Three boys are standing on a circular boundary of a fountain. They are at equal distance from each other. If the radius of the boundary is 5 m, the shortest distance between any two boys is

### Concept & Equilateral Triangle Properties Since the three boys are at equal distances from each other on a circular boundary, they form the vertices of an equilateral triangle inscribed within the circle. The relationship between the side $a$ (distance between the boys) and the circumradius $R$ of an equilateral triangle is: $$R = \frac{a}{\sqrt{3}} \implies a = R\sqrt{3}$$ ### Step-by-Step Solution 1. **Identify the Given Information:** The radius of the circular boundary (circumradius) is $R = 5$ m. 2. **Apply the Formula:** We need to find the side length $a$ of the equilateral triangle. $a = R \times \sqrt{3}$ 3. **Calculate:** $a = 5 \times \sqrt{3} = 5\sqrt{3}$ m. ### Exam Strategy & Shortcut Memorizing the $a = R\sqrt{3}$ relationship for inscribed equilateral triangles turns this into a 2-second read-and-mark question. No calculation is needed beyond attaching $\sqrt{3}$ to the given radius. ### Common Pitfall Some students might attempt to calculate the arc length between the boys instead of the straight-line distance, incorrectly choosing a distance involving $\pi$. The phrase "shortest distance" always implies a straight line. ### Final Answer Therefore, the correct answer is **$5\sqrt{3}$ m**.