Let's define the cost of the red shirt as \( r \).
According to the problem, the blue shirt costs $13, and it is $2.50 more than the price of the red shirt. Therefore, we can write the equation as:
\[ 13 = r + 2.50 \]
To find \( r \), we can solve the equation by subtracting \( 2.50 \) from both sides:
\[ r = 13 - 2.50 \]
Calculating the right side gives:
\[ r = 10.50 \]
Thus, the cost of the red shirt is \( \boxed{10.50} \).