Geometry

Question Analysis  

The main focus is on using Heron's formula to find the perimeter. Given the area and the ratio of the triangle's sides (3:3:4), we'll represent the sides with a variable, calculate the semi-perimeter, apply Heron's formula to solve for the variable, and then determine the perimeter.

 

Key Concept Explanation  

Heron's Formula: For a triangle with side lengths , , , the area , where  is the semi-perimeter. This formula allows us to find the area when all side lengths are known. When side lengths are in a ratio, we can use a common variable to represent them and then apply the formula.

 

Step-by-Step Solution  

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

Let the sides of the triangle be , , and .  

The perimeter , so the semi-perimeter .  

2. Apply Heron's formula:  

Substitute into Heron's formula: