Geometry

Question Analysis  

The task is to determine the length of the shortest side of the larger polygon, given the ratio of perimeters of two similar polygons (1:3) and the length of the shortest side of the smaller polygon (4 units). The key is applying the property that the ratio of corresponding side lengths of similar polygons equals the ratio of their perimeters.

Key Concept Explanation  

For similar polygons, if the perimeter ratio is , then the ratio of any pair of corresponding side lengths is also . This is because perimeter, a linear measurement, scales proportionally with side lengths.

Step-by-Step Solution  

1. Set up the proportion:  

Let  be the length of the shortest side of the larger polygon. Given the perimeter ratio , we can set up the proportion .

2. Solve the proportion:  

Cross - multiply: .  

So,

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