A triathlete competes in a triathlon in which the swimming, biking, and running segments are all of the same length. The triathlete swims at a rate of 2.5 kilometers per hour, bikes at a rate of 18 kilometers per hour, and runs at a rate of 9.2 kilometers per hour. What is the triathlete's average speed, in kilometers per hour, for the entire race?

Let the length of each segment be d kilometers.

The time taken to complete the swimming segment is t_s = \frac{d}{2.5} hours.

The time taken to complete the biking segment is t_b = \frac{d}{18} hours.

The time taken to complete the running segment is t_r = \frac{d}{9.2} hours.

The total time taken to complete the entire race is t_{total} = t_s + t_b + t_r .

The total distance covered in the entire race is 3d kilometers.

Therefore, the average speed for the entire race is given by:

\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}} = \frac{3d}{t_{total}} = \frac{3d}{t_s + t_b + t_r}

Substitute the expressions for t_s , t_b , and t_r into the above equation:

\text{Average speed} = \frac{3d}{\frac{d}{2.5} + \frac{d}{18} + \frac{d}{9.2}}

\text{Average speed} = \frac{3}{\frac{1}{2.5} + \frac{1}{18} + \frac{1}{9.2}}

\text{Average speed} = \frac{3}{0.4 + 0.0556 + 0.1087}

\text{Average speed} = \frac{3}{0.5643}

\boxed{\text{Average speed}\approx5.32\,\text{km/hr}}