In triangle \( ABC \), we know that:
- \( CD \) is the altitude from \( C \) to \( AB \).
- \( CD = 3 \) (the length from point \( C \) to point \( D \)).
- \( DB = 4 \) (the length from point \( D \) to point \( B \)).
We want to find the length of segment \( AD \).
Since triangle \( ABC \) is a right triangle, we can denote:
- \( AD = x \)
- \( DB = 4 \)
From this, it follows that the entire length of \( AB \) is:
\[ AB = AD + DB = x + 4 \]
By applying the geometric mean theorem (which states that the length of the altitude from the right angle in a right triangle is the geometric mean of the segments it divides the hypotenuse into), we have:
\[ CD^2 = AD \cdot DB \]
Substituting the known values into the equation gives:
\[ 3^2 = x \cdot 4 \]
This simplifies to:
\[ 9 = 4x \]
Now, solve for \( x \):
\[ x = \frac{9}{4} = 2.25 \]
Thus, the length of segment \( AD \) is:
\[ AD \approx 2.25 \]
Rounding this to two decimal places, we find:
\[ \boxed{2.25} \]