Geometry

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

Key Concept Explanation:
In an isosceles triangle, two sides are equal.
The triangle - inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Step - by - Step Solution:
Case 1: If the equal sides have a length of cm.
Let the sides of the triangle be 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: If the equal sides have a length of cm.
Let the sides of the triangle be cm, cm, and cm.
Check the triangle - inequality:

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