Step 1 : Understanding the Question
This geometry problem involves an equilateral triangle, where all three sides and all three interior angles ($60^\circ$) are equal. We are given the total area and asked to find the 'in-radius.' The in-radius is the radius of the incircle, which is the largest circle that can be drawn inside the triangle, touching all three sides. To find this, we must first determine the length of the triangle's side.
Step 2 : Key Formulas and approach
The problem is solved in two main steps. First, we use the area formula for an equilateral triangle to solve for the side length 'a'. Second, we apply the specific formula that relates the side of an equilateral triangle to its in-radius.
Key Formulas:
1. $\text{Area of Equilateral Triangle} = \frac{\sqrt{3}}{4} a^2$
2. $\text{In-radius (r)} = \frac{a}{2\sqrt{3}}$
Step 3 : Detailed Explanation
Setting up the Area Equation: We are told the area is $169\sqrt{3}$. Using the formula, we write: $\frac{\sqrt{3}}{4} a^2 = 169\sqrt{3}$.
Solving for Side 'a': We cancel $\sqrt{3}$ from both sides, leaving $\frac{a^2}{4} = 169$. Multiplying both sides by 4 gives $a^2 = 169 \times 4$. Taking the square root of both sides, $a = \sqrt{169} \times \sqrt{4} = 13 \times 2 = 26$ cm.
Calculating In-radius: Now that we know the side length is 26 cm, we plug it into the in-radius formula: $r = \frac{26}{2\sqrt{3}}$.
Simplifying the Fraction: Dividing 26 by 2 gives 13. The expression simplifies to $r = \frac{13}{\sqrt{3}}$.
Final Comparison: We check the calculated value against the options provided. The result $\frac{13}{\sqrt{3}}$ matches perfectly.
Step 4 : Final Answer
The in-radius of the equilateral triangle is $13/\sqrt{3}$ cm, which is option (B).