To find the distance between the points \((4,9)\) and \((2,4)\), we can use the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, \((x_1, y_1) = (4, 9)\) and \((x_2, y_2) = (2, 4)\).
Substituting in the coordinates:
\[ d = \sqrt{(2 - 4)^2 + (4 - 9)^2} \]
Calculating the differences:
\[ d = \sqrt{(-2)^2 + (-5)^2} \]
Calculating the squares:
\[ d = \sqrt{4 + 25} \]
Adding the values inside the square root:
\[ d = \sqrt{29} \]
Thus, the distance between the points \((4, 9)\) and \((2, 4)\) is
\[ \sqrt{29} \]
As a fully simplified radical expression, the final answer is
\[ \sqrt{29} \]