Math Problem Statement

Sort the descriptions of the polygons. All of the side lengths shown are in inches. Each of these side lengths can create at least one polygon. Sort the descriptions into the correct category. 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 5, 7, 8, and 11 Side lengths of 9.5, 11.8, 14.4 Side lengths of
8 1 ​ ,
3 1 ​ ,
7 6 ​ , and
7 3 ​

Solution

Let's analyze each set of side lengths to categorize them into "Would allow you to make many different polygons" or "Would make one unique polygon."

1. Side lengths of 3, 5, 6

  • Analysis: A triangle with sides 3, 5, and 6 can form different triangles because there is no restriction on the angles between these sides. This means various triangles could be formed with these side lengths.
  • Category: Would allow you to make many different polygons

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

  • Analysis: Four side lengths can form multiple types of quadrilaterals (like trapezoids, rectangles, or parallelograms) with different internal angles, leading to a variety of shapes.
  • Category: Would allow you to make many different polygons

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

  • Analysis: Similar to the previous case, this set of side lengths can form many different types of quadrilaterals, as there are no strict constraints on the angles.
  • Category: Would allow you to make many different polygons

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

  • Analysis: Again, this set of side lengths can form different quadrilaterals since they do not dictate the internal angles, leading to multiple possibilities.
  • Category: Would allow you to make many different polygons

5. Side lengths of 9.5, 11.8, 14.4

  • Analysis: This is a set of three side lengths, which would form triangles. These side lengths can form different triangles depending on the angles, as there are no constraints leading to a unique configuration.
  • Category: Would allow you to make many different polygons

6. Side lengths of 8, 13, 7, and 7

  • Analysis: These side lengths could potentially form a quadrilateral, like an isosceles trapezoid, with varying internal angles, so different shapes can be formed.
  • Category: Would allow you to make many different polygons

Summary

Would allow you to make many different polygons:

  • 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 5, 7, 8, and 11
  • Side lengths of 9.5, 11.8, 14.4
  • Side lengths of 8, 13, 7, and 7

Would make one unique polygon:

  • None of the descriptions are in this category, as all the given sets can form multiple polygons.

Would you like more details on any of these, or have any questions about how I arrived at these conclusions?

Related Questions:

  1. How do you determine whether a set of side lengths forms a unique polygon?
  2. What is the concept of different polygons having the same side lengths?
  3. How can angles affect the uniqueness of polygons formed by given side lengths?
  4. What is the triangle inequality theorem and how does it apply to polygon formation?
  5. Could any of these side lengths form a specific type of polygon (e.g., rectangles, squares)?

Tip: For any polygon, the sum of the lengths of any two sides must always be greater than the third side to form a valid shape. This is known as the triangle inequality for triangles and a similar principle for other polygons.

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Math Problem Analysis

Mathematical Concepts

Polygon Geometry
Triangle Inequality
Quadrilateral Properties

Formulas

Sum of internal angles for polygons
Triangle inequality theorem

Theorems

Triangle inequality theorem
Properties of quadrilaterals

Suitable Grade Level

Grades 6-8