To find the length of segment \( TU \), we can use the Power of a Point theorem. According to the theorem, if a point \( S \) is outside a circle and secants \( SQ \) and \( SU \) intersect the circle, then the following relationship holds:
\[ SR \cdot SQ = ST \cdot SU \]
Here, we know:
- \( SR = 4 \)
- \( SQ = 10 \)
- \( ST = 2 \)
- We need to find \( TU \) so that \( SU = ST + TU \).
Thus, we can express \( SU \) as: \[ SU = ST + TU = 2 + TU \]
Now, substituting the known values into the Power of a Point formula: \[ SR \cdot SQ = ST \cdot SU \] \[ 4 \cdot 10 = 2 \cdot (2 + TU) \] Calculating the left side: \[ 40 = 2(2 + TU) \]
Next, we expand the right side: \[ 40 = 4 + 2TU \]
Now, we will isolate \( TU \): \[ 40 - 4 = 2TU \] \[ 36 = 2TU \] Dividing both sides by 2 gives: \[ TU = 18 \]
Thus, the length of segment \( TU \) is \( \boxed{18} \).