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Would allow you to make many different polygons Would make one unique polygon

Side lengths of 3, 5, 6
Side lengths of 3, 5, 6, and 9
Side lengths of 8, 10, 11, and 14
Side lengths of 8.1, 6.9, 13.4
Side lengths of 5, 7, 8, and 11
Side lengths of
6
1
​
,
7
2
​
,
6
5
​
, and
2
1
​

1 answer

To determine whether a certain set of side lengths can make many different polygons or one unique polygon, we can analyze each set of side lengths based on whether they can form multiple polygons or are constrained to a single polygon type.

Analysis of Each Set of Side Lengths

  1. Side lengths of 3, 5, 6

    • These side lengths can create various triangles by varying the angles between the sides (as long as they satisfy the triangle inequality theorem). Thus, many different polygons can be created.

  2. Side lengths of 3, 5, 6, and 9

    • With four sides, these can form different quadrilaterals. Since the sides do not restrict the shape to one unique polygon, they can create many different polygons.

  3. Side lengths of 8, 10, 11, and 14

    • Similar to the previous example, these four sides can be rearranged to form various quadrilaterals; hence, they also allow for many different polygons.

  4. Side lengths of 8.1, 6.9, 13.4

    • With three distinct lengths, different triangles can be formed. Therefore, this set would also allow for many different polygons.

  5. Side lengths of 5, 7, 8, and 11

    • As a set of four distinct side lengths, this can generate various shapes and configurations; hence, it supports many different polygons.

  6. Side lengths of 6, 7, 6, and 2 (assuming the format presents four sides but looks like there are two pairs)

    • Here we have two sides of length 6, one of length 7, and one of length 2. This could allow for a unique polygon (specifically a kite shape where the equal sides offer symmetry), but it's also possible to vary angles, allowing for some multiplicity. However, due to the uneven sides present, this could restrict configurations. Still, it likely permits many different polygons as a primary shape structure.

Summary

  • For all side sets given, they provide significant possibilities for constructing multiple polygons due to the varying combinations of dimensions, angles, and arrangements. Thus, we conclude that each set typically allows for many different polygons rather than being restricted to a unique polygon.