Rigid vs Flexible Shapes Math Example 2

Solution

1. 1
   Step 1: Triangle (5, 7, 8): by SSS, all three sides fixed uniquely determine the triangle. Only $1$ possible shape. Degrees of freedom $= 0$ (rigid).
2. 2
   Step 2: Quadrilateral (5, 6, 5, 6): fixing four side lengths still leaves one angle free to vary (e.g., the rectangle $5\times6$ and various parallelograms). The shape has $1$ degree of freedom — it can be sheared continuously.
3. 3
   Step 3: In general, a polygon with $n$ sides has $n - 3$ degrees of freedom when side lengths are fixed. For $n=3$: $0$ (rigid); for $n=4$: $1$ (flexible).