulations would be one since the triangle could be rotated throughout the quadrilateral, pentagon, and hexagon.




C. The previous page of this assignment defines the terms concave and convex in regards to polygons. Find the definitions of convex polygon and concave polygon in the textbook (chapter 3). What is the major distinction between a convex and a concave polygon according to the textbook?

Convex polygons are polygons that can have lines drawn from the vertices and the lines remain inside the shape. Concave polygons can have lines drawn from each of the vertices and the lines go outside the shape.






D. If a polygon is no longer required to be convex, what effect does this have on the number of triangulations? Determine the number of triangulations of a quadrilateral, pentagon, and hexagon if the polygons are no longer convex.





E. If we continued to search for the number of ways of triangulating convex polynomials with more vertices, we would find that the number of ways increases very significantly very quickly. There is a distinct pattern in the number of triangulations. This pattern is denoted by the sequence known as the “Catalan numbers.”

Search the internet for the “Catalan numbers.” You will likely encounter a Wikipedia entry on the subject. In any case, identify the first eight terms of the Catalan number sequence. Do the terms of the sequence match your answers above?

The first eight Catalan numbers are 1,1,2,5,14,42,132,429 and yes these numbers match my answers.

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