Geometry

Question Analysis:
This question combines the properties of isosceles triangles and the triangle - inequality theorem.
The main focus is on determining the valid side - length combinations for an isosceles triangle and then calculating its perimeter.

Key Concept Explanation:
1. In an isosceles triangle, two sides are equal.
2. The triangle - inequality theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
That is, if the sides of a triangle are , , and , then a + b > c, a + c > b, and b + c > a.

Step - by - Step Solution:
Case 1: Assume the equal sides have a length of cm.
Let cm, cm, and cm.
Check the triangle - inequality: cm, and .
So, a triangle with side lengths cm, cm, and cm does not exist.
Case 2: Assume the equal sides have a length of cm.
Let cm,

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