Geometry

Question Analysis  

Given the area of a triangle and the ratio of its sides, the key is to represent the sides using a common variable, calculate the semi-perimeter, and then apply Heron's formula to find the variable and ultimately the perimeter.

 

Key Concept Explanation  

Heron's Formula: For a triangle with side lengths , , , the area , where  is the semi-perimeter.

Ratio Representation: When sides are in a ratio like , we can represent them as , , and  to simplify calculations.

 

Step-by-Step Solution  

1. Represent the sides and calculate the semi-perimeter:  

Let the sides be , , .  

The perimeter , so the semi-perimeter .

2. Apply Heron's formula:  

Substitute into Heron's formula:  

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