An insignia of football team consists of a metal frame in the shape of an equilateral triangle. A circle is inscribed in the triangular frame. What is the ratio of the side of the triangle to that of the diameter of the circle ?

Let the side of the equilateral triangle be s. The height h of an equilateral triangle is (√3/2)s. The inradius r (the radius of the inscribed circle) is 1/3 of the height:
r = (1/3) × (√3/2)s = s/(2√3).
The diameter d of the circle is 2r:
d = 2 × (s/(2√3)) = s/√3.
The ratio of the side of the triangle to the diameter of the circle is:
s : (s/√3) = 1 : (1/√3) = √3 : 1.
Multiplying both terms of the ratio by √3 gives the equivalent ratio:
(√3 × √3) : (1 × √3) = 3 : √3.
In the provided options, 'T' and '13' appear to be OCR errors for √3 and 3. If 'T' represents √3, then Option A (3 : T) correctly represents 3 : √3, which simplifies to √3 : 1. Option B (T : 13) likely represents √3 : 3 (or 1 : √3). Thus, A is the correct choice.