To categorize the side lengths into "Would allow you to make many different polygons" and "Would make one unique polygon," we need to consider the conditions governing the formation of polygons.
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Many different polygons: A set of side lengths can create many different polygons if the side lengths meet the triangle inequality for any combination of three sides and do not result in a degenerate polygon. This means that the sum of the lengths of any two sides must be greater than the length of the remaining side.
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One unique polygon: This typically occurs when the side lengths are the same or meet specific criteria that restrict their variations to a single shape (例えば, in the case of a triangle where all its sides and angles must be equal).
Now let's sort the side lengths based on these criteria:
Would allow you to make many different polygons:
- Side lengths of 3, 5, 6, and 9: Several combinations and variations are possible.
- Side lengths of 5, 7, 8, and 11: Several combinations since no two sides are equal, and they satisfy the triangle inequality.
- Side lengths of 6.3, 14.8, 13.4: These can also create varying triangles, given none are equal and they satisfy the triangle inequality.
- Side lengths of 8, 10, 11, and 14: Again, varying combinations can be formed here.
- Side lengths of 8, 7, 5, and 7: This set can also create multiple different shapes.
Would make one unique polygon:
- Side lengths of 3, 5, 6: These lengths can potentially form one unique triangle since they meet the triangle inequality, creating one specific triangle shape.
Based on the above analysis, here's the sorted categorization:
Would allow you to make many different polygons:
- Side lengths of 3, 5, 6, and 9
- Side lengths of 5, 7, 8, and 11
- Side lengths of 6.3, 14.8, 13.4
- Side lengths of 8, 10, 11, and 14
- Side lengths of 8, 7, 5, and 7
Would make one unique polygon:
- Side lengths of 3, 5, 6