Geometry

Question Analysis  

The task is to find the length of the longest side of the smaller polygon using the given perimeter ratio (7:12) and the longest side of the larger polygon (36 units). The key is applying the property that the ratio of corresponding side lengths of similar polygons equals their perimeter ratio.  

Key Concept Explanation  

For similar polygons, if the perimeter ratio is , the ratio of corresponding side lengths is also . Thus, .  

Step-by-Step Solution  

1. Set up the proportion:  

Let  = length of the longest side of the smaller polygon.  

The perimeter ratio is , so: